Abstract Booklet

International Conference on Differential Equations and Control Problems: Modeling, Analysis and Computations(ICDECP19)

June 17-19, 2019

ABSTRACT BOOKLET

Editted by Dr. Muslim Malik Assistant Professor, School of Basic Sciences & Compiled by A Team of Research Scholar School of Basic sciences, India Institute of Technology Mandi

Organised by:

School of Basic Sciences Indian Institute of Technology Mandi, Mandi, H.P.

June 12, 2019

ii Contents

Plenary Talks 1 PDE: Classical and Modern (Prof. A. K. Nandkumaran)...... 1 Set Differential Equations: An Overview and Recent Developments (Gnana Bhaskar Tenali)..1 Mathematical modeling and numerical simulation of particles in ﬂuids of complex geometries (S. Sundar , Nityananda Roy, Thomas Goetz)...... 1 NUMERICAL METHODS TO CAPTURE δ SHOCKS ARISING IN THE SYSTEMS OF PRES- SURELESS GAS DYNAMICS (G.D. Veerappa Gowda)...... 2 Chaotic behaviour in Differential and Difference Equations with their Fractal Nature (Mohammad Sajid)...... 2 A New Higher Order Nonliear PDE Model for Effective Image Denoising and its Fourier Spectral Accurate Solution (B.V. Rathish Kumar)...... 3 Application of Operator Theory in Controllability Analysis (Raju K. George)...... 3 Parameter Uniform Numerical Schemes for a class of Singularly Perturbed Problems (Kapil Ku- mar Sharma)...... 5 Can parallel bearing surfaces support load- A Tribological Study (Prawal Sinha)...... 6 AN OVERVIEW ON INVERSE PROBLEMS ARISING IN FLUID DYNAMICS (JAIME H. ORTEGA)...... 6 Differential quadrature in the study of vibrations of non-uniform FG circular plates under hydrostatic peripheral loading and thermal environment (Roshan Lal)...... 7

Invited Speakers 9 A Domain Decomposition Method for Singularly Perturbed Parabolic Reaction-Diffusion Prob- lems (S.C.S. Rao1 and S. Kumar2)...... 9 A Finite element approach to a moving boundary problem with variable thermal conductivity (Rajeev and Rishabh Daal)...... 9 Approximation Method for Generalized Fractional Derivatives with Application (Rajesh Pandey) 10 Entropy Stable Schemes For Relativistic Hydrodynamics Equations (Harish Kumar)...... 10 Nonlinear dynamics of Some Complex Ecological Sytems (R. P. Gupta)...... 10 A high-order quasi-variable meshes two-level implicit compact scheme for solving three-dimensional nonlinear non-stationary advection-diffusion equation (Navnit Jha)...... 11 Delayed Mathematical Models of HIV (Saroj Kumar Sahani)...... 11 Numerical Solution to Blood Model Navier-Stokes Equations in the Entrance Region of Concen- tric Annuli with Rotating Inner Wall (Srinivasa Rao Nadiminti1; N. Gayathri Devi1 and A.Kandasamy)...... 12 On the nondegenerate soliton solutions of Manakov system (M. Senthilvelan)...... 12 Noise-Induced extinction in an ecological model (Partha Sarathi Mandal)...... 13 Numerical Investigation of Natural Convection of Casson Fluids in a Square Porous Cavity under the Effects of Thermal Radiation. (Sapna Sharma)...... 13 A PARAMETER-UNIFORM IMPLICIT SCHEME FOR TWO-PARAMETERS SINGULARLY PERTURBED PARABOLIC PROBLEMS (Devendra Kumar)...... 14

iii Modeling Lithiation Induced Stresses in High-Capacity Electrode Particles with Concentration Dependent Properties (Poornesh Kumar Koorata)...... 14 Understanding molecular transport in a non-conserving systems: The role of interactions (Arvind Kumar Gupta)...... 15 SH-wave propagation in monoclinic medium with linearly varying inhomogeneity ( Sumit Kumar Vishwakarma, Tapas Ranjan Panigrahi, Rupinderjit Kaur.)...... 15 Stabilization and chaos control in multispecies system (Anuraj Singh)...... 16 A fourth-order orthogonal spline collocation method to fourth-order boundary value problems ( Anil)...... 16 Asymptotic Analysis of Steady Stokes Equations in an Oscillating Domain ( Bidhan Chandra Sardar)...... 17 Periodic solutions of vector disease model with harvesting term (Shilpee Srivastava)...... 17 Mixed Virtual Element Methods for fourth order nonlinear parabolic problems (P. Dhanumjaya, K. Balaje)...... 18 Generation of Uniform Magnetic Flux using different Coil and Plate Arrangement (Md Tarikul Islam, Md Ataur Rahman Khan, Md. Moniruzzaman Bhuyan, Mohammad Anwar Hossain, Md. Mehedi Hasan Bhuiyan.)...... 18 Optimization methods and algorithms for non-parallel support vector machines (M. Tanveer).. 18 Elliptic problems with critical growth sign changing nonlinearities. (Sarika)...... 19 Modeling and Simulation of dc-dc Converter Based Power Electronics Learning : a review (In- dresh Yadav)...... 19 Solution of two-parameter singularly perturbed one dimensional parabolic equations using non- polynomial spline ( Shahna, Talat Sultana and Arshad Khan)...... 19 Creeping flow of a sphere in non-concentric spherical container using slip condition ( M. Krishna Prasad)...... 20 Two-warehouse inventory model with quantity discount policy ( Himanshu Rathore)...... 20 A Comparative Study of Natural Nanouid Convection for Different Initial and Boundary Condi- tions ( Jyoti Sharma)...... 21 Application of finite fractional Hankel-type transformation in Dirichlets problem (V. R. Lakshmi Gorty)...... 21 Existence Results for Fractional Impulsive Delay Differential Equations (Renu Chaudhary ).. 22 OPTIMAL CONTROL IN STOCHASTIC LANDAU-LIPSCHITZ-GILBERT EQUATION (Ananta K. Majee)...... 22 Survey of Nature Inspired Optimization Algorithms in Fuzzy Control Systems (Gaurav Saxena, Gomathi Bhavani, Shilpee S. Saxena)...... 23 A study of approximate controllability for abstract nonlocal neutral integro-differential equations with finite delay (Kamaljeet)...... 23 Complex plankton dynamics induced by adaptation and defense (Nilesh Kumar Thakur and Archana Ojha)...... 23 CONVERGENCE ANALYSIS OF TIKHONOV REGULARIZATION FOR NON-LINEAR STA- TISTICAL INVERSE LEARNING PROBLEMS (ABHISHAKE, GILLES BLANCHARD AND PETER MATHE)...... 24 CONTROLLABILITY OF A CLASS OF FRACTIONAL IMPULSIVE DIFFERENTIAL EQUA- TIONS IN A BANACH SPACE (Abdur Raheem)...... 24 APPROXIMATE CONTROLLABILITY OF SECOND ORDER SEMILINEAR CONTROL SYS- TEM WITH NONLOCAL CONDITIONS (Urvashi Arora and N. Sukavanam)...... 25 BOUNDEDNESS AND COMPACTNESS OF WEIGHTED DIFFERENTIATION COMPOSI- TION OPERATORS BETWEEN SOME WEIGHTED SPACES (Zaherr Abbas)...... 25 Optimization criteria for performance of heat engines working under different constraints ( Renuka Rai and Ramandeep Singh Johal)...... 25

iv Wind turbine blade section optimization using a quantitative study (M. Balachandar)...... 26 Mathematical modeling in health science (Jagdev Singh)...... 26 Fractional exothermic reactions models having constant heat source in porous media with different kind of memories (Devendra Kumar )...... 26 Piezothermoelastic continuum subjected to point mechanical load (Anita Devi Thakur)..... 27 Dynamical Analysis of Michaelis-Menten Enzyme Reactions (B S Lakshmi and S S Phulsagar). 27 Truss Topology Optimization With Static And Dynamic Constraints Using AISC-ASD ( Ghan- shyam G. Tejani)...... 28 Inuence of Temperature Jump and Concentration Slip on inclined MHD Bioconvection past a vertical porous plate in the presence of Nanoparticles and Gyrotactic Microorganism (Rakesh Choudhary and Shalini Jain)...... 28 Numerical Solution of Linear and Higher order Delay Differential Equations using Coded Differ- ential Transform Method (Giriraj Methi And Anil Kumar)...... 29 Studies of Hyperloop Vehicle for Transportation: A Review (V. K. Srivastav; Aditya Priyanka; Abhishek Kumar; Shudhanshu Kumar; Anand Raj.)...... 29 Effect of active case nding on dengue control: Implications from a mathematical model (Pankaj Kumar Tiwari)...... 29 Change in Stability Behavior of Spatiotemporal Phytoplankton Dynamics with Different Types of Functional Response (Randhir Singh Baghel)...... 30 Modelling of duct based photovoltaic thermal (PVT) air collector (Rohit Tripathi1; G. N. Tiwari; Deepak Sharma)...... 30 Mathematical model of unstable self-limiting thermo chemical temperature oscillations in Aus- tralian Cycades (Akash Bhavsar)...... 31 On the existence and uniqueness of solutions to discontinuous dynamic equation on time scales ( Sanket Tikare)...... 31 Generalized energy inequality for weak solutions to Damped Navier-Stokes equations (Rajib Haloi and Subha Pal)...... 32 Dynamical study of a delayed SEIRS model with saturated incidence and impulsive vaccination: Effect of household wastes ( Kunwer Singh Mathur)...... 32 Thermal Convection in Oldroydian Nanouid Layer Saturating a Porous medium with Rigid-free and Rigid-rigid Boundaries ( Abhilasha)...... 33 Inclined MHD Williamson Fluid Flow with Slip Boundary and Heat and Mass Transfer due to Porous and Melting Stretching Surface with Non-Linear Radiation and Heat Source ( Amit Parmar)...... 33 (Sandip Rakshit)...... 34 An overview to investigate role of rheology on thermal convection in ferroﬂuids (Veena Sharma) 34 (Sheetal Dharmatti )...... 34

Paper Presentation 35 Numerical Solution of Lane-Emden type Equations Using Multilayer Perceptron Neural Network Method (Akanksha Verma, Manoj Kumar)...... 35 Computational Simulation for Time-Fractional Diffusion Equation with Neumann Boundary Con- ditions (A.S.V. Ravi Kanth, Neetu Garg)...... 35 On Solution of Fractional Order Advection-Diffusion Equation in Porous Media (Prashant Pandey) 36

An exact l1 penalty function method for multi-dimensional ﬁrst-order PDE constrained control optimization problem (Preeti)...... 36 Numerical solution of Lane-Emden type equations using Adomian decomposition method with unequal step-size partitions (UMESH, Manoj Kumar)...... 37 Solution of Riemann problem for non-ideal magnetogasdynamic ﬂow (Pooja Gupta)...... 37

v Wave interaction with a tunnel in a sea with bottom undulation (MANISHA, Dr. RAMANABABU KALIGATLA)...... 37 Subgrid multiscale stabilized finite element analysis for various transport equations (Manisha Chowdhury, B.V. Rathish Kumar)...... 38 Numerical solution for two dimensional space-time fractional reaction diffusion equation (Sachin Kumar)...... 38 Mathematical Modeling of Surface wave transference in a piezo-composite media using WKB technique (Sonal Nirwal, Sanjeev A. Sahu)...... 39 Modelling of Aeration Efficiency At Gabion Weir (KM. Luxmi, Nand Kumar Tiwari, Subodh Ranjan Vajesnayee)...... 39 Modelling of scour around spur dykes (Amit kumar, Subodh ranjan vajesnayee, and Nand kumar Tiwari)...... 40 Modelling the delay dynamics of malware propagation (Sangeeta Kumari)...... 40 Analysis of a density dependent model with discrete delays (Anuraj Singh, Ankit Parwaliya and Ajay kumar)...... 41 Approximate analytical solution for shock wave in rotational axisymmetric perfect gas: Isother- mal flow (G. Nath, Sumeeta Singh)...... 41 MULTIBODY MODELLING OF A RAIL VEHICLE USING MR SUSPENSION SYSTEM (Deepak Goyal, Sultan Singh, Anil Kumar)...... 41 Size-dependent vibration of microplate resonators based on the modified couple stress theory and three-phase-lag heat conduction model (Harendra Kumar and Santwana Mukhopadhyay). 42 Stability Analysis of a Delay Induced Dynamical Model on Oncolytic Virotherapy (Hitesh K. Singh and Dwijendra N. Pandey)...... 42 On Existence of Solution of First Order Retarded Differential equations with piecewise constant delays (Aradhana Bandekar, Y. S. Valaulikar )...... 43 Evolution of weak shock wave in two-dimensional steady supersonic flow in dusty gas (Rahul Kumar Chaturvedi)...... 43 Existence and regularity of solutions of fractional differential equations involving Hilfer fractional derivative of order 1 < α < 2 and type 0 ≤ β ≤ 1 (Anjali Jaiswal, D. Bahuguna).. 43 On First Integral Method and Lie Symmetry of meta-mKdV equation (Mahima Poonia, K. Singh) 44 On Monotone Method for a First Order Neutral Differential Equation (Mamta Kumari, Y. S. Valaulikar)...... 44 Space time fractional nonlinear partial differential system: Exact solution and conservation laws (Baljinder Kour, Sachin Kumar)...... 44 Solution of Partially Singularly Perturbed System of Initial and Boundary Value Problems Using Non-Uniform Haar Wavelet (Akmal Raza, Arshad Khan)...... 45 Existence, Uniqueness and Regularity of Mild Solutions of Fractional Order Navier-Stokes Equa- tions with Finite Delay (Md Mansur Alam, Shruti Dubey)...... 45 A new approach of operational matrices for hyperbolic partial differential equations (Somveer Singh, Mani Mehra)...... 46 Analytical Solution of 1-D Advection-Dispersion Equation with an Additional Source/Sink term in the Semi-infinite Aquifer using Dispersion Theory (RohitKumar, Manish Chaudhary and Mritunjay Kumar Singh)...... 46 Exponentiable objects in Q-TOP (Harshita Tiwari)...... 46 Lp Spectra of Strongly Carleman Pseudo Differential Operators associated with integral transform (Pragya Shukla )...... 47 Metrical fixed point theorems via locally finitely T-transitive binary relations under certain control functions (Aftab Alam, Mohammad Arif and Mohammad Imdad)...... 47 Some Results on Summation-Integral-type Operators and Their Properties (Rishikesh Yadav, Ra- makanta Meher, Vishnu Narayan Mishra)...... 47

vi Kantorovich type generalization of modided Szasz-Mirakjan´ Operators (Ankita R Devdhara, Vishnu Narayan Mishra)...... 48 θ∗-WEAK CONTRACTIONS AND DISCONTINUITY AT THE FIXED POINT (ATIYA PER- VEEN)...... 48 A New Type of Paranorm Intuitionistic Fuzzy Zweier I-convergent Double Sequence Spaces (Hira Fatima)...... 48 Relation-theoretic multi-valued θ-contraction]Relation-theoretic fixed point results for Multi-valued (θ, R)-contractions with an Application (Mohammad Imdad , Md Hasanuzzaman and Waleed M. Alfaqih)...... 49 Optimal control analysis of an e-epidemic model including firewall effect (Prerna Singh, Ranjit Kumar)...... 49 Solution of Differential Algebraic Equations using Coded Differential Transform Method (Anil Kumar and Giriraj Methi)...... 50 Dynamic modeling and control of divided wall distillation multicomponent separation (Manali Kokare, C. S. Mathpati, Ajit Kumar, S. S. Jogwar)...... 50 ASYMPTOTIC ANALYSIS OF BOUNDARY OPTIMAL CONTROLPROBLEM ON A GEN- ERAL BRANCHED STRUCTURE (S. AIYAPPAN,A. K. NANDAKUMARAN , AND ABU SUFIAN)...... 51 Approximate controllability of multi-term time-fractional differential inclusions with nonlocal conditions (Ashish Kumar, Dwijendra N. Pandey)...... 51 Approximate Controllability of Semilinear Fractional Evolution Systems with multiple Delays in Control (Abdul Haq, N. Sukavanam)...... 52 CONTROLLABILITY OF NONLOCAL FRACTIONAL ORDER INTEGRO-DIFFERENTIAL SYSTEMS WITH TIME VARYING DELAY (Ajay Kumar, N. Sukavanam)...... 52 Development of Higher-order Implicit-Explicit Robert-Asselin Type Time Filters (Praveen K. Maurya, Manoj K. Rajpoot)...... 52 Modelling hydraulic characteristics of Gabion weir by soft computing techniques (Siddharth Sonkar1, N.K.Tiwari2, and Subodh Ranjan Vajesnayee3)...... 53 Effect of radiative heat transfer on the growth and decay of acceleration waves in non-ideal magnetogasdynamics (Shobhit Kumar Srivastava)...... 53 Wind turbine blade section optimization using a quantitative study (M.Balachandar, B.U Raja Ramakrishnaa and N.Ramanan)...... 53 Hybrid impulsive effects on quasi-synchronization of neural networks with parameter mismatch and mixed time-varying delays (Rakesh Kumar)...... 54 Novel divergence measure for refined single valued neutrosophic sets and its utility in decision making (Adeeba Umar, R. N. Saraswat)...... 54 Analysis of Surface Seismic Waves in Piezomagnetic Layered Structure (Suman Goyal, Sanjeev Anand Sahu )...... 55 Velocity Profile of Shear Horizontal (SH) surface waves in Bi-layered FGPM/Porous Piezoelec- tric Plate (Shreeta Kumari, Sanjeev A. Sahu1, Kamlesh K. Pankaj)...... 55 Thermal analysis of convective-radiative pin fin with MATLABs inbuilt tool Pdepe considering temperature dependent properties (Sarvjeet Singh, Rohit K. Singla)...... 56 Effect of Solar Flare on climate change by Solar Flare Wave Model and Its Application (Sumit Bainjwan, Vishal Dhakane )...... 56 Modelling Aeration Efficiency of Hydraulic Jump at Under Sluice Gate (Nirali Vashishth, Subodh Ranjan Vajasneyee, and N K Tiwari)...... 57 Interference of Closely Placed Bridge Piers On Local Scour (Anuj Kataria1 and Baldev Setia2). 57 The influence of vegetation type and cover on rain garden hydrological performance ( Anuj kumar1, Krishna Kumar Singh2)...... 58

vii Non-linear Deformation of Thin Elastic Model Membrane Driven by Electrostatic Forces (Amar Shrivastava and Paritosh Mahata)...... 58 Integral Equation technique for solution of diffraction of obliquely incident water waves by rectangular asymmetric trench (1Amandeep Kaur, 1S. C. Martha, 2A. Chakrabarti)...... 59 Life Cycle Analysis (LCA) of Low Volume Rural Hill Roads (Akhilesh Nautiyala, Sunil Sharmab) 59 Flow Characteristics at the Conﬂuence of WJC and SYL Canals (Pradeep kumar1 and Baldev Setia2)...... 60 TREND ANALYSIS OF HYDROLOGICAL PARAMETERS OF TWO AGRARIAN DISTRICTS OF HARYANA, INDIA (Mridula Sharma1, Arun Goel)...... 60 GEOMETRIC PROPERTIES OF THE EXTENDED τ GAUSS HYPERGEOMETRIC FUNC- TION (R. ROY AND R. K. JANA)...... 61 Theoretical Investigation of Networks of Interacting Exclusion Processes (Tripti Midha, Arvind Kumar Gupta)...... 61

CNA 63 On solvability of some nonlinear functional-integral equations with applications (Amar Deep and Deepmal)...... 63 Effect of Solar Flare on climate change by Solar Flare Wave Model and Its Application (Sumit Bainjwan, Vishal Dhakane )...... 63 Entropy Generation in the Flow of Sisko Nanofluids over a Stretching Sheet (Ankita Bisht and Rajesh Sharma)...... 64 Adomian Decomposition Method for the solution of a Parial Differential Equations of Fractional Order ( Pratibha Verma and Manoj Kumar)...... 64 A new efficient numerical scheme for variable order fractional sub-diffusion equation ( Sarita Nandal, Dwijendra Narain Pandey)...... 65 Transverse Hydromagnetic and Media Permeability Effect on Mixed Convective Flow in a Chan- nel Filled by Porous Medium with asymmetric wall heating condition ( Km. Renu, Ashok Kumar)...... 65 Numerical study of entropy generation in porous medium vertical channel subjected to mixed convection ( Paresh Vyas, Kusum Yadav)...... 66 Curvelet optimized method for solving partial differential equations on general manifolds ( Deepika Sharma, Kavita Goyal)...... 66 Solution of inverse fractional Fisher’s equation by differential quadrature method ( G. Arora, Pratiksha)...... 67 Uncertainty propagation using Wiener-Bspline wavelet expansion (Navjot Kaur and Kavita Goyal) 67 Entropy Generation Analysis For A Micropolar Fluid Flow Due To A Moving Surface (Paresh Vyas, Manvi Adha)...... 67 Mathematical analysis of surface wave propagation in Functionally Graded Material using WKB approximation (Sonali Mondal, Sanjeev A. Sahu)...... 68 Data bounded WENO reconstructions of high order schemes (Ritesh Kumar Dubey, Sabana Parvin) 68 Solution of singular fractional Lane-Emden type equations by an analytical technique (Anoop Kumar, Seema)...... 69 NUMERICAL APPROACH FOR A COUPLED SYSTEM OF SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEM WITH NON-SMOOTH DATA (Aarthika K, V. Shanthi)...... 69 THRESHOLDING FUNCTIONS INVOLVED IN THE WAVELETBASED DENOISING METHOD (Princess Raina and Zaheer Abbas )...... 69 One-dimensional solute migration model with first-order production term in semi-infinite porous media (Affreen Akhter, Mritunjay Kumar Singh )...... 70

viii STUDY OF FRACTIONAL THERMOELASTIC PROBLEM WITH MOVING HEAT SOURCE (Jaya Bikram, G.D. Kedar)...... 70 Numerical Solution of Fractional Order Non-Conservative Advection-Diffusion Equation (Anup Singh and S. Das)...... 71 Finite element analysis of semilinear time-fractional diffusion equation (Dileep Kumar∗a, Sud- hakar Chaudharyb, V.V.K Srinivas Kumar∗)...... 71 Effect of Aspect Ratio on Natural Convective flow in a Rectangular Enclosure Occupied by Anisotropic Porous Medium (Ashok Kumar, Ajay Kumar∗1, Km. Renu and M. S. Rawat).. 72 NUMERICAL SOLUTION OF DAMPED FORCED OSCILLATOR PROBLEMS BY OPER- ATIONAL MATRIX OF INTEGRATION (Dr. Mithilesh Singh1, Seema Sharma2, Sunil Rawan3)...... 72 Flow of a Hydromagnetic Fluid through a porous medium between permeable beds with damping effects. (Ravi Kumar and B.G. Prasad)...... 73 Finite element solution of a problem on coupled thermoelasticity for functionally graded material by two different approaches for time domain (Om Namha Shivay∗ and Santwana Mukhopadhyay)...... 73 Simulation of Heat transfer of Ferro fluid in Cylindrical Micro-Channel (Ramesh kumar1∗, Harry Garg2, S.K Dhiman1)...... 74 Computational Analysis of Bed of a Mobile Channel (Atul Ailawadhi1 Baldev Setia2,)...... 74 Numerical study on MHD flow of nanofluid due to a rotating disk with heat generation and partial slip effect (V. K. Chaurasiya, Ramayan Singh and Rajat Tripathi)...... 75 Fibonacci collocation method to solve nonlinear space-time fractional order advection-reaction- diffusion equation (Kushal Dhar Dwivedi)...... 75 Abhishek Verma,N K Singh 1 1Department of Mechanical Enginee (Abhishek Verma,N K Singh ) 76 Numerical study of inclined stretchable partially heated enclosure filled with nanofluid (Pentyala Srinivasa Rao, Anil Kumar)...... 76

CP 77 Dynamic modeling and control of divided wall distillation multicomponent separation (Manali Kokare, C. S. Mathpati, Ajit Kumar, S. S. Jogwar)...... 77

DE 79 Existence of mild solutions for neutral fractional functional integro-differential equations with non instantaneous impulses of order α ∈ (1, 2) (Pallavi Bedi, Anoop Kumar)...... 79 CHARACTERIZATION OF POLARIZED SHEAR WAVES IN POROUS-PIEZOELECTRIC MEDIUM OVER A HETEROGENEOUS ELASTIC SUBSTRATE CONTAINING POINT SOURCE (SUBHASHIS KARMAKAR*, SANJEEV A. SAHU)...... 80 Smooth stable manifold for delay equations with arbitrary growth rates (Lokesh Singh*, Dhiren- dra Bahuguna)...... 80 Approximation of fixed points and the solution of delay differential equation via new iterative scheme (Javid Ali, Faeem Ali)...... 81 Classification of Some Functional Differential Equations with Constant Coefficients to Solvable Lie Algebras (Jervin Zen Lobo, Y.S Valaulikar)...... 81

MD 83 Speech analysis based emotion-aware healthcare system (Akshita Abrol, Praveen K. Lehana).. 83 Analysing the efﬁciency of hexagonal microﬂuidic fuel cell (Jyoti Lalotra, Praveen K. Lehana). 83 Varying Trends of Smart Bandaging (Priti Rajput, Praveen K. Lehana)...... 84 Varying Trends of Smart Bandaging (Verasis Kour, Praveen K. Lehana)...... 84 Impact of rigid surface on dispersion and damping characteristic of Love-type wave propagating in an initially stressed piezoelectric layer (Shishir Gupta, Rachaita Dutta, Soumik Das).. 85

ix Discharge characteristic of multi-cycle triangular labyrinth weir (Subhankar Das, Parthajit Roy) 85 Geometrical changes in journal bearing due to piezoelectric actuators (Aayush Trivedia, Wolfgang Seemannb, and Mohammad Talhac)...... 86 Studies of Hyperloop Vehicle for Transportation: A Review (V.K. Srivastav1, Aditya Priyanka1, Abhishek Kumar1, Shudhanshu Kumar1, Anand Raj1)...... 86 Studies of Hyperloop Vehicle for Transportation: A Review (V.K. Srivastav1, Aditya Priyanka1, Abhishek Kumar1, Shudhanshu Kumar1, Anand Raj1)...... 86 Bivariate Bernstein-Schurer-Stancu type GBS operators based on (p, q)-integers (Mohd. Ahasan and M. Mursaleen)...... 87 Uncertain eigenvalue analysis of finite element modelled functionally gradient arches (Moham- mad Amir#, Mohammad Talha∗)...... 87 Studies of Hyperloop Vehicle for Transportation: A Review (V.K. Srivastav1, Aditya Priyanka1, Abhishek Kumar1, Shudhanshu Kumar1, Anand Raj1)...... 87 Second-order statistics of the elastic buckling of skewed functionally gradient plates with material uncertainties (Mohammed Shakir1∗ and Mohammad Talha 2)...... 88 Estimation of mean rainfalls using Geostatistical techniques in Kabul River Basin, Afghanistan (Shamsullah Sultani1 1∗, Arun Goel 2)...... 88 COMPUTATIONAL AND EXPERIMENTAL STUDIES OF FLY ASH BRICK (V. K. Srivastav1 R. K. Singh1, R. Kumar1, A Kumar1, A. K. Chhotu1, A. R. Paul2)...... 89 3D Stress Analysis of Reinforced Concrete Sleeper (Tejas M. Gondhalekar1 and S. K. Panigrahi2) 89 Triple-diffusive convection with more realistic two temperature model for nanofluids (Urvashi Gupta)...... 90 Design of wind turbine for the application of desalination process (S.Ramachandran1 ,S.Vasanth1∗,S.Devarajan1) 90 Enhancing software reliability through the generalized inflection S-shaped failure rate with testing effort in growth model (Vishal Pradhan1, Ajay Kumar1 and Joydip Dhar1)...... 91 Experimental and numerical study of blood ow characteristics in human coronary artery with plaque (Wasim Saliha,Pradeep Kumarb, Parmod Kumarc, Mohammad Talha ∗)...... 91 Numerical simulation for radial stress and displacement of jet engine Exhaust pipe (Y.K.Singh1 and S. K. Panigrahi2)...... 92

MM 93 Cross-diffusion induced Turing and non-Turing patterns in Rosenzweig-MacArthur model (Nayana Mukherjee)...... 93 POPULATION DYNAMIC CONSEQUENCES OF FEARFUL PREY IN A SPATIOTEMPO- RAL PREDATOR-PREY SYSTEM (Ranjit Kumar Upadhyay, Swati Mishra )...... 93 A Mathematical model for Human Papillomavirus and its impact on cervical cancer in India (R Praveen Kumar, K Murugesana)...... 94 Predator-prey system: Herd behavior and anti-predator traits contribute in enriching the evolution of stronger prey defence (Rajat Kaushik, Sandip Banerjee)...... 94 Effect of porosities with and without fractures on propagation of SH wave: case wise study (Shishir Gupta, Soumik Das, Rachaita Dutta )...... 95 Dynamical behaviour of predator-prey systems with variation of Allee function in predator’s growth: Structural sensitivity analysis (Deeptajyoti Sen)...... 95 MODELLING OF SEDIMENT EXCLUDER (Dibyendu Das1, N.K.Tiwari2,Subodh Ranjan vajesnayee3 )...... 96 FEM Analysis of Adhesively Bonded Composite Patches (V.Divakar1 and S. K. PanigrahiR2).. 96 Solving a variational inclusion problem with its corresponding resolvent equation problem involving XOR-Operation (1Rais Ahmad,2Javid Iqbal,2Shakeel Ahmed,1Saddam Husain).. 97 Modeling and Simulation of Nylon Liner Shaped Charge Jets (Yadav Ombir Singh∗, Nimje S.V and Choudha P.K.)...... 97

x Modeling the impact of sanitation and awareness on the spread of infectious diseases (Rajanish Kumar Raia, A.K. Misraa∗, Yasuhiro takeuchib)...... 98 Modelling of scour around Cylindrical Piers (M Vignesh1, Subodh Ranjan Vajesnayee2, and N K Tiwari3)...... 98

NA 99 Continuous wavelet transform of Schwartz tempered distributions in S0(Rn) (Jay Singh Maurya) 99 Continuous fractional wave packet system in Sobolev type space (Manab Kundu and Akhilesh Prasad)...... 99 Cechˇ proximity relation and Rough Set Theory (Pankaj Kumar Singh, Surabhi Tiwari)..... 100 A Comparative Study of Monitor function in Mesh Reconstruction (Prabhat Mishra and Ritesh Kumar Dubey)...... 100 Studies of Hyperloop Vehicle for Transportation: A Review (V.K. Srivastav1, Aditya Priyanka1, Abhishek Kumar1, Shudhanshu Kumar1, Anand Raj1)...... 101 MESHLESS METHOD FOR THE NUMERICAL SOLUTION OF SPACE AND TIME FRAC- TIONAL WAVE EQUATION (Hitesh Bansu ,Sushil Kumar)...... 101 Semi-Analytical Solution of Fractional Convection-Dispersion Equation by using Conformable Derivative Approach and Homotopy Analysis Method (Manish Chaudhary∗, Rohit Kumar and Mritunjay Kumar Singh)...... 101 Convergence Analysis of New Hybrid Scheme for Singularly Perturbed Parabolic Problems with Interior Layers (Mr. Narendra Singh Yadav1 Dr. Kaushik Mukherjee)...... 102 Fractional calculus for k-Mittag-Leffler function of two variables with the kernel F3 (Owais Khan) 102 Approximation By Modified Lupas Operators Based On (p, q)Integers (M. Mursaleena, Zaheer Abbasb and Mohd Qasimb)...... 103 HURWITZ-LERCH ZETA FUNCTION AND SOLUTION OF FRACTIONAL KINETIC EQUA- TION (Raghib Nadeem)...... 103 Computation of fractional integrals and derivatives for the product of Mathieu-type series and generalized Mittag-Leffler function (Mohd Saif )...... 103 Turing patterns in a Prey-Predator Model with Hassell-Varley Functional Response (Vikas Kumar and Nitu Kumari)...... 104

Invited Talks 105

Author Index 107

xi xii Plenary Talks

PDE: Classical and Modern Prof. A. K. Nandkumaran Department of Mathematics, Indian Institute of Science, Bangalore- 560012, India. Email: [email protected] In this talk, we brieﬂy present the developments of Partial Differential Equations (PDE) as a modelling for classical problems. We, then realize the importance of analysis in the study of PDEs. We see how the weak solutions appears naturally and some of the developments. We touch upon various equations arising in the literature, conservation laws, calculus of variations to optimal control problems. We also see the notions like weak solutions, Sobolev spaces, viscosity solutions etc.

Set Differential Equations: An Overview and Recent Developments Gnana Bhaskar Tenali Department of Mathematical Sciences Florida Institute of Technology Florida, USA. Email: gtenali@ﬁt.edu The study of Set Differential Equations (SDEs) is fast evolving as an independent discipline. In this talk, we give an outline of fundamental and recent results on the existence and uniqueness of solutions of SDEs in Banach spaces and in Frechet spaces. We also discuss the stability theory for the solutions of SDEs, using the standard methods as well as methods based on geometry of convex bodies and the theory of mixed volumes.

Mathematical modeling and numerical simulation of particles in fluids of complex geometries S. Sundar , Nityananda Roy, Thomas Goetz HEAD, Department of Mathematics Indian Institute of Technology Madras (IIT Madras) Chennai 600 036, India. Email: [email protected] We are inquisitive to study on the topic regarding the behavior of soft and hard particles in complex flow system. In case of soft particles like droplet and bubble we applied population balance equation coupled with hydrodynamics and for hard particles we applied Newtons and Eulers equations of motion. For soft particles our motive is to develop mathematical model as well as numerical methods to predict the number density, volume fraction of particles in complex flow system. In case of hard particles we are trying to construct a model of particles sedimentation considering that the particles attraction force works between the particle and the obstacle wall. Our purpose is to constitute a model about the behavior of non spherical particles in complex flow system.

1 NUMERICAL METHODS TO CAPTURE δ SHOCKS ARISING IN THE SYSTEMS OF PRESSURELESS GAS DYNAMICS G.D. Veerappa Gowda TIFR Centre for Applicable Mathematics PB No:6503, Yelahanka New Town,Bengaluru-560065. Email: [email protected]

In this study, a class of Godunov-type solvers is formulated for a weakly hyperbolic pressureless gas dynamics system. In one dimension the system is written as

ρt + (ρu)x = 0, 2 (ρu)t + (ρu )x = 0. (0.0.1)

An Engquist-Osher type solver is constructed utilizing the homogeneity property. Again on introducing a new variable v = ρu in the second equation of (0.0.1), one can treat flux function ρu2 = v2/ρ as a convex 1 function in v with discontinuous coefficient ρ . Now by using the idea of the discontinuous flux, a Godunov- v2 type interface flux is constructed for which is again utilized in the first equation to determine its interface ρ flux. The results obtained from this conservative Godunov type solver generate steady shocks with almost of double the strength as compared to other Godunov type solvers. Since the considered systems satisfy the generalized Rankine-Hugoniot conditions, a non-conservative version of the above schemes are proposed and tested on various numerical examples. In particular, non-conservative Godunov-type solver developed here outclass other well-known solvers in capturing stationary shock waves. These Godunov-type solvers are also extended to a two-dimensional pressureless system. Computational results clearly show a non- conservative Godunov-type solver captures steady shocks considerably better than third order DG scheme in two dimensions. It is also shown for augmented Burgers systems non-conservative version captures even moving delta-shocks with exceptional strengths.

Chaotic behaviour in Differential and Difference Equations with their Fractal Nature Mohammad Sajid College of Engineering, Qassim University, Saudi Arabia Email: [email protected]

The real problems and issues on the frontiers of modern scientific, technological, economical, and social researches are nonlinear in nature. It seems that it is a nonlinear world. Most nonlinear systems are not possible to solve analytically or much harder to analyze. During the last three decades of the 20th century, the excessive studies of nonlinear dynamics shows that chaos occurs widely. Before this period, the chaos was generally regarded as a nuisance and designed out of the model. Many researches show that chaotic phenomena are completely deterministic and characteristic for typical nonlinear systems. Numerous researches are resolved by using chaos theory in the last few decades. Chaos exists everywhere in the world since most of the problems are nonlinear in nature. Chaos is developing in a new way that influences the world around us, and consequently also influences our ways of approaching, analyzing and solving problems. As we are aware that the technological advances require a deep understanding of physical processes in engineering and science. A variety of such physical process can be modelled as differential equations and difference equations. The main purpose of this presentation is to explore chaotic behaviour in differential and difference equations with their fractal nature; and brings attention of mathematicians and engineers for those who are interested in chaotic behaviour and want to introduce something new and advanced methodology. Fortunately, chaos is one of them. To achieve our target here, we provide some techniques to study chaos in mathematical modeling arising from differential and difference equations. In many cases, chaos

2 can be effectively visualized by using fractals. The demonstration of some mathematical models associated to chaos with their applications is explored here.

Keywords: Chaos, Difference Equations, Differential Equations, Fractals.

A New Higher Order Nonliear PDE Model for Effective Image Denoising and its Fourier Spectral Accurate Solution B.V. Rathish Kumar Department of Mathematics, IIT kanpur, India. Email: [email protected]

In this talk we will introduce a new higher order nonlinear PDE model for image denoising and demonstrate its ability to denoise without any staircase effect as routinely noticed with lower order PDE models. We use convexity splitting based fourier spectral scheme for the computation of the denoised version of a given noisy image. Fourier spectral method is both accurate and faster than other standard approaches. Performance of the model and the method will be discussed based on the bench mark test images and the related computational metrics.

Application of Operator Theory in Controllability Analysis Raju K. George Department of Mathematics, Indian Institute of Space Science and Technology, Thiruvananthapuram - 695 547, Kerala, India. E-mail: [email protected]

Controllability is one of the fundamental properties of a controlled dynamical systems which ensures the ability of a dynamical system to steer the state of the system from any initial state to a desired ﬁnal state in a given time interval. R.E. Kalman [3] introduced the notion of controllability in 1960’s and established the complete characterization of controllability for linear systems in ﬁnite dimensional spaces. Subsequently many researchers extended the controllability notion for nonlinear systems, especially for semilinear systems, ( see [2]), of the form:

dx = A(t)x + B(t)u + f(t, x(t)) 0 ≤ t < t ≤ t < ∞, (0.0.2) dt , 0 1 x(t0) = x0

n m where, for each t ∈ [t0, t1], the state vector x(t) ∈ R , the control vector u(t) ∈ R , A(t),B(t) are matrix valued functions and the function f : I × Rn → Rn is a nonlinear function. Many of the powerful tools of nonlinear analysis like fixed point theory, monotone operator theory, set-valued function theory, Lie Group approach etc. are employed to establish controllability of such systems, refer to [3]. In this talk, we discuss the notion of controllability for both linear and nonlinear systems in finite dimensional and infinite dimensional settings. We show how the theory of monotone operators and fixed point theory are used to establish controllability results.

Bibliography

[1 ] R.W. Brockett, Finite Dimensional Linear Systems, SIAM, Philadelphia (2015).

3 [2 ] R. K. George, Approximate Controllability of Non-autonomous Semilinear Systems. Nonlinear Analysis, Theory, Methods and Application, Vol.24, pp.1377-93,( 1995). [3 ] J. Klamka, Controllability and Minimum Energy of Control, Springer, (2019).

Parameter Uniform Numerical Schemes for a class of Singularly Perturbed Problems Kapil Kumar Sharma Department of Mathematics, South Asian University, New Delhi, India. Email: [email protected] (Jointly with) Pankaj Mishra Research Scholar, SAU, New Delhi, India. Amiya K Pani, Indian Institute of Technology, Mumbai, India. Graeme Fairweather, Mathematical Reviews, American Mathematical Society, USA. The singularly perturbed differential equations are ubiquitous in mathematical modeling of several real life phenomena and provide a realistic simulation of the phenomena. The singularly perturbed differential equation is characterized by a parameter, which is multiplied in a highest order derivative term. The solution of this class of differential equations exhibits layer behavior in narrow regions. These narrow regions are known as layer region and rest part of the domain is known as outer region. In this talk, there is an attempt to introduce the audience with singularly perturbed problems and challenges in the development of the numerical methods to solve this class of problems. Further, the parameter uniform numerical schemes based on orthogonal spline collocation are presented.

Can parallel bearing surfaces support load- A Tribological Study Prawal Sinha Department of Mathematics and Statistics Indian Institution of Technology Kanpur, India. E-mail: [email protected] Tribological studies are as old as the wheel. For hydrodynamic lubrication it is imperative that the bearing surfaces be inclined. However over the last 70 years studies have suggested that even parallel surfaces can support some load. Several researchers have experimentally investigated the thermal influence on the load carrying capacity of parallel slider bearings. The results of all these researchers show the existence of a lifting force (load capacity) even when parallel bearings are in operation. However, the precise causes which are responsible for this phenomenon are not precisely understood. Rodkiewicz and Sinha [1] provided an orderly analysis which elaborates on the mechanisms that may be responsible for the fluid generated lifting force. It is indicated that the consideration of the fore region pressure together with the density variation may lead to a useful load support with a reduced friction, even for parallel sliding bearing. In recent years researchers have focused attention on thermohydrodynamic analysis of rough surfaces. Recently Sinha and Getachew [2] numerically analyzed the combined effect of thermal and surface roughness on the performance of an infinitely long slider bearing using stochastic approach. In their study two types of roughness: longitudinal roughness and transverse roughness were considered. The analysis indicated that for parallel sliders some load capacity may be generated due to the combined effect for both types of roughness. All of the works that has appeared in literature do not seem to conform to the experimental results obtained for parallel sliders. It seems natural that as a consequence of heating, there would be a thermal expansion in asperities and probably a distortion of the slider surface. Thus in this study the effect of thermal distortion of the slider and asperities on different characteristics of an infinitely tilted pad rough slider bearing is analyzed using stochastic approach.

References

4 [1 ] Rodkiewicz, C.M., Sinha, Prawal.: On the Lubrication Theory: A Mechanism Responsible for Generation of the Parallel Bearing Load Capacity, Trans. ASME, J. Lub. Tech. 115, 584-590 (1993)

[2 ] Sinha, Prawal and Getachew A. (2009), THD analysis for slider bearing with roughness: special reference to load generation in parallel sliders, Acta Mechanica, 207, Issue 1, pp 11.

AN OVERVIEW ON INVERSE PROBLEMS ARISING IN FLUID DYNAMICS JAIME H. ORTEGA Department of mathematical Engineering and Center for Mathematical Modeling, UMI 2807-CNRS UCHILE UNIVERSIDAD DE Chile Santiago- Chile. Email : [email protected]

Inverse problems became an interesting topic of study in several areas of knowledge. The main idea is to recover some parameters of characteristic of the media which can not be obtained directly. Some areas of application are optic, radar medical imaging, signal processing, nondestructive testing, astronomy, among others. In this talk we will present some ideas on the called inverse geometrical problems, in which we are interested in to recover some geometrical properties of the media by means of boundary measurements, arising in ﬂuid dynamics. In the last years, there is an extensive literature in this topic. We will present some ideas on the identiﬁability and stability results, we will present also some numerical experiments.

Differential quadrature in the study of vibrations of non-uniform FG circular plates under hydrostatic peripheral loading and thermal environment Roshan Lal Department of Mathematics, IIT Roorkee, India [email protected]

In the present study, the effect of exponential thickness variation and hydrostatic peripheral loading have been analyzed on axisymmetric vibrations of functionally graded circular plate subjected to non-linear temperature rise along the thickness on the basis of Kirchoffs plate theory. The plate is graded in thickness direction and controlled by a power law index. The mechanical properties of the such a plate material are assumed to be temperature-dependent. Keeping the uniform thermal environment over the top and bottom surfaces, the equations for thermo-elastic equilibrium and axisymmetric motion for such a plate model have been derived by Hamiltons principle. Employing generalized differential quadrature method, the frequency equation for clamped boundary condition has been obtained, which has been solved numerically using the MATLAB. The lowest three roots so obtained have been retained as the frequencies for the first three modes of vibration. The influence of various parameters such as thickness parameter, power law index, in-plane force parameter and temperature difference has been analyzed on the vibration characteristics of the plate. By allowing the frequency to approach zero, the critical buckling loads with varying values of other parameters have been computed. A study with linear as well as uniform temperature rise has also been performed. The validity of the present technique is confirmed by comparing the results with the published work.

Keyword : Differential quadrature, vibration, functionally graded circular plates, exponential thickness variation, temperature-dependent material, Non-linear temperature distribution.

5 6 Invited Speakers

A Domain Decomposition Method for Singularly Perturbed Parabolic Reaction-Diffusion Problems S.C.S. Rao1 and S. Kumar2 1Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, India. 2Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, India. Emails: [email protected], [email protected]

In this work we consider a singularly perturbed parabolic reaction-diffusion problem that exhibits parabolic boundary layers. To solve this problem numerically, we develop a domain decomposition method of Schwarz waveform relaxation type. We also provide error analysis of method. The numerical approximations generated from the method are proved to be uniformly convergent, having ﬁrst order in time and almost fourth order in space. Moreover, faster convergence of the algorithm is established for small perturbation parameter. The method is then extended to a coupled system of singularly perturbed parabolic reaction-diffusion problems. Numerical results are given in support of theoretical ﬁndings.

A Finite element approach to a moving boundary problem with variable thermal conductivity Rajeev and Rishabh Daal Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University) Email: [email protected]

Moving boundary problems are nonlinear problems and involves a moving interface/ boundary which makes it difficult to get its exact solution. In this article, we consider a mathematical model of a moving boundary problem that includes variable thermal conductivity and a constant temperature at the left boundary of the domain. The temperature distribution and moving boundary involved in the model are calculated with the aid of similarity variables and Galerkin finite element method. To show the accuracy of the calculated solution, the comparisons between our solution and an analytical solution (for a particular case) are demonstrated through tables. In this study, it is seen that the proposed approach is simple and sufficiently correct to estimate the solution of various moving boundary problems governing the process of phase change.

Keywords: Phase change problem, variable thermal conductivity, similarity variable, Galerkin residual approach.

AMS subject classiﬁcations : 35R37, 80A22, 80M10.

7 Approximation Method for Generalized Fractional Derivatives with Application Rajesh Pandey Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi-221005, India. Email: [email protected]

We discuss generalized fractional derivatives and approximation methods for these fractional derivatives. In special case, generalized fractional derivative reduces to Riemann-Liouville and Caputo factional derivatives. For application of the approximation schemes, fractional advection diffusion equation is considered.

Entropy Stable Schemes For Relativistic Hydrodynamics Equations Harish Kumar Department of Mathematics, IIT Delhi, India. [email protected]

In this work, we propose high order finite difference schemes for the equations of relativistic hydrodynamics, which are entropy stable. The crucial components of these schemes are a computationally efficient entropy conservative flux and suitable high order entropy dissipative operators. We first design a higher order entropy conservative flux. For the construction of appropriate entropy dissipative operators, we derive entropy scaled right eigenvectors. This is then used with ENO based sign preserving reconstruction of scaled entropy variables, which results in higher order entropy stable schemes. Several numerical results are presented up to fourth order to demonstrate entropy stability and performance of these schemes.

Nonlinear dynamics of Some Complex Ecological Sytems R. P. Gupta Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi-221005, India. [email protected]

The purpose of this presentation is to offer a detailed nonlinear dynamical behavior of some ecological system. The proposed systems consider different types of functional response and nonlinear harvesting functions. Positivity and boundedness of the solutions of the systems are guaranteed. The uniform persistence of the systems are discussed. The stability and bifurcation analysis of various steady states of these systems are analyzed in detail. The local existence and stability of limit cycle emerging through Hopf- bifurcation are given. The emergence of homoclinic loop has been shown through simulation when the limit cycle arising though Hopf-bifurcation collides with a saddle point. Using numerical examples it is shown that these systems admit periodic, quasi-periodic and chaotic solutions. Many forms of complexities such as chaotic bands (including periodic windows, period-doubling bifurcations, period-halving bifurcations and attractor crisis) and chaotic attractors have been observed. Sensitivity analysis is carried out and it is observed that the solutions are highly dependent to the initial conditions. It is pointed out that even if the systems are simple it may exhibit chaotic dynamics.

8 A high-order quasi-variable meshes two-level implicit compact scheme for solving three-dimensional nonlinear non-stationary advection-diffusion equation Navnit Jha Faculty of Mathematics and Computer Science, South Asian University, Chanakyapuri,New Delhi, India-110 021 [email protected]

A two-level implicit compact-scheme on quasi-variable meshes is reported for solving three-dimensional second-order mildly nonlinear parabolic partial differential equations. The new nineteen-point compact scheme exhibit fourth-order accuracy in space and second-order in time on a variable spacing mesh network as well as on uniformly spaced mesh points. We have also developed an operator- splitting technique to im- plement the alternating direction implicit method for computing the advection-diffusion equation. Thomas algorithm computes each tri-diagonal matrix that arises from alternating direction implicit steps in minimal computing time. The operator-splitting method is unconditionally stable. The improved accuracy is achieved at a lower cost of computation and stor- age, because the spatial mesh parameters tune the meshes location according to the behaviour of solution values. The new method successfully applied to the Navier- Stokes equation, advection-diffusion equation and Burger’s equation for the computational illustrations that corroborate the order, accuracies, and robustness of the new high-order implicit compact scheme. The main highlight of the present work lies in obtaining a fourth-order scheme on a quasi-variable mesh network, and their superiority over the corresponding uniform meshes high-order compact scheme.

Keywords:Quasi-variable mesh network; Compact scheme; ADI scheme; Advection-diffusion equation; Navier-Stokes equation; Burger’s equation; Stability. Mathematics Subject Classiﬁcation : 35G61; 65M06; 65M12; 65N06; 35Q35.

Delayed Mathematical Models of HIV Saroj Kumar Sahani Department of Mathematics, South Asian University, Akbar Bhawan Chankyauri, New Delhi-110021 [email protected]

n this lecture, we consider a delayed HIV model with apoptosis of cells that resembles a lot of characteristics HIV infection. The delayed variable considered here are immunological and intracellular delay which have been introduce to make the model more logical and practical. The underlying model has been studied theoretically using linear stability analysis. To ascertain the global dynamics of the model, a Lyapunov functional approach has been performed. The stability switch criteria taking the delay as the bifurcating parameter, leading to Hopf bifurcation has been studied. We have explored the possibility of the transition of the system from order to chaos using the numerical simulations. The analytical results obtained have been veriﬁed by numerical simulations with the parametric values available in literatures. The results thus can be used to describe the extensive dynamics exhibited by the model introduced in this article. The effects of apoptosis on viral load has been explored with the help of numerical simulations.

9 Numerical Solution to Blood Model Navier-Stokes Equations in the Entrance Region of Concentric Annuli with Rotating Inner Wall Srinivasa Rao Nadiminti1; N. Gayathri Devi1 and A.Kandasamy Division of Mathematics, Department of Science and Humanities, Vignan’s Foundation for Science, Technology and Research (VFSTR) Vadlamudi, Guntur, Andhrapradesh - 522213, India Department of Mathematical and Computational Sciences National Institute of Technology Karnataka, Surathkal Mangalore, Karnataka-575025, India. [email protected]

Blood model Navier-Stokes equation in the entrance region of concentric annuli has been studied numerically. We assumed that the inner cylinder is rotating with a constant angular velocity and the outer cylinder is stationary. A finite difference analysis method is used to obtain the velocity profiles in the various directions and pressure variations along the radial direction. Under the Prandtl’s boundary layer assumptions, the continuity and momentum equations are solved iteratively using a finite difference analysis method. Computational results are obtained for various non-Newtonian flow parameters and geometrical considerations. Comparison of the present results with the results available in literature for various particular cases like Newtonian fluid has been done and found to be in good agreement.

Keywords:Concentric Annuli; Entrance Region Flow; Casson Fluid; Rotating Wall; Finite Difference Method.

On the nondegenerate soliton solutions of Manakov system M. Senthilvelan Department of Nonlinear Dynamics School of Physics Bharathidasan University Tiruchirappalli - 620 024 [email protected]

In this talk, we demonstrate that the celebrated Manakov system can admit a more general type of nondegenerate fundamental solitons corresponding to different wave numbers, undergo collisions without any energy redistribution. We show that the previously known soliton solutions which allow energy redistribution among the modes turns out to be a special case corresponding to solitary waves with identical wave numbers in both the modes and travelling with the same velocity. We also discuss how these nondegenerate soliton solutions exhibit various symmetric and asymmetric wave pro les in detail. Finally, we present the collision scenario in detail for all the cases.

Keywords:Degenerate soliton, Nondegenerate soliton, Manakov equation, Hirota bilinear method

10 Noise-Induced extinction in an ecological model Partha Sarathi Mandal Department of Mathematics, NIT Patna [email protected]

In this talk, we study a stochastically forced predator-prey model with strong Allee effect in prey population. In the deterministic case, we show that the model exhibits the stable interior equilibrium point or limit cycle corresponding to the co-existence of both species. We investigate a probabilistic mechanism of the noise-induced extinction in a zone of stable interior equilibrium point. Computational methods based on the stochastic sensitivity function technique are applied for the analysis of the dispersion of random states near stable interior equilibrium point. This method allows to construct an elliptic domain and estimate the threshold value of the noise inten- sity for a transition from the coexistence to the extinction.

Keywords: Allee effect; Noise-induced transition; Stochastic sensitivity function.

Numerical Investigation of Natural Convection of Casson Fluids in a Square Porous Cavity under the Effects of Thermal Radiation. Sapna Sharma Thapar Institute of Engineering and Technology,Patiala [email protected]

In nature most of the fluids exhibit non-Newtonian behavior such as oils, emulsions, blood etc. Vis- coplastic fluids are class of non-Newtonian fluids which possess some yield stress. These fluids deform or flow only when value of applied stress is more than yield stress. To understand the behavior of such fluids various rheological model are developed. One such model is Casson fluid which was studied by Casson in 1959 to understand the characteristics of pigments of printing ink. In the present study, free convection in a square porous cavity filled with casson fluid under the influence of thermal radiation is investigated. The governing equations of the flow problem are converted into finite element equations and solved by using Penalty Finite Element method. The square porous cavity has been partially heated from below by a heating source and symmetrically cooled from both side walls. The effects of yield stress of the fluid on heat and fluid transport inside the cavity is studied for different values of temperature difference, across the hot and cold surfaces. Also, the effects of different lengths of heated zone is investigated for three different values of heated zone. All important results have been expressed in terms of Casson fluid parameter, Darcy number, Rayleigh number, Prandtl number, Radiation parameter. It is observed that with rise in Casson fluid parameter rate of heat transfer and fluid flow rate enhances.

Keywords:Casson ﬂuid; Radiation; Penalty Finite Element method; square porous cavity

11 A PARAMETER-UNIFORM IMPLICIT SCHEME FOR TWO-PARAMETERS SINGULARLY PERTURBED PARABOLIC PROBLEMS Devendra Kumar Department of Mathematics, Birla Institute of Technology and Science, Pilani, Rajasthan-333031, India. Email: [email protected]

A parameter-uniform implicit scheme for two-parameter singularly perturbed boundary value problems is constructed. Sharp bounds on the solution derivatives are derived. The solution is also decomposed into the sum of regular and singular components and the bounds on the derivatives of these components which are used in the convergence analysis are also drawn. The numerical method comprises the backward Eulers scheme based on the Rothes technique in the temporal direction and the Crank-Nicolson scheme on a predeﬁned Shishkin mesh in the spatial direction. Through rigorous analysis, the theoretical results for two 2 2 different cases: Case I. 1/2 → 0 as 2 → 0, and Case II. 2/1 → 0 as 1 → 0 which show that the method is convergent irrespective to the size of the parameters 1, 2 are provided. The order of accuracy in the ﬁrst and second cases are shown O((∆t)2 + N −1(lnN)2) respectively. Two test problems are encountered to verify the computational results with theoretical results.

Modeling Lithiation Induced Stresses in High-Capacity Electrode Particles with Concentration Dependent Properties Poornesh Kumar Koorata Applied Solid Mechanics Lab, Department of Mechanical Engineering, NITK Surathkal, Mangalore 575025 [email protected]

The lithium ion intercalation in high-capacity electrode particle is modeled in terms of concentration ux over particle’s radius that is governed by differential expression for concentration dependent diffusion. The itercalation effect is modeled in terms of governing equations for stresses. The combined equation of elasticity and ion concentration diffusion is solved for stress distribution to better understand the particle’s response to intercalation. In this article the set of PDEs are solved for a spherical particle using Crank-Nicolson scheme. The interesting portion of the article is dedicated for predicting the response under realistic circumstances with concentration dependent variation in elastic modulus which is linearly varied from 100 GPa to 30GPa from zero to fully lithiated condition, respectively. The results are obtained in terms of radial and circumferential stresses and effective and hydrostatic stresses normalized over yield limit of high-capacity particle.

12 Understanding molecular transport in a non-conserving systems: The role of interactions Arvind Kumar Gupta Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar, Punjab, India-140001. [email protected]

Many non-equilibrium biological processes such as intracellular transport, cellular organization, cellular motility, etc are supported by the enzymatic molecules called molecular motors/motor proteins. These motors carry vesicles (≥ 50 nm) and typically move with 5nm steps along microtubules. For performing the mechanical work, they convert the chemical energy derived from the hydrolysis of ATP. Experiments suggest that they behave in a cooperative manner and interact locally among them. The interaction energy (E) is either attractive (E > 0) or repulsive (E < 0). Similar kind of interactions is also visualized in vehicular traffic. Driven diffusive systems provide a fruitful framework for studying the statistical properties of such non equilibrium realistic processes. In the presence of driven external field, they reach a non equilibrium steady state (NESS) characterized with non-vanishing particle current. Totally asymmetric simple exclusion process (TASEP) is the minimal model for describing the unidirectional motion of particles along a lattice, where each particle occupies and covers only a single site of the lattice. We explain the collective behavior of interacting particles using a variant of TASEP that consolidates the interactions in the thermodynamically consistent procedure. These interacting molecular motors or interacting vehicles move along linear filaments (tracks) and can reversibly associate/dissociate from them. It is shown that even for weak interactions, theoretical predictions from simple mean-field approach deviate significantly from Monte Carlo simulation results. To overcome this issue, we proposed a new theoretical method namely correlated cluster mean-field theory that takes into account some correlations. The effect of interactions on stationary phase diagrams, particle currents and densities are explicitly evaluated. The theoretical results are further tested with extensive computer simulations.

Keywords: Exclusion processes; Monte Carlo simulation; Mean-ﬁeld approximation; Correlations; In- teraction.

Mathematics Subject Classiﬁcation :82C70; 82C80; 82C22.

SH-wave propagation in monoclinic medium with linearly varying inhomogeneity Sumit Kumar Vishwakarma, Tapas Ranjan Panigrahi, Rupinderjit Kaur. Department of Mathematics, BITS-Pilani, Hyderabad Campus, Hyderabad-500078, India. [email protected]

The present work studies SH-wave propagation in a monoclinic layer lying over a monoclinic half space. The elastic constants of the layer and the half space have been assumer to varies linearly with depth. Using suitable boundary conditions, the dispersion equation has been deduced in closed form. The results are compared with the isotropic case for the validation of the study. The effect of inhomogeneity parameter associated with the elastic constants have been studied numerically and presented graphically. It has been found that inhomogeneity associated with medium has a great bearing on the phase velocity of the SH-wave.

Keywords: SH wave, monoclinic medium, inhomogeneity, dispersion equation, phase velocity.

Mathematics Subject Classiﬁcation : 83C10; 74J15

13 Stabilization and chaos control in multispecies system Anuraj Singh ABV-Indian Institute of Information Technology and Management Gwalior, M.P., India. [email protected]

In this work different mechanisms are used to stabilize and chaos control in the models. To appreciate the challenge of chaos control, there are two categories in control algorithm: feedback and non-feedback. By approximate linearization approach, a feedback control law is obtained which stabilizes the closed loop system. On the other hand, by suitable change of coordinates in the state space, a feedback control law is obtained. This feedback control renders the complex nonlinear system to be linear controllable system such that the positive equilibrium point of the closed-loop system is globally asymptotically stable. Another approach is for chaotic system in which the control is applied to the system so as the controlled system admits a stable attractor which may be an equilibrium point or a limit cycle. The bounded feedback is used to achieve the stabilization of unstable ﬁxed point of the uncontrolled chaotic system. Delayed feedback control is used to control the chaos to periodic orbits. Numerical results substantiate the analytical ﬁndings.

A fourth-order orthogonal spline collocation method to fourth-order boundary value problems Anil Department of Mathematics, Birla Institute of Technology and Science, Goa. [email protected]

In this article, we study the orthogonal spline collocation methods (OSCM) for fourth-order linear and nonlinear boundary value problems. Cubic monomial basis functions and piecewise Hermite cubic basis functions are used to approximate the solution u for both linear and nonlinear boundary value problems, respectively. Earlier, several numerical experiments were performed to fourth-order linear and nonlinear boundary value problems using ﬁnite difference methods, ﬁnite element methods, B-spline technique, Sinc-Galerkin method and decomposition method. All the methods lead to the order of convergence only optimal or sub-optimal at the nodal points with more computational cost. In this paper, we use orthogonal cubic spline collocation methods and obtained optimal order of convergence for Also, super-convergent result is achieved for norm. We use MATLAB version of Almost block diagonal (ABD) solver to mini- mize the computational cost. It requires only order of n operations. We discuss dynamics of the stationary SwiftHohenberg equation for different values of α.

Keywords: Orthogonal cubic spline collocation methods (OCSCM), Fourth-order linear and nonlinear boundary value problem, Cubic monomial basis functions, Piecewise Hermite cubic basis functions and Almost block diagonal (ABD) matrix, MATLAB version of ABD solver.

14 Asymptotic Analysis of Steady Stokes Equations in an Oscillating Domain Bidhan Chandra Sardar Dr. Bidhan Chandra Sardar IIT Ropar. [email protected]

In this talk, we consider the steady Stokes system in a n-dimensional domain with a rapidly oscillating (n -1) dimensional boundary prescribed with Neumann boundary condition and periodicity along the lateral sides is considered. Our aim is to study the limiting analysis (as → 0) of the steady Stokes problem and identify the limit problem in a ﬁxed domain. Finally, show the weak convergences of velocities are improved to strong convergence by introducing corrector terms.

Keywords: Differential equations, Nano uids, Natural convection, Rayleigh number, Metallic and non- metallic nanoparticles.

Periodic solutions of vector disease model with harvesting term Shilpee Srivastava Department of Mathematics Birla Institute of Technology and Sciences Pilani, Dubai campus Dubai, 345055, UAE. [email protected]

System of delay differential equations plays a vital role in the ﬁeld of science and are now widely used for prediction and analysis in various areas of life sciences for example immunology, population dynamics, physiology, neural networks and epidemiology and atmospheric sciences, e.g. NPZ models. Time delays occur in these models due to duration of certain hidden process like stages of a life cycle, duration of infectious period, the immune period and so on. Introduction of time delays in models increase the complexity of these models and therefore it is essential to study the qualitative behavior of such models using stability or bifurcation analysis. To understand some of the outcomes here the qualitative behaviour of vector disease model with constant delay has been analysed. In this paper, we have studied the existence of positive peiodic solution of vector disease model proposed by Kenneth L. Cooke with effect of harvesting term. y0(t) = −c(t)y(t) + b(t)y(t − τ(t))(1 − y(t)) − I(t, y(t)) An example has been given to illustrate the outcome. Also by simulation, assuming parameters a function of time we have shown how delay term affect the periodicity of the solution.

Keywords: Periodic Solution, Positive Solution, vector disease model, harvesting term.

Mathematics Subject Classiﬁcation : 34K13, 34K15, 39A10, 39A12.

15 Mixed Virtual Element Methods for fourth order nonlinear parabolic problems P. Dhanumjaya, K. Balaje Department of Mathematics, BITS-Pilani K K Birla Goa Campus, Zuarinagar, GOA-403 726, India. [email protected]

Virtual element method (VEM) is a recent numerical technique which is a generalisation of the finite element method on polygons. The difference between the various finite element methods and the virtual element method is that it admits non-polynomial basis functions which are not required to be computed in practice. Instead, the degrees of freedom are chosen so that the stiffness matrix is computed exactly without explicitly knowing the basis functions. This enables us to work with more general meshes that the finite element method cannot handle in general. In this paper, we discuss mixed virtual element method (VEM) for a class of fourth order nonlinear parabolic problems. We prove some theoretical results including a priori bounds, optimal error estimates for the semi-discrete and completely discrete schemes. Finally, we perform some numerical experiments to validate the theoretical results.

Keywords: Fourth order nonlinear parabolic problems, mixed virtual element method, a priori bounds, semidiscrete Galerkin method, completely discrete Galerkin method, optimal error estimates and numerical experiments.

Generation of Uniform Magnetic Flux using different Coil and Plate Arrangement Md Tarikul Islam, Md Ataur Rahman Khan, Md. Moniruzzaman Bhuyan, Mohammad Anwar Hossain, Md. Mehedi Hasan Bhuiyan. Department of Electrical and Electronic Engineering, Southern University Bangladesh, Chittagong, Bangladesh [email protected]

In this paper we have studied the generation of uniform magnetic flux Coil-Plate arrangement. Here we have shown by simulation that if we use cupper plate and cupper coil, the magnetic flux increases with the increase of coil number. But at one point the magnetic flux start to decrease with the increase of coil number because the magnetic flux reaches at its saturation point. We observe that if we gradually increase the coil number up to six coil the magnetic flux reaches its peak value. If we further increase the coil number above six the magnetic flux decreases. On the other hand we also observe the phenomena for gold plate and coil but the magnetic flux is comparatively low. Our proposed coil-plate arrangement may have potential application in Magnetic Resonance Imaging (MRI).

Keywords: Coil; Conductive Plate; Simulation; Magnetic Flux.

Optimization methods and algorithms for non-parallel support vector machines M. Tanveer Department of Mathematics, IIT Indore [email protected]

In this talk, I will discuss twin support vector machines (TWSVM), a binary SVM classiﬁer that determines two nonparallel planes by solving two related SVM-type problems, each of which is smaller than in a conventional SVM. The twin SVM formulation is in the spirit of proximal SVMs via generalized eigenval- ues. Further, few recent variants of TWSVM will be discussed to show their performances and applications to real-world problems.

16 Elliptic problems with critical growth sign changing nonlinearities. Sarika Bennett University, Greater Noida [email protected],

The talk is concerned about the existence and multiplicity results of elliptic equations involving polynomial/exponential type nonlinearities with sign changing weight functions. The bering map analysis and Nehari method play an important role to obtain the existence and multiplicity of weak solutions for such type of elliptic equations. I will explain this analysis and also would like to discuss the extension of this technique to non-local elliptic equations involving p-fractional Laplace operator.

Modeling and Simulation of dc-dc Converter Based Power Electronics Learning : a review Indresh Yadav

Solution of two-parameter singularly perturbed one dimensional parabolic equations using non-polynomial spline Shahna, Talat Sultana and Arshad Khan Department of Mathematics, Jamia Millia Islamia, New Delhi-25, India. Department of Mathematics, Lakshmibai College, University of Delhi, New Delhi-52, India. [email protected]

In this paper, a class of second order singularly perturbed interior layer problems is examined. Non- polynomial mixed spline is used in this paper to develop the tridiagonal scheme. Error analysis is also carried out. The method is shown to converge point-wise to the true solution with second as well as fourth order. Linear and nonlinear second order singularly perturbed boundary value problems have been solved by the presented method. Numerical illustrations are given to demonstrate the efﬁciency of proposed method..

Keywords: Heat engines, Optimization, Non-equilibrium thermodynamics.

Mathematics Subject Classiﬁcation : 80A05, 82-08.

17 Creeping ﬂow of a sphere in non-concentric spherical container using slip condition M. Krishna Prasad Department of Mathematics, National Institute of Technology, Raipur-492010, Chhatisgarh, India. [email protected]

Creeping axisymmetric ow caused by a sphere in viscous uid within non-concentric spherical container is investigated using slip condition on the surface of the sphere. The boundary conditions on the spherical container are Cunningham’s (Mehta-Morse) and Kvashnin’s models. Previously Faltas and Saad ( Math. Meth. Appl. Sci. 2011, 34, 1594-1605) considered the same problem using Happel and Kuwabara models. In the limit of small Reynolds number, a general solution is constructed from the superposition of the solutions in the two spherical coordinate systems based on the inner sphere and spherical container. Numerical results for the hydrodynamic drag force exerted on the inner sphere are obtained with good convergence for various values of the relative distance between the centers of the sphere and spherical container, slip coefﬁcient of the sphere, and the volume fraction. The hydro-dynamic drag results are in good agreement with the existing solutions in the literature.

Keywords: Creeping ﬂow; slip sphere; hydrodynamic drag force; spherical container.

Mathematics Subject Classiﬁcation : 76A05; 76D07; 76S05.

Two-warehouse inventory model with quantity discount policy Himanshu Rathore Manipal University Jaipur, Jaipur, Rajasthan, India [email protected]

Keywords: advertisement frequency, preservation, quantity discount, two-warehouse.

Mathematics Subject Classiﬁcation : 49-XX.

18 A Comparative Study of Natural Nanouid Convection for Different Initial and Boundary Conditions Jyoti Sharma U.I.E.T., Panjab University, Chandigarh-160014, India. [email protected]

In present study a two-warehouse inventory model is established for deteriorating items. Controllable deterioration rate under the effect of preservation technologies. The market environment is such that the demand is received according to linear function of advertising frequency and selling price. Quantity discount policy is presented to keep bond of customer solid. The selling price, total cycle length and preservation cost parameters are taken as decision variables. The study is numerically veriﬁed with suitable graphical representation.The paper presents partial differential equations based on conservation laws of mass, momentum and energy which are analyzed to study the onset of convection in a nanouid layer for different initial and boundary conditions. The equations are solved to get steady state solution and perturbations are added to the original variables. The obtained partial differential equations are converted into ODE by seek- ing solution which is exponentially varying in time and space. Further these ordinary differential equations are solved to get expression for thermal Rayleigh number which is studied ana lytically and numerically. A detailed comparative analysis of the problem is undertaken for various initial and boundary conditions which leads to different equations and hence affect the convective nature of the system broadly. The impact of various nanouid parameters on the instability of the system are explored using metallic and non-metallic nanoparticles.

Keywords: Differential equations, Nano uids, Natural convection, Rayleigh number, Metallic and non- metallic nanoparticles.

Application of ﬁnite fractional Hankel-type transformation in Dirichlets problem V. R. Lakshmi Gorty SVKMs NMIMS University, MPSTME [email protected]

In this paper, finite fractional Hankel-type transformation has been defined depending on three real parameters. Operational calculus properties are shown in an application problem at the end of the paper. The classical concept on a more generalized version of finite fractional Hankel transforms is studied and applied to Dirichlets problem at the end of the study showing application in engineering field.

19 Existence Results for Fractional Impulsive Delay Differential Equations Renu Chaudhary GD Goenka University, Gurugram, India. [email protected]

In this paper, we apply the monotone iterative technique coupled with the method of lower and upper solutions to obtain the existence of extremal mild solutions for delay fractional integrodifferential equations with non-instantaneous impulses in an ordered Banach X  cDνy(t) = Ay(t) + F (t, y , R t a(t, s, y )ds), t ∈ ∪m (k , t ], m ∈ ,  t 0 s n=0 n n+1 N m y(t) = Gn(t, yt), t ∈ ∪n=1(tn, kn], (0.0.3)  y(t) = θ(t), t ∈ (−∞, 0], where cDν is the Caputo fractional derivative of order ν, 0 < ν < 1. A is a closed densely deﬁned linear operator which generates a strongly continuous semigroup {S(t)}t>0 of bounded linear operators on Banach space X. 0 = t0 = k0 < t1 < k1 < t2 < ··· < tm < km < tm+1 = b are impulsive points. The function yt :(−∞, 0] → X express the time history of the function y from −∞ to the present time t and determined by yt(φ) = y(t + φ) for φ ∈ (−∞, 0] which belongs to some abstract phase space B. The nonlinear functions a : ∆ × B → X, F : [0, b] × B × X → X and Gn :(tn, kn] × B → X with ∆ = {(t, s) : 0 6 s 6 t 6 b}, satisﬁes certain assumptions to be mentioned later. The impulses starts suddenly at the points tn and their effect remains on the interval [tn, kn]. More precisely, at the points tn, the function y experience an abrupt impulse and follow different rules in the two subintervals (tn, kn] and [kn, tn+1]. The function y is continuous at the points kn, n = 0, 1, 2, . . . , m, m ∈ N. At last, an application is discussed to show the applicability of the obtained results.

Keywords: Non-instantaneous impulses, Monotone iterative technique, Measure of noncompactness, Semigroup theory.

Mathematics Subject Classiﬁcation : 34A08, 34G20, 34K30, 34K45, 47D06.

OPTIMAL CONTROL IN STOCHASTIC LANDAU-LIPSCHITZ-GILBERT EQUATION Ananta K. Majee IIT Delhi [email protected]

In this presentation, we study an optimal control problem for the stochastic LandauLipschitz-Gilbert equation on a bounded domain in Rd (d = 1; 2; 3). We first establish existence of a relaxed optimal control for relaxed version of the problem. As the control acts linearly in the equation, we then establish existence of an optimal control for the underlying problem. Moreover, convergence of a structure preserving finite element approximation for d = 1 and physically relevant computational studies will be discussed. Further- more, we study an optimal control problem of N interacting ferromagnetic particles which are immersed into a heat bath through the application of a distributed exterior field minimizing a quadratic cost functional. With the dynamic programming principle, we show the existence of a unique strong solution of the optimal control problem. With a Hopf-Cole transformation the related nonlinear Hamilton-Jacobi-Bellman equation is recast into a linear PDE on the manifold M = (S2)N, whose classical solution is represented with a Feynman-Kac formula. We propose to use this probabilistic representation to numerically study optimal switching dynamics with Monte Carlo simulations.

20 Survey of Nature Inspired Optimization Algorithms in Fuzzy Control Systems Gaurav Saxena, Gomathi Bhavani, Shilpee S. Saxena Birla Institute of Technology and Sciences Pilani, Dubai campus Dubai, 345055, UAE. [email protected]

Several Non-deterministic Polynomial-time (NP) hard problems are nowadays solved with metaheuristic, nature inspired optimization algorithms. However, the problem of parameter adaptation continues to plague these methods. To remedy this, many approaches using Fuzzy logic have been suggested. Fuzzy logic controllers play a signiﬁcant role in control system design and their major objective is to tackle uncertainty and imprecision which could be data or process driven. For exact modelling of process parameters and control, it is vital that the fuzzy controller parameters have to be chosen and tuned in a way that they capture the process dynamics quite accurately. In this paper, we endeavor to present advances in nature inspired computing with reference to their contribution to control systems application. Various nature inspired techniques have been explored and surveyed with applications to optimization of parameters in control systems. The scope of fuzzy logic systems in solving the parameter optimization problem and thereupon potentially improving the performance of some nature inspired optimization methods also has been presented. We also review many nature inspired metaheuristic algorithms for optimization of fuzzy control parameters.

Keywords: Ant Colony Optimization Algorithms, Fuzzy logic controller, Meta- heuristics, Nature In- spired Optimization Algorithms, Particle Swarm Optimization

A study of approximate controllability for abstract nonlocal neutral integro-differential equations with ﬁnite delay Kamaljeet

Complex plankton dynamics induced by adaptation and defense Nilesh Kumar Thakur and Archana Ojha Department of Mathematics, National Institute of Technology Raipur, India. Emails: [email protected], [email protected]

In this paper, we have investigated a model for phytoplanktonzooplankton interaction and incorporated the adaptation (dormancy of the predators such as resting eggs) in zooplankton. The dormant stages are usually better equipment to withstand harsh environmental conditions than active ones. It has also been observed that the phytoplankton produces toxin in order to provide a self-defense from predators. We have studied how (i) adaptations allow an organism to be successful in a particular harsh environment, (ii) toxin spread surrounding the water surface provide defense. Analytically we have studied the local stability of the model system. To understand the effect of adaptation and defense on plankton dynamics we have plotted the time series and spatiotemporal pattern. Our numerical investigation reveal that the adaptation can suppress the ﬂuctuation in population densities, and system shows a transient complex spatiotemporal pattern which is either a mixture of spatially periodic steady states or travelling/standing waves by increasing the time and space.

Keywords: Adaptation dormancy plankton spatiotemporal pattern.

AMS subject classiﬁcations : 92D25, 34K20, 37M05

21 CONVERGENCE ANALYSIS OF TIKHONOV REGULARIZATION FOR NON-LINEAR STATISTICAL INVERSE LEARNING PROBLEMS ABHISHAKE, GILLES BLANCHARD AND PETER MATHE Institute of Mathematics, University of Potsdam, Karl-Liebknecht-Strasse 24-25, 14476 Potsdam, Germany [email protected]

We study a non-linear statistical inverse learning problem, where we observe the noisy image of a quantity through a non-linear operator at some random design points. We consider the widely used Tikhonov regularization (or method of regularization, MOR) approach to reconstruct the estimator of the quantity for the non-linear ill-posed inverse problem. The estimator is defined as the minimizer of a Tikhonov functional, which is the sum of a data misfit term and a quadratic penalty term. We develop a theoretical analysis for the minimizer of the Tikhonov regularization scheme using the ansatz of reproducing kernel Hilbert spaces. We discuss optimal rates of convergence for the proposed scheme, uniformly over classes of admissible solutions, defined through appropriate source conditions.

Keywords:Statistical inverse problem; Tikhonov regularization; Reproducing kernel Hilbert space; General source condition; Minimax convergence rates. Mathematics Subject Classiﬁcation : 65J20; Secondary: 62G08, 62G20, 65J15, 65J22.

CONTROLLABILITY OF A CLASS OF FRACTIONAL IMPULSIVE DIFFERENTIAL EQUATIONS IN A BANACH SPACE Abdur Raheem AMU, Aligarh [email protected]

In the present paper, we considered the following class of controlled fractional impulsive differential equations in a Banach space X:

α cD (t) = Ax(t) + Bu(t) + f(t, x(t), x(ν(t))), t ∈ [0,T0], t 6= ti, i = 1, 2, ...q ˆ ∆x(ti) = Ii(x(ti)), i = 1, 2, ..., q 0 ˆ ∆x (ti) = Ji(x(ti)), i = 1, 2, ..., q x(t) = h(t), x0(t) = g(t) t ∈ [−τ, 0]

2 where the state variable takes values in Banach space X and the control function u· is given in L ([0; T0]: α U), the Banach space of admissible control functions with U a Banach space, cD denotes the fractional derivative of Caputo of order α ∈ (12], −A is the infinitesimal generator of a strongly continuous α- 2 I J order cosine family {Cα(t)}t≥0 on a Banach space X. The maps f : [0,T0] × X → X and maps i ;i defined on X satisfy some suitable conditions, and the function v : [0; T0] → [0; T0] is continuous such that 0 ≤ vi(t) ≤ t; ti → [0,T0] for all i = 1, 2, .., q such that t1 < t2 < ... < tq; and m, q → N,T0 > 1 0.h, g → C ([−τ; 0]; X) i.e. h, g are continuously differentiable on [−τ; 0]. Let I = [0,T0]. In this paper an associated integral equation is obtained by using fractional integral and the family of cosine or sine of linear operators and then by using measure of non compactness and Monch’s fixed point theorem, we prove that considered problem is controllable on [0; T0]. We included an example to illustrate the abstract results. Keywords:Controllability; Fractional differential equations; Impulsive conditions; Monch FIxed point theorem.

Mathematics Subject Classiﬁcation : 93B05, 34A08, 35R12, 47H08.

22 APPROXIMATE CONTROLLABILITY OF SECOND ORDER SEMILINEAR CONTROL SYSTEM WITH NONLOCAL CONDITIONS Urvashi Arora and N. Sukavanam Department of Mathematics Bennett University, Greater Noida, India. Email: [email protected])

This paper is concerned with the approximate controllability of second order semilinear control system with nonlocal conditions in Hilbert spaces. Sufﬁcient conditions for the approximate controllability are established by assuming the approximate controllability of the corresponding linear system. The results are obtained when the nonlinear term satisﬁes monotone condition which is a weaker condition than the Lip- schitz continuity. Some approaches made by earlier authors led to certain inequality conditions involving various system constants. But, in the proposed approach, there is no need of any inequality condition to prove the approximate controllability results. Finally, an example is provided to illustrate the application of the obtained results.

Keywords:Approximate controllability, Semilinear systems, Monotone condition, Nonlocal Condi- tions.

Mathematics Subject Classiﬁcation : 34K30; 34K35; 93C25.

BOUNDEDNESS AND COMPACTNESS OF WEIGHTED DIFFERENTIATION COMPOSITION OPERATORS BETWEEN SOME WEIGHTED SPACES Zaherr Abbas Department of Mathematical Sciences, Baba Ghulam Shah Badshah University Rajouri [email protected]

The paper shall begin with an introduction to function spaces, composition operators, weighted composition operators and product of composition operators with differentiation. Afterwards, some classical operators will be shown to be composition (or weighted) operators, thus making the class of composition operators to be rich. Then we shall discuss the main aims of studying composition operators. Finally, boundedness and compactness of weighted differentiation composition operators between some weighted spaces of analytic functions will be discussed.

Optimization criteria for performance of heat engines working under different constraints Renuka Rai and Ramandeep Singh Johal Department of Applied Sciences, University Institute of Engineering and Technology, Panjab University, Chandigarh-160014, India. Department of Physical Sciences, IISERMohali, Mohali, India. [email protected]

Although Carnot efficiency gives an upper bound for thermal efficiency of heat engines, it has limitations in understanding the performance of real heat engines. In practice, heat engines are irreversible. Here the thermodynamic processes take place at finite rate, infinite time and may have nite sizes of source and sink. In this work, we compare the optimization criteria and the resulting bounds on efficiency of heat engines working under different constraints. The resulting optimization problem for maximum work extraction

23 in finite time involves solving the corresponding Euler-Lagrange equation for the control variables. The equation is simplified assuming tight-coupling conditions. The nite time output and efficiency are compared with the corresponding exergy analysis in the quasi-static case.

Keywords: Heat engines, Optimization, Non-equilibrium thermodynamics.

Mathematics Subject Classiﬁcation : 80A05, 82-08.

Wind turbine blade section optimization using a quantitative study M. Balachandar Sri Sai Ram Institute of Technology, Chennai.

Mathematical modeling in health science Jagdev Singh Department of Mathematics, JECRC University, Jaipur-303905, Rajasthan, India. Email: [email protected]

A mathematical model is a representation of a system by employing mathematical concepts, logic and language. The procss of developing a mathematical model is named as mathematical modelling. In this talk, we will discuss about the concept of mathematical modelling and about a giving up smoking model pertaining to a new fractional derivative non-singular kernel in nature. We will explore the numerical results which are obtained with the aid of iterative scheme. The talk will cover the existence and uniqueness of the solution by applying the ﬁxed- point theorem. Further, we will discuss about the numerical results of new smoking model associated with new fractional derivative and will compare with other derivatives.

Keywords: Mathematical modeling; Smoking model ; New fractional derivative ; Fixed-point theorem.

Fractional exothermic reactions models having constant heat source in porous media with different kind of memories Devendra Kumar Department of Mathematics, University of Rajasthan, Jaipur-302004, Rajasthan, India. Email: [email protected]

In this work, we study the exothermic reactions models having constant heat source in the porous media with different kind of memories. The patterns of heat flow profiles are very essential for heat transfer in every kind of the thermal insulation. In this work, we focus on the driving force problem due to the fact that temperature gradient is assumed. The mathematical equation of the mathematical model is confined in a fractional energy balance equation (FEBE), which furnishes the temperature portrayal in conduction state having uniform heat source on steady state. An iterative algorithm is used to derive the numerical solution of the FEBE. Some numerical results are given in the form of graphs and tables to see the effects of different parameters and variables on temperature profiles.

Keywords: Fractional exothermic reactions models; Porous media; Heat source; Fractional derivatives; Iterative method

24 Piezothermoelastic continuum subjected to point mechanical load Anita Devi Thakur Vallabh Govt. College Mandi, Himachal Pradesh, India. Email: [email protected] In this paper two-dimensional problem of piezothermoelasticity has been considered to investigate the disturbance in homogeneous, transversely isotropic (6mm class) generalized cylindrical piezothermoelastic continuum subjected to continuous mechanical load acting on thermally insulated and electrically shorted surface. The Laplace and Hankel transforms technique have been employed to express the boundary conditions in the transformed domain. The formal solutions are employed to obtain the system of simultaneous linear algebraic equations. These systems of equations are solved by using Gauss elimination process for the unknowns. These values of unknowns are used to ﬁnd the expressions of displacements, temperature change, electric potential, stresses and electric displacement in the transformed domain. The inverse transform integrals are evaluated by using numerical technique. Temperature, normal stress and shear stress so obtained in the physical domain, are computed numerically from the relevant expressions and relations code for PZT-5A material. Finally, the illustration of the results for classical and non-classical models of thermoelasticity has been presented graphically.

Keywords: Piezoelectric; Integral transforms; Relaxation time; Pyroelectric.

Dynamical Analysis of Michaelis-Menten Enzyme Reactions B S Lakshmi and S S Phulsagar Department of Mathematics and Statistics, Dr. Harisingh Gour Vishwavidyalaya, Sagar, M.P. [email protected] Enzymes are biological catalysts that alter the rates of reactions in cells without being changed them- selves during the course of a reaction. A biochemical reaction almost invariably has an output which is not necessarily linear. Such nonlinear phenomena involving enzymes have been explained by several people, amongst them Michaelis and Menten are worth mentioning. A simple form of Michaelis-Menten enzyme equation is [S] V0 = Vmax Km + [S] where V is the velocity of the reaction, V = V0 at t = 0, Vmax is the maximum velocity of the reaction, [S] is the concentration of the substrate S, Km is the Michaelis constant. Following the law of mass action, there are many mathematical models with a Michaelis Menten type output. With an output of this kind one could consider the following set of equations dx dy cy = a − bx − xpyq, = xpyq − dτ dτ y + 1 cy This equation when viewed as Predator-Prey equations, the term y+1 represent the type II functional response of Holling. Taking some particular values of the parameters a, b, p, q a detailed analysis of the system is taken up. The system is analysed by studying the associated differential equations, phase plane analysis and bifurcation analysis. Some of the oscillating chemical reactions viz. Belousov-Zhabotinsky reaction has a similar form of enzyme reactions.

Keywords:Enzyme reaction, Michaelis-Menten, system of differential equations, equilibrium points, phase plane, bifurcation analysis.

Mathematics Subject Classiﬁcation : 80A30, 92C45.

25 Truss Topology Optimization With Static And Dynamic Constraints Using AISC-ASD Ghanshyam G. Tejani School of Technology, GSFC University, Vadodara, Gujarat, India. [email protected]

In this study, simultaneous size and topology optimization of planar and space trusses subjected to static and dynamic constraints are investigated. All the benchmark trusses consider discrete cross-sectional areas from the American Institute of Steel Construction - Allowable Strength Design (AISC-ASD) to consider the practical aspect of manufacturing. Moreover, the trusses are seen with multiple loading conditions and subjected to constraints for natural frequencies, element stresses, nodal displacements, Euler buckling criteria, and kinematic stability conditions. Truss Topology Optimization (TTO) can be accomplished by the removal of super uous elements and nodes from the ground structure, and results in the saving of the mass of the truss. In this method, the difﬁculties arise due to singular solution and unnecessary analysis; therefore, FEA model is reformed to resolve these difﬁculties. The static and dynamic responses to the TTO problems are challenging due to its search space, which is implicit, non-convex, non-linear, and often leading to divergence. This study compares performance of four meta-heuristics such as TeachingLearning-Based Optimization (TLBO), Heat Transfer Search (HTS), Symbiotic Organisms Search (SOS), and Water wave optimization (WWO) for solving discrete TTO problems.

Keywords: Non-linear problem Structural optimization; Metaheuristic; Planar and space trusses; AISC- ASD; Discrete sections.

Mathematics Subject Classiﬁcation : 65-05:Experimental papers; 46N10: Applications in optimization; 90C30: Nonlinear programming.

Inuence of Temperature Jump and Concentration Slip on inclined MHD Bioconvection past a vertical porous plate in the presence of Nanoparticles and Gyrotactic Microorganism Rakesh Choudhary and Shalini Jain Bhartiya Skill Development University, Jaipur India. University of Rajasthan, Jaipur, Rajasthan, India [email protected]

In this paper, we examined the effects of temperature jump and concentration slip on inclined MHD bioconvection past a vertical porous plate through porous media in the presence of both nanoparticles and gyrotactic microorganism. The governing partial differential equations are reduced into ordinary differential equation with using suitable similarity transformation. A numerical scheme, called Runge-Kutta fourth fth order Fehlberg method (RKF45) has been used to solve above ordinary differential equations. The effects of pertinent parameter for variation in the velocity profile, velocity profile at far eld, temperature profile, concentration pro le and motile microorganism density pro le has been obtained. We have validate the result obtained from current study with existing results.

Keywords:Gyrotactic microorganism; Temperature jump; Concentration slip; Inclined MHD; Nanopar- ticles; RKF-45.

Mathematics Subject Classiﬁcation : 76-XX.

26 Numerical Solution of Linear and Higher order Delay Differential Equations using Coded Differential Transform Method Giriraj Methi And Anil Kumar Department of Mathematics and Statistics, Manipal University Jaipur-303003, Rajasthan, India

Aim of the paper is to obtain numerical solutions of linear and higher order delay differential equations (DDEs) using Coded Differential Transform Method (CDTM). We have numerically solved some initial value delay differential equations by coded differential transform method in Mathematica version 11 and compared the solutions with the exact solutions. We have illustrated few examples to show efﬁcacy of our method. We have obtained various terms of series solution by CDTM. Our solutions are approximately close to exact solutions. The CDTM may be one of the suitable method to solve DDEs.

Keywords:Delay Differential Equations, Coded Differential transform Method, Numerical Solution, Mathematica.

Mathematics Subject Classiﬁcation : 39B99; 65Q20; 65Q30

Studies of Hyperloop Vehicle for Transportation: A Review V. K. Srivastav; Aditya Priyanka; Abhishek Kumar; Shudhanshu Kumar; Anand Raj. Motihari College of Engineering, Motihari, Bihar [email protected]

This paper presents a critical review of hyperloop vehicle that is useful for next generation transportation. The Hyperloop concept is proposed as a quicker, cheaper alternative to high-speed rail. It is seen from the literature that computational simulation play an important role to simulate the air flow around the capsule considering the effect of the axial-flow air compressor and the levitation cushion on the air flow. In this paper, we provide all the boundary conditions in tabular form that was used in both computational as well as experimental papers. The present work will also compare different hyperloop models used by the researchers.

Keywords: Hyperloop, Computational Fluid Dynamics (CFD), Tube transport system, Tube vehicle

Effect of active case nding on dengue control: Implications from a mathematical model Pankaj Kumar Tiwari Department of Mathematics, University of Kalyani, Kalyani [email protected]

Dengue control in India is a challenging task due to complex healthcare settings. In yesteryears, an am- plication of dengue infections in India posed the need for introspection of existing dengue control policies. Prior understanding of the impacts of control interventions is necessary for their future implementation. In this paper, we propose and analyze a compartmental model of dengue to assess the impact of active case nding (ACF) on dengue disease transmission. Currently, primary prevention of dengue is possible only with vector control and personal protection from the bites of infected mosquitoes. Although a few experimental studies are performed to assess ACF in dengue disease, but this is the rst attempt to represent and study the dynamics of disease using ACF as a control strategy. Local and global dynamics of the system are

27 studied. We use sensitivity analysis to see the effects of controllable parameters of the model on the basic reproduction number and total number of infective population. We nd that decrease in the biting rate of mosquitoes, and increase in the rate of hospitalization and/or notication, death rate of mosquitoes and ACF for asymptomatic and symptomatic individuals play crucial role for the reduction of disease prevalence. We calibrate our model to the yearly dengue cases in eight dengue endemic states of India. The results of our study show that ACF of symptomatic individuals will have signicant effect on dengue case reduction but ACF of asymptomatic individuals cannot be ignored. Our ndings indicate that the healthcare organizations must focus on ACF of symptomatic as well as asymptomatic individuals along with personal protection and mosquitoes control to achieve rapid reduction of dengue cases in India.

Change in Stability Behavior of Spatiotemporal Phytoplankton Dynamics with Different Types of Functional Response Randhir Singh Baghel Department of Mathematics, Poornima University, Jaipur, Rajasthan,India [email protected]

In this paper, we focus on the comparison between two types of functional responses, namely, Holling type-II and ratio-dependent functional response interaction on the phytoplankton dynamics with susceptible and infected class of populations with diffusion. We observed that the system dynamical behavior affected by the response functions and the system is more stable in the case with ratio dependent functional response. Furthermore, we explore the higher-order stability analysis of the system for both linear and no-linear system with respect to both response functions.

Keywords: Diffusion driven instability, Pattern formation, Higher-order stability analysis.

Modelling of duct based photovoltaic thermal (PVT) air collector Rohit Tripathi1; G. N. Tiwari; Deepak Sharma chool of Electrical, Electronics and Communication Engineering, Galgotias University, G. Noida, U.P., India. Centre for Energy Studies, Indian Institute of Technology Delhi, Houz Khas, New Delhi, India [email protected]

In present study, photovoltaic thermal collector has been proposed with duct where the semitransparent silicon solar cell based photovoltaic module is used. The area of PV module is considered as 0.605 m2. This PV module generates 35 Peak Watts electrical power. Just below the PV module, one closed duct is placed and small portion on duct, back to PV module, and one DC fan is placed. This DC fan can be operation with 12 volts and 0.5 A. The comparative study has been analyzed for one convectional Flat Plate collector, one photovoltaic module and proposed duct based PVT collector with air. In this study, it is found that the proposed system is generating more electrical power to convectional photovoltaic module and more thermal energy from convectional photovoltaic thermal system, 20% and 36%, respectively. The limitation is that the cost of proposed system is slightly high to others but it is self-sustainable system and this thermal energy can be further used for heating purpose also, like: building heating, drying etc.

Keywords:PV Module, FPC, Energy, Exergy and Power.

Mathematics Subject Classiﬁcation :

28 Mathematical model of unstable self-limiting thermo chemical temperature oscillations in Australian Cycades Akash Bhavsar School of Technology, GSFC University, Vadodara, Gujarat, India. [email protected]

In this study, mathematical models have been developed to explain unstable, self-limiting thermo chemical temperature oscillations, which are observed in Australian cycads macrozamia lucida and M macleayi during the pollination period. These cones develop daily midday thermogenic temperature rises as high as 12 C above ambient temperature, during their approximately two week pollination period. The cone temperature response model is developed to accurately predict the cone temperatures over multiple days, based on simulations of experimental results from 28 thermogenic events from 3 different cones, each simulated for either 9 or 10 sequential days. Determined the optimum values of parameters that are responsible for unstable, self-limiting thermo chemical temperature oscillations. Determined the simple possible model that can explain the temperature responses obtained in the ﬁeld study and lab experiments, for complex systems like this more than one solution is possible to simulate the system.

Keywords: Mathematical modeling; Thermal system optimization; Inverse thermo-chemical system; Bio chemical system; System parameters identiﬁcation.

Mathematics Subject Classiﬁcation : 65-05:Experimental papers; 46N10:Applications in optimization; 90C05:linear programming

On the existence and uniqueness of solutions to discontinuous dynamic equation on time scales Sanket Tikare Department of Mathematics, Ramniranjan Jhunjhunwala College, Ghatkopar (W), Mumbai (MS), India - 400 086. [email protected]

In this paper, we study the existence and uniqueness of solutions to the dynamic equation x∆(t) = f t, x(t) , t ∈ T, where x : T → R, f : T × R → R and T = [a, b]T is a ﬁnite time scale interval with min T = a and max T = b. Here we do not assume any sort of continuity about f. The Arzela–Ascoli lemma and Banach’s ﬁxed point theorem are used to investigate the existence of solutions. We obtain the maximum interval for the existence of solution under certain growth conditions. We prove uniqueness of solution under certain conditions.

Keywords: Existence and uniqueness; Dynamic equations; Time scales; Caratheodory function; Fixed point theorem.

Mathematics Subject Classiﬁcation :26E70; 34A36; 34N05.

29 Generalized energy inequality for weak solutions to Damped Navier-Stokes equations Rajib Haloi and Subha Pal Department of Mathematical Sciences, Tezpur University, Assam, India [email protected]

Existence of weak solutions to the damped NavierStokes equation with no slip boundary condition in a bounded domain in R3 are proved and generalized energy inequality for the solutions are discussed.

Keywords: Navier Stokes equation, Semi-discretization method, No slip boundary condition.

Mathematics Subject Classiﬁcation :76D05, 35Q30,76D03.

Dynamical study of a delayed SEIRS model with saturated incidence and impulsive vaccination: Effect of household wastes Kunwer Singh Mathur Department of Mathematics and Statistics, Dr. Harisingh Gour Vishwavidyalaya, Sagar, M.P. [email protected]

With the increasing of urban population, the accumulation of household waste and its disposal has become an increasingly arduous issue. The household wastes cause spreading several kinds of deadly diseases and hence aroused the attention from all sectors of society. In this paper, a delayed SEIRS model with a saturated incidence rate is proposed to study the dynamics of diseases spread by household wastes. To control the spread of diseases, vaccinations into susceptible population and disinfection of bacteria from environment are done at every fixed moment of time in an impulsive manner. It is obtained that the model has a disease-free unique positive T-periodic solution, which is globally asymptotically stable under certain conditions. Further, the comparison theorem of impulsive differential equations is used to prove the permanence of the system. Finally, the numerical simulation confirms and validates these theoretical findings. This study provides useful information for decision-makers to select appropriate choices in controlling of diseases spread by household waste and save cost.

Keywords: Household waste, impulse, delay, SEIRS model, global stability, permanence.

Mathematics Subject Classiﬁcation : 92Bxx, 92B05.

30 Thermal Convection in Oldroydian Nanouid Layer Saturating a Porous medium with Rigid-free and Rigid-rigid Boundaries Abhilasha Department of Mathematics and Statistics, Himachal Pradesh University, Shimla-171005 INDIA. [email protected]

The present paper investigates the effect of rheology on the onset of convection in top- heavy nanofluid heated from below saturating an isotropic and homogeneous porous medium. The rheology of the nanofluid is described by Oldroyd model (a relation between shear thining and stress thickening). The employed model incorporates the effects of Brownian motion and thermophoresis due to the presence of nanoparticles. The governing nonlinear partial differential equations of the dynamical system are reduced to non- dimensional ordinary linear differential equations by using the linear theory, normal mode technique and the non-dimensional variables. The mathematical analysis is concerned with the solutions of the characteristic value problem for two cases: i) rigid-free and ii) rigid-rigid boundaries, respectively. The deciding stability parameter, the non-dimensional thermal Rayleigh number for each of these two cases is derived using trial functions satisfying the appropriate boundary conditions in each of these cases. The variation of thermal Rayleigh number with respect to wave number for certain fixed permissible values of other parameters are computed numerically using the software MATHEMATICA-5.2. It is found that the stationary convection is independent of both Pramdlt number and particle density Rayleigh number. The stabilizing effects of medium porosity, strain retardation and the destabilizing effects of stress relaxation, Lewis number, concentration Rayleigh number, modified diffusivity ratio are also shown graphically. These results are compared favorably with those obtained earlier.

Keywords: Nanoﬂuid, Oldroyd model, Thermal Convection, Porous Medium, Thermal Rayleigh number.

Mathematics Subject Classiﬁcation : 76Exx, 76Sxx.

Inclined MHD Williamson Fluid Flow with Slip Boundary and Heat and Mass Transfer due to Porous and Melting Stretching Surface with Non-Linear Radiation and Heat Source Amit Parmar Poornima College of Engineering Rajasthan, India [email protected]

In this article, we have investigated a problem for momentum, thermal energy and mass transfer behavior of inclined Williamson fluid due to melting stretching surface in the presence of non-linear thermal radiation and heat source with suction and slip boundary condition. Williamson is a real fluid. A real fluid has both minimum viscosity and maximum viscosity depending upon the molecular shape of the fluid. By using suitable transformation, the governing equations are converted into non-linear coupled ODEs and solved by using RungeKutta based shooting technique with MATLAB. The effect of various parameters on velocity and temperature profiles are discussed and display graphically. Local Nusselt number and skin friction coefficient are tabulated.

31 Sandip Rakshit

An overview to investigate role of rheology on thermal convection in ferroﬂuids Veena Sharma Department of Mathematics and Statistics, Himachal Pradesh University, Shimla-171 005, India. Email: [email protected]

A fascinating smart fluid magnetorheological fluid (MRF) has the potential in designing hydraulic systems (active shock and vibration dampers, actuators, values, automobile suspensions, seats), very fast, powerful computers of less cost as well as convenient softwares and are being used in material research, and bio-medications etc. The various rheological models of ferrofluids are synthesized and prepared by the researchers in the present era so as to examine to control the stability and the form of convective motions (whether stationary cellular motion or oscillatory motion) both experimentally and theoretically, in the considered physical system. Therefore, there is compelling need to develop new and improved MRF to lower their production cost through improved manufacturing processes and develop MR fluid based application devices that will demonstrate the engineering feasibility of the MR fluid concept and will highlight the implementation challenges in near future. An important application of MRF also lies in biomedicine area where the carrier liquid is blood which is known to have special rheological properties. In the present talk we will discuss the survey of the gradual developments from the research and applications point of view. A special thrust in the discussion will be given to the ultra precision polishing of ceramics using MRF and diamond abrasives which have promising potential in aerospace applications (silicon nitride bearings are used in main engines of NASAs space shuttle and thruster in Rocket engine). In addition to this, we will also examine how the rheological behavior of ferrofluids affects the stability of convective motions in physical systems using Galerkin finite element of weighted residues and truncated Fourier series method.

Sheetal Dharmatti Data assimilation type Optimal control problem for Cahn Hilliard Navier Stokes’ system.

Department of Mathematics, Indian Institute of Science Education and Research Thiruvananthapuram, In- dia. Email: [email protected] This work is concerned about some optimal control problems associated to the evolution of two isothermal, incompressible, immiscible fluids in a two-dimensional bounded domain. The Cahn-Hilliard-Navier- Stokes model consists of a NavierStokes equation governing the fluid velocity field coupled with a convective CahnHilliard equation for the relative concentration of one of the fluids. A distributed optimal control problem is formulated as the minimization of a cost functional subject to the controlled nonlocal Cahn-Hilliard-Navier-Stokes equations. We establish the first-order necessary conditions of optimality by proving the Pontryagin maximum principle for optimal control of such system via the seminal Ekeland variational principle. The optimal control is characterized using the adjoint variable. We also study another control problem which is similar to that of data assimilation problems in meteorology of obtaining unknown initial data using optimal control techniques when the underlying system is same as above. Keywords: AMS subject classifications :

32 Paper Presentation

Numerical Solution of Lane-Emden type Equations Using Multilayer Perceptron Neural Network Method Akanksha Verma, Manoj Kumar Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Prayagraj-211004, (U.P.) India [email protected]

In this paper, we discuss the Multilayer perceptron artiﬁcial neural network method for the solution of second order non-linear singular differential equations of Lane-Emden type. Our aim is to produced optimal solution of Lane-Emden equations with less computation using multilayer perceptron artiﬁcial neural network method in comparison to other existing methods. Several test examples have been considered to illustrate the proposed method. The results obtained to prove that this method has capability to become an effective approach for solving Lane-Emden problems with less computing time and memory space.

Key words: Multilayer perceptron Artiﬁcial Neural Network, Singular Initial Value Problem, Lane- Emden equation, Error back propagation, Quasi Newton method.

AMS Subject Classiﬁcation: 65N20

Computational Simulation for Time-Fractional Diffusion Equation with Neumann Boundary Conditions A.S.V. Ravi Kanth, Neetu Garg Department of Mathematics National Institute of Technology Kurukshetra, Kurukshetra-136119, India Email: [email protected], [email protected]

In this paper, computational simulation for a class of time-fractional diffusion equation with Neumann boundary conditions is presented. We construct a numerical scheme by combining L1 approximation for Caputo time-fractional derivative and exponential B-spline approximation for spatial derivatives. The proposed scheme is unconditionally stable. The numerical examples are presented to conﬁrm the efﬁciency and applicability of the proposed scheme.

Keywords Time-fractional diffusion equation, Caputo derivative, Exponential B-spline method, Stabil- ity. AMS subject classiﬁcation: 35R11. 65M12.

33 On Solution of Fractional Order Advection-Diffusion Equation in Porous Media Prashant Pandey Department of Mathematical Sciences Indian Institute of Technology (BHU), Varanasi, 221005, India address: [email protected]

In the present article, an operational matrix method with Laguerre polynomials is applied to solve a space-time fractional order non-linear Cahn-Hilliard equation, which is used to calculate chemical potential and free energy for a non-homogeneous mixture. Constructing operational matrix for fractional differentiation, the collocation method is applied to convert Cahn-Hilliard equation into an algebraic system of equations, which have been solved using Newton method. The salient features of the article is to ﬁnding the stability analysis of the proposed method and the pictorial presentations of numerical solution of the concerned equation for different particular cases and showcasing of the effect of advection and reaction terms on the nature of solute concentration of the considered mathematical model for different particular cases..

Keywords: Fractional Calculus, Convergence Analysis, Sub-diffusion, Porous Media, Laguerre Poly- nomials

AMS subject classiﬁcations :35Q35, 35B35, 35B30

An exact l1 penalty function method for multi-dimensional ﬁrst-order PDE constrained control optimization problem Preeti Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad-826004, Jharkhand, India. [email protected]

In this paper, we use the exact l1 penalty function method to solve a multi-dimensional ﬁrst-order PDE constrained control optimization problem. The relationships between an optimal solution to the aforesaid problem and its associated penalized problem with the exact l1 penalty function are established. Further, we show that an optimal solution to the considered problem is a minimizer of its associated penalized problem under the hypothesis of convex Lagrange functional. In addition, the theoretical results are justiﬁed with some examples.

Keywords: Exact l1 penalty function method Multi-dimensional control optimization problem Neces- sary optimality conditions.

AMS subject classiﬁcations : 26A51, 49J15.

34 Numerical solution of Lane-Emden type equations using Adomian decomposition method with unequal step-size partitions UMESH, Manoj Kumar Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Prayagraj-211004 (U.P) India. [email protected], [email protected]

In this paper, Modiﬁed Adomian decomposition method (MADM) for solving Lane-Emden type equations with unequally space partitions is presented. To prove the robustness and effectiveness of the proposed method various examples are considered. This method overcomes the singular behavior of the problems and exhibits the approximations of high accuracy with a large effective region of convergence.

Keywords: Modiﬁed Adomian decomposition method, Adomian polynomials, Singular ordinary differential equations

AMS subject classiﬁcations : 65N20

Solution of Riemann problem for non-ideal magnetogasdynamic ﬂow Pooja Gupta Department of Mathematical Sciences Indian Institute of Technology (B.H.U.) Varanasi 221005, India, E-mail ID: [email protected]

The present paper concerns with the analytical solution of the Riemann problem for magnetogasdynamic equations governing an inviscid unsteady one-dimensional flow of non-ideal polytropic gas subjected to the transverse magnetic field with infinite electrical conductivity. By using the Lax entropy condition and R-H conditions, we derive the elementary wave solutions i.e. shock wave, simple wave and contact discontinuities without any restriction on the magnitude of initial data states and discussed about their properties. Fur- ther, the density and velocity distribution in the flow field for the cases of compressive wave and rarefaction wave is discussed. Here we also compare/contrast the nature of solution in non-ideal magnetogasdynamic flow and ideal gas flow.

Keywords: Riemann problem; Non-ideal; Magnetogasdynamics; Shock wave; Simple wave

Wave interaction with a tunnel in a sea with bottom undulation MANISHA, Dr. RAMANABABU KALIGATLA Department of Applied Mathematics IIT (ISM) DHANBAD Dhanbad- 826004, Jharkhand, India, Emails: [email protected], [email protected]

A boundary value problem for linear wave interaction with a tunnel in a sea with bottom undulation is studied by the method of eigenfunction expansion and modified mild-slope equation. The tunnel is assumed to be rectangular shape and, trapezoidal and circular undulations are considered as breakwaters in the present study. The effect of bottom undulation on waves and thereby effect on tunnel is analyzed. Solu- tions are obtained by means of separation of variables and those are matched at the interface boundaries by the conservation of mass flux and continuity of pressure and velocity. The important scattering coefficients such as reflection and transmission coefficients are obtained by solving system of equations. Forces are on tunnel are calculated and depicted for wave and structural parameters. Moreover, a definite range of wave

35 incident angles is demonstrated for obtaining the critical angle which provides the least reﬂection and most force on the tunnel.

Keywords: Submerged tunnel, Submerged breakwater, Eigenfunction expansion, Modified mild-slope equation, Reflection coefficient.

AMS subject classiﬁcations : 34L10

Subgrid multiscale stabilized ﬁnite element analysis for various transport equations Manisha Chowdhury, B.V. Rathish Kumar Indian Institute of Technology Kanpur Kanpur, Uttar Pradesh, India. [email protected]

Finite element method (FEM) is a well-known numerical scheme for solving partial differential equations of boundary value problems in the areas of structural analysis, heat and mass transfer, fluid flow etc. Finding analytical solution of various types of transport equation such as advection-diffusion-reaction (ADR) equation with variable coefficients, coupled Stokes and ADR equations, coupled Navier Stokes and ADR equations are very difficult and challenging task. Due to wide range of applications of these transport problems in the fields of bio-medical engineering, chemical engineering, environmental sciences etc, analysis of numerical schemes,such as FEM for the approximate solution of these equations are attractive topics of research. In this talk a stabilized finite element method for solving transport equations will be presented. Here subgrid scale approach along with algebraic approximation to the subscales has been chosen to stabilize the Galerkin finite element method. Both a priori and a posteriori finite element error estimates in L2 norm will also be discussed. Numerical results will be presented to verify the theoretically established expressions.

Keywords : Transport equation Subgrid multiscale ﬁnite element method a priori error estimate a posteriori error estimate

Numerical solution for two dimensional space-time fractional reaction diffusion equation Sachin Kumar Department of Mathematical Sciences, Indian Institute of Technology(B.H.U) Varanasi 221005, India. E-mail address: [email protected]

In this article, an operational matrix method with Genocchi polynomials is applied to solve a two dimensional space-time fractional order nonlinear reaction-diffusion equation. An operational matrix for fractional order differentiation is derived. Applying collocation method and using the said matrix, fractional non-linear partial differential equation is reduced to a system of algebraic equations, which can be solved using Newton iteration method. The salient features of the article are the pictorial presentations of the numerical solution of the concerned equation for different particular cases to show the effect of reaction term on the solution proﬁle and also the change of its behavior when the system approaches from standard order to fractional order. The accuracy of our proposed method is validated through the error analysis of the obtained numerical results with the existing analytical results for two spatial fractional order nonlinear equation.

36 Keywords: Fractional Calculus, Diffusion equation, Sub-diffusion, Porous Media, Genocchi Polyno- mials.

AMS subject classiﬁcations : 35Q35, 35B35, 35B30

Mathematical Modeling of Surface wave transference in a piezo-composite media using WKB technique Sonal Nirwal, Sanjeev A. Sahu Department of Applied Mathematics, IIT (ISM) Dhanbad, Jharkhand, 826004, India, Email: [email protected]

This paper proposes an analytical solution (using asymptotic approximation) for the propagation of surface waves in a smart piezo-composite structure. The considered structure is a three-layered system in which the uppermost piezoelectric (PE) layer is accomplished by a functionally graded piezoelectric material (FGPM) layer and piezomagnetic (PM) half-space. The Wentzel-Kramers-Brillouin (WKB) asymptotic method is adopted to solve the differential equations. Inﬂuence of various affecting parameters on phase velocity curve of different modes has been shown graphically.

Keywords: Asymptotic approximation, FGM, Surface wave.

AMS subject classiﬁcations : 74J15, 74A40, 74D05, 74G10, 74H10.

Modelling of Aeration Efﬁciency At Gabion Weir KM. Luxmi, Nand Kumar Tiwari, Subodh Ranjan Vajesnayee National Institute of Technology Kurukshetra, Haryana-136119, India. [email protected], nand [email protected], [email protected]

Gabion weir consists of a porous medium filled with different shape and size of coarser materials. The turbulence generated by gabion weir will promote the aeration efficiency in the form of large number and bigger forms of bubbles.Aeration is defined as the process of oxygen transfer from atmosphere to water. From this process, the amount of oxygen in water is enhanced and measured as dissolved oxygen (DO) in terms of ppm. Dissolved oxygen is one of the best indicators of water quality. This paper attempts to investigate the prediction of aeration efficiency of gabion weir by using AI- based modelling techniques. The output values of aeration efficiency through gabion weir were computed using ANN, Linear regression, Gaussian process and randomforest by taking mean size, porosity, discharge, drop height as input parameters. Data is taken by conducting experiments in NIT laboratory. Comparing these modelling techniques, it was found that ANN has been giving better results than other considered techniques. The findings of this paper will help in choosing better gabion weir model and modelling technique for best result.

Keywords: Gabion Weir; Aeration Efﬁciency; Dissolved Oxygen (DO); Porosity; Gaussian Process; Random Forest; ANN; Linear Regression.

37 Modelling of scour around spur dykes Amit kumar, Subodh ranjan vajesnayee, and Nand kumar Tiwari M.Tech Civil Engineering, National Institute of Technology Kurukshetra, Haryana-136119, India. [email protected], [email protected], [email protected]

Spur dykes are typical man-made hydraulic structures and widely used in allu- vial rivers to defend against the effects of disaster and helps in river restoration. Spur dykes are generally built perpendicular or at an angle to the channel bank to protect it against scouring. Spur dykes impact the ow and bed load dynamics around itself and creates variation in ow pattern and bottom pro le according to the relationship between the ow characteristics and bed con gurations. This paper attempts to investigate the scouring around the spur dykes by increasing the roughness and changing the angle of spur dykes. By using the AI- based modelling technique, It has been predicted the scouring around the spur dykes by changing the roughness, and dimension of spur dykes. In this study, the output value of scour around spur dykes due to increasing its surface roughness were predicted using ANN and Gaussian Process by taking parameters like mean size(d50), velocity, angle, and contraction ratio of spur dykes.

Keywords: ANN, Gaussian process, scouring, spur dykes, roughness, D50, constriction ratio.

Modelling the delay dynamics of malware propagation Sangeeta Kumari Department of Applied Mathematics, Indian Institute of Technology(Indian School of Mines), Dhanbad, India. [email protected]

To understand the transmission dynamics of malware propagation in wireless sensor networks, an attempt has been made. An e-epidemic delay model with the non linear incidence rate and sigmoid type recovery rate has been proposed. Existence and stability analyses of the equilibria are performed. Attention has been paid to the occurrence of Hopf bifurcation. Numerical simulations is performed for validating the theoretical analysis with the help of MATLAB. The impact of the control parameters on the system dynamics is investigated. Most effective measures are suggested based on the obtained results to control the propagation of malicious entities.

Keywords: Delay differential equation, Wireless sensor network, Hopf bifurcation, Malware propagation, Stability Analysis.

AMS subject classiﬁcations : 37C75, 65P30, 37M10, 37M05.

38 Analysis of a density dependent model with discrete delays Anuraj Singh, Ankit Parwaliya and Ajay kumar ABV-Indian Institute of Information Technology and Management, Gwalior, M.P., India. [email protected], [email protected], [email protected]

A two dimensional model with ﬁnite number of discrete delay has been investigated.The Permanence,persistence,positivity,and boundedness of the delayed model has been determined under certain conditions.Phenomenon of Hopf bifurcation has been determined for different combination of delays. It is resulted that delay cause Hopf bifurcation and further complex dynamics in the system. By using central manifold theorem and normal theory the direction and stability of Hopf bifurcation is determined. The analytical ﬁndings have been validated by exhaustive numerical simulation which exhibits a wide range of complex dynamics in the system.

Keywords: Delay, Global Stability, Hopf bifucation, Permanence.

Approximate analytical solution for shock wave in rotational axisymmetric perfect gas: Isothermal ﬂow G. Nath, Sumeeta Singh Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Prayagraj 211004, Uttar Pradesh, India, Email: [email protected]

The propagation of cylindrical shock wave in rotational axisymmetric perfect gas under isothermal flow condition is investigated. Mathematical model for the considered problem using PDEs is presented. Dis- tribution of gasdynamical quantities are discussed. The density, azimuthal fluid velocity and radial fluid velocity are assumed to be varying according to power law with distance from the axis of symmetry in the undisturbed medium. Approximate analytical solutions are obtained by expanding flow variables in power series of (C/U)2, where C is the sound speed in undisturbed fluid and U is the propagation velocity of shock wave. Zeroth and first order approximations are discussed by the aid of power series method. Solutions for zeroth order approximation are constructed in approximate analytical form. The effect of flow parameters namely: adiabatic exponent and ambient density variation index q are studied on the flow variables. Key- words: Shock wave, Power series method, Isothermal flow, Perfect gas, Rotating medium, Modeling using PDEs

AMS subject classiﬁcations : 76L05, 76U05, 76M55.

MULTIBODY MODELLING OF A RAIL VEHICLE USING MR SUSPENSION SYSTEM Deepak Goyal, Sultan Singh, Anil Kumar Department of Mechanical and Industrial Engineering, IIT Roorkee, India. [email protected]; [email protected]; [email protected]

The improvement in critical speed of rail vehicles has been ever needed. This article presents a multibody dynamics approach for evaluating critical speed of a rail vehicle. 17 degrees of freedom are used for mathematical modeling of a LHB rail vehicle by considering lateral vehicle dynamics. The degrees of freedom represent lateral and yaw movements of all the 4 wheelsets, lateral, yaw and roll movements of bogies and car body. Eigen value approach is opted for predicting the stability behavior of themodel. In uence of design parameters on critical speed is investigated. A sensitivity analysis hasbeen performed to

39 assess the in uence of all the dampers on critical speed. The most suitable position for replacing the existing passive damper with magnetorheological (MR) damper has been identi ed. Bouc-wen model for MR damper is implemented in MATLAB/SIMULINK. MR model is coupled with multibody model of the rail vehicle, as a result the behavior of the rail vehicle equipped with MR damper is analysed. A comparative study for critical speed between MR damper and passive damper suspension is done.Improvement in critical speed of rail vehicle has been observed.

Keywords: LHB COACH , CRITICAL SPEED , SENSITIVITY ANALYSIS, VEHICLE STABLITY, MR DAMPER.

Size-dependent vibration of microplate resonators based on the modiﬁed couple stress theory and three-phase-lag heat conduction model Harendra Kumar and Santwana Mukhopadhyay Department of Mathematical Sciences, Indian Institute of Technology (B.H.U.), Varanasi-221005. harendra.rs.,[email protected]

Thermoelastic damping (TED) is a significant energy loss factor at room temperature in micro-scale resonators. The prediction of TED is important in the designing of high quality of micro-electromechanical system (MEMS) resonators. In the present work, an analytical expression for the quality factor (Q) of TED is presented by applying modified couple stress theory (MCST) considering plane stress condition and the three-phase-lagging (TPL) heat conduction model. As case study, the effect of the length-scale parameter on the quality factor of TED in Kirchhoff microplate resonators are discussed in detail. To study the behavior of TED, the material of the microplate resonator is considered as Silicon. The variation of TED as functions of the normalized frequency, microplate thickness, reference temperature have been investigated. The effect of phase-lag parameters on TED has also been shown. The results of the present model are compared to those obtained by the classical continuum theory. The current results show that when the material length scale parameter increases, the quality factor increases significantly from the classical continuum theory.

Keywords: Thermoelastic damping; Quality factor; Size effects; Microplate resonator; Modiﬁed couple stress theory; Three-phase-lag heat conduction model.

Stability Analysis of a Delay Induced Dynamical Model on Oncolytic Virotherapy Hitesh K. Singh and Dwijendra N. Pandey Department of Mathematics Indian Institute of Technology Roorkee Roorkee, 247667, India. [email protected], [email protected]

In this paper, we have studied a non-linear delayed dynamical model that illustrates the interaction between an oncolytic virus and the cancerous cells. An oncolytic virus is a virus that selectively kills tumor cells without harming healthy cells. The model includes two kind of cancerous cells, infected cells (infected by an oncolytic virus) and uninfected ones. We have thoroughly analyzed the stability of all the equilibrium points of the system by ﬁnding the roots of the corresponding characteristic polynomials which are the exponential polynomials with the coefﬁcients involving delay variable. The necessary stability conditions are established in order to stabilize the system in equilibrium. The existence of Hopf bifurcation is shown with delay as bifurcating parameter. In order to verify the analytical results, numerical simulations are carried out using two MATLAB solvers namely ode45 and dde23.

Keywords:

40 On Existence of Solution of First Order Retarded Differential equations with piecewise constant delays Aradhana Bandekar, Y. S. Valaulikar Goa University,Taleigao Plateau ,Panaji, Goa-403206, India. [email protected], [email protected] In this paper, a nonlinear boundary conditions has ben proposed and analysed. We discusses the existence of a solution for a ﬁrst order Retarded Differential equations with piecewise constant delays and with Nonlinear Boundary Conditions between Lower and Upper Solutions. It is observed that there exists atleast one solution between coupled lower and upper solution and also between coupled lower and upper solution in reverse order. This results were obtained by using Arzela Ascoli Theorem , Schauder’s Fixed Point Theorem and L1-Caratheodary conditions. We have examined all possible cases that the equation could go through.

Keywords: Retarded Differential equations,piecewise constant delays, Nonlinear Boundary Conditions ,Lower and Upper Solutions.

AMS subject classiﬁcations : 28A75,33E30, 34B15,,34K10

Evolution of weak shock wave in two-dimensional steady supersonic flow in dusty gas Rahul Kumar Chaturvedi Department of Mathematical Sciences Indian Institute of Technology (B.H.U.) Varanasi 221005, India. [email protected] The present paper concerns with the propagation of weak shock waves in a dusty gas governed by the quasilinear hyperbolic PDEs. Using the method of wavefront analysis the transport equations governing the evolution of weak discontinuities are derived which lead to determine the shock formation distance and conditions which ensure that there will not evolve any shock wave on the wavefront. Also the influence of dust particles, ratio of specific heats and upstream flow Mach number on the shock formation distance is discussed.

Keywords: Hyperbolic PDEs; Weak shock; Wavefront analysis; Dusty gas.

Existence and regularity of solutions of fractional differential equations involving Hilfer fractional derivative of order 1 < α < 2 and type 0 ≤ β ≤ 1 Anjali Jaiswal, D. Bahuguna Department of Mathematics, Indian Institute of Technology Kanpur, Kanpur-208016, India. [email protected] In this paper we investigate the regularity of a solution of a linear problem involving Hilfer fractional derivative. We define the mild solution of an abstract Cauchy problem and obtain conditions under which a mild solution becomes a strong solution. We also study a semilnear fractionl evolution equation and give a suitable definition of a mild solution and establish some existence results for a mild solution. MSC 2010: 34G10, 34A08, 34G20, 34A12. Keywords: Hilfer fractional derivative, strong solution, mild solution, solution operator, fixed point theorem.

41 On First Integral Method and Lie Symmetry of meta-mKdV equation Mahima Poonia, K. Singh Department of Mathematics Jaypee University of Information Technology Waknaghat, Distt.-Solan-173234(H.P.), India. [email protected] In this paper, the meta-mKdV equation is investigated using the first integral method and Lie symmetry. In Lie symmetry, similarity variables are constructed from the Lie symmetry generators which lead to the ordinary differential equation. Using this method, the governing equation is reduced to the Abels equation of second kind. The First integral method provides polynomial first integrals. Using these first integrals the exact solutions of given equation are derived. Applying this method to the meta-mKdV equation, two implicit and some exact travelling wave solutions are obtained. These solutions are completely new contributions.

Keywords: The ﬁrst integral method; Lie Symmetry; meta-mKdV equation.

AMS subject classiﬁcations : 35Qxx, 35E99, 35G25

On Monotone Method for a First Order Neutral Differential Equation Mamta Kumari, Y. S. Valaulikar Department of Mathematics, Shree Damodar College of Commerce & Economics, Comba, Margao, Salcete, Goa 403 601, India. [email protected] This paper discusses the existence of solution for periodic boundary value problem of a ﬁrst order neutral differential equation with piecewise constant deviating argument by using the monotone iterative technique.

Keywords: Neutral differential equations, piecewise constant deviating argument, positive solution, periodic boundary value, monotone iterative technique.

AMS subject classiﬁcations : 34K20, 34K40.

Space time fractional nonlinear partial differential system: Exact solution and conservation laws Baljinder Kour, Sachin Kumar Department of Mathematics and Statistics, Central University of Punjab, Bathinda-151001, Punjab, India. [email protected]; [email protected] The object of the present article is to study space time fractional generalized Hirota Satsuma coupled Korteweg-de Vries (HSCKdV) system for exact solution using power series method corresponding to Lie symmetry reduction of HSCKdV system. The exact solution obtained in power series form further analyzed for convergence. Conservation laws of the HSCKdV system are constructed by using the new conservation theorem and generalized fractional Noether’s operator.

Keywords: Fractional generalized Hirota Satsuma coupled Korteweg-de Vries equations, power series solution, conservation laws.

AMS subject classiﬁcations : 35R11, 34A05, 35C10, 35L65.

42 Solution of Partially Singularly Perturbed System of Initial and Boundary Value Problems Using Non-Uniform Haar Wavelet Akmal Raza, Arshad Khan Department of Mathematics, Jamia Millia Islamia, New Delhi-110025, India. [email protected],[email protected]

An efficient non-uniform Haar wavelet method is proposed for the numerical solution of system of first order linear partially singularly perturbed initial value problem on piecewise uniform Shishkin mesh and ρ−mesh. Further, we apply same technique for solving system of second order linear partially singularly perturbed boundary value problem on piecewise uniform Shishkin mesh and q−mesh. Our method produces better results in comparison to uniform Haar wavelet, classical finite difference operator method and parameter uniform methods. We demonstrated two test problems to support the theory, accuracy and efficiency of the non-uniform Haar wavelet method.

Keywords: System of Differential Equations; Non-Uniform Haar wavelet; Shishkin Mesh; Singular perturbation; Initial and Boundary Value Problems.

Existence, Uniqueness and Regularity of Mild Solutions of Fractional Order Navier-Stokes Equations with Finite Delay Md Mansur Alam, Shruti Dubey Department of Mathematics Indian Institute of Technology Madras Chennai-600 036, India. [email protected], [email protected].

In this work, we consider time-fractional Navier-Stokes equations (NSE) with the external force involving ﬁnite delay over a bounded domain Ω ⊂ R3 having sufﬁciently smooth boundary. We transform the system of equations (NSE) to an abstract Cauchy problem and then investigate local existence and unique- 1 ness of the mild solutions for the initial datum φ ∈ C [−r, 0]; D(A 2 ) , where A is the Stokes operator. With some suitable restriction on initial datum we establish the global existence and regularity of the mild solutions. We use semigroup theory, some tools of fractional calculus and Banach contraction mapping principle to establish our results.

Keywords: Fractional calculus, Navier-Stokes equations, delay differntial equations, analytic semigroup, Mild solutions, fractional power of operators.

MSC 2010: 34A08, 34K37, 35D99, 76D05, 76D03

43 A new approach of operational matrices for hyperbolic partial differential equations Somveer Singh, Mani Mehra Department of mathematics, IIT Delhi, India [email protected],[email protected]

A new approach based on operational matrices of Legendre wavelets is introduced for the class of ﬁrst order hyperbolic partial differential equations with the given initial conditions. Operational matrices of integration of Legendre wavelets are derived and utilized to transform the given PDE into the linear system of equations by combining collocation method. Convergence analysis and error estimation the presented technique are also investigated. Some numerical experiments are performed to demonstrate accuracy and efﬁciency of the proposed method

Analytical Solution of 1-D Advection-Dispersion Equation with an Additional Source/Sink term in the Semi-inﬁnite Aquifer using Dispersion Theory RohitKumar, Manish Chaudhary and Mritunjay Kumar Singh Department of Applied Mathematics Indian Institute of Technology (Indian School of Mines) Dhanbad - 826004 (Jharkhand). [email protected]

Mathematical model describing pollutant transport in semi-in nite aquifer is often represents advection-dispersion equation (ADE). In this present problem, ADE is considered with a source/sink term incorporated in one of the functional form. The aquifer is initially contaminant free and one additional source term is considered at the inlet boundary. Flux type boundary condition is at the outlet boundary. Time-dependent velocity and dispersion is considered to add more re- alism in this problem. A closed form solution is obtained using Laplace transform and Matlab is used to obtain the graphical representation. A numerical solution is obtained by Crank-Nicolson scheme. The results are compared among both the solutions and found a good agreement between them. The effect of source/sink term as a function in the 1-D ADE is explained. The proposed model may be used as a prelim- inary predictive tool in a groundwater resource and management.

Keywords: Aquifer, Advection dispersion equation, Contamination, Source-Sink, Time-dependent dispersion and velocity. AMS subject classiﬁcations : 76Rxx, 76Sxx

Exponentiable objects in Q-TOP Harshita Tiwari Department of Mathematical Sciences Indian Institute of Technology (B.H.U.) Varanasi 221005, India, E-mail: [email protected]

In this paper, we have defined the exponential Q-topology on the power-set of two Q-topological spaces in the category Q-TOP of Q-topological spaces and Q-continuous maps (where Q is a fixed member of a fixed variety of -algebras), and studied the exponentiable objects in the category Q-TOP.

Keywords: Exponentiable object; Q-topology; -algebra.

44 Lp Spectra of Strongly Carleman Pseudo Differential Operators associated with integral transform Pragya Shukla Department of Mathematical Sciences Indian Institute of Technology (B.H.U.) Varanasi 221005, India, E-mail: [email protected] In this paper, we have deﬁned Lp Spectra of Strongly Carleman Pseudo Differential Operators associated with integral transform and studied its various properties like minimal, maximal operators and essential spectrum.

Keywords: Pseudo Differential Operators; Minimal operators; Maximal operators. AMS subject classiﬁcations :

Metrical fixed point theorems via locally finitely T-transitive binary relations under certain control functions Aftab Alam, Mohammad Arif and Mohammad Imdad Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India. [email protected], [email protected] and [email protected] In this paper, we extend relation-theoretic contraction principle due to Alam and Imdad to a nonlinear contraction using a relatively weaker class of continuous control functions employing a locally finitely T - transitive binary relation, which improves the corresponding fixed point theorems especially due to: Alam and Imdad (J. Fixed Point Theory Appl. 17 (2015) 693-702), Agarwal et al. (Applicable Analysis, 87 (1) (2008) 106-116), Berzig and Karapinar (Fixed Point Theory Appl. 2013:205 (2013) 18 pp), Berzig et al. (Abstr. Appl. Anal. 2014:259768 (2014) 12 pp) and Turinici (The Sci. World J. 2014:169358 (2014) 10 pp).

Keywords: locally ﬁnitely T - transitive binary relations; control functions; R-connected sets.

Some Results on Summation-Integral-type Operators and Their Properties Rishikesh Yadav, Ramakanta Meher, Vishnu Narayan Mishra Applied Mathematics and Humanities Department, Sardar Vallabhbhai National Institute of Technology Surat (Gujarat-395007), India Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Anuppur, Madhya Pradesh 484887, India Emails : [email protected], meher [email protected], [email protected] In this paper, we study the Szasz-Mirakyan- Kantorovich type operators and obtain the rate of convergence with the help of local approximation results by using modulus of smoothness, second order modulus of continuity, Peetre’s K-functional and functions belonging to the Lipschitz class. The weighted approximation properties are discussed for computing the order of approximation and related theorems are also proved. To check the asymptotic behavior of the said operators, we prove the Voronovskaya type theorem. The convergence of the Szasz-Mirakyan- Kantorovich type operators is discussed. Graphical approach is also given and the convergence shown via graphically and numerically. At last we take example to show the approximation of the operators to the function by numerically, which is illustrated by table and comparison of the said operators with the Szasz-Mirakyan- Kantorovich operators is took place in sense of absolute numerical error.

Keywords: Szasz-Mirakjan- Kantorovich, Korovkin-type approximation results, modulus of smoothness, Peetre’s K-functional, weighted modulus of continuity.

45 Kantorovich type generalization of modided Szasz-Mirakjan´ Operators Ankita R Devdhara, Vishnu Narayan Mishra SVNIT, Surat, India. [email protected], [email protected]

In this paper, we generalize the modided Sza´ asz-Mirakjan operators in Kantorovich form. We discuss moments and central moments of the operators. We prove some local approximation results for the operators.

Keywords: Szasz-Mirakyan´ operators, Korovkin’s theorems, Rate of convergence, Voronovskaya result. AMS subject classiﬁcations :

θ∗-WEAK CONTRACTIONS AND DISCONTINUITY AT THE FIXED POINT ATIYA PERVEEN Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India, Email: [email protected]

In this paper, the notion of -weak contraction is introduced, which is utilized to prove some fixed point results. These results are helpful to give a positive response to certain open question raised by Kannan [Amer. Math. Monthly 76:1969] and Rhoades [Contemp. Math. 72:1988] on the existence of contractive definition which does not force the mapping to be continuous at the fixed point. Some illustrative examples are also given to support our results. As applications of our result, we investigate the existence and uniqueness of a solution of integral equations of Volterra type.

A New Type of Paranorm Intuitionistic Fuzzy Zweier I-convergent Double Sequence Spaces Hira Fatima Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India. [email protected]

Fuzzy set theory was introduced by Zadeh [Fuzzy sets, Inform Control, (8)(1965), 338-353] which is a powerful hand set for modelling uncertainty and vagueness in various problems arising in ﬁeld of science and engineering and has a wide range of applications. Fuzzy topology is one of the most important and useful tools and it proves to be very useful for dealing with such situations where the use of classical theories breaks down. In this article, we introduce the paranorm type intuitionistic fuzzy Zweier I-convergent double I I sequence spaces 2Z(µ,ν)(p) and 2Z0(µ,ν)(p) for p = (pij) a double sequence of positive real numbers and study the fuzzy topology on the said spaces.

46 Relation-theoretic multi-valued θ-contraction]Relation-theoretic ﬁxed point results for Multi-valued (θ, R)-contractions with an Application Mohammad Imdad , Md Hasanuzzaman and Waleed M. Alfaqih Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India. Department of Mathematics, Hajjah University, Hajjah, Yemen. [email protected], [email protected] and [email protected]

The aim of this paper is to introduce a relatively new concept of multi-valued (θ, R)-contractions and utilize the same to prove some ﬁxed point results in metric spaces endowed with an amorphous binary relation. Illustrative examples are also furnished to exhibit the utility of our results proved herein. Finally, we utilize some of our results to investigate the existence and uniqueness of a positive solution for Volterra type integral equation.

Keywords: Fixed point, θ-contractions, multi-valued θ-contractions, binary relations, integral equations.

Optimal control analysis of an e-epidemic model including ﬁrewall effect Prerna Singh, Ranjit Kumar IIT(ISM) Dhanbad, India. [email protected], [email protected]

A mathematical model has been proposed that studies the virus propagation of a distributed attack on a targeted network. The model is composed of two different classes of computer nodes-attacking and targeted. A simpler model is further obtained from these systems by nondimensionalisation, and its dynamical properties are discussed. The ﬁrewall security is taken as a media coverage factor in this work and it is dis- covered that it helps to diminish the virus propagation in the network upto some extent. We then carry out the optimal control analysis of the model system and give results regarding the effect of control measures in controlling the virus propagation in a network. The aim of introducing control theory in this work is to provide a measure for minimizing the virus transmission in a network. Numerical experiments are carried out to justify the analytical ﬁndings.

Keywords: Firewall, distributed attack, optimal control.

AMS subject classiﬁcations : 92D30; 34D08; 34D23; 34D45; 65L07.

47 Solution of Differential Algebraic Equations using Coded Differential Transform Method Anil Kumar and Giriraj Methi Department of Mathematics & Statistics, Manipal University Jaipur-303003, Rajasthan, India [email protected]; [email protected]

Aim of the paper is to obtain numerical solutions of some differential algebraic equations using Coded Differential Transform Method (CDTM). We have compared our series solutions with other researcher using multi quadric method. We have used Mathematica for numerical solutions and graphical illustrations. CDTM avoids linearization, discretization, complex calculations and save lots of time.

Keywords: Differential algebraic equations, Coded differential transform Method, Numerical solution, Mathematica

AMS subject classiﬁcations : MS34-04; MS34L99; MS37-04

Dynamic modeling and control of divided wall distillation multicomponent separation Manali Kokare, C. S. Mathpati, Ajit Kumar, S. S. Jogwar Department of Chemical Engineering, Institute of Chemical Technology, Mumbai. [email protected], [email protected], [email protected],[email protected]

Distillation is one of the most widely used separation method in chemical and allied industries. The design of distillation unit involves system of ordinary differential equations related to overall mass, component, energy balance and thermodynamic equilibria of the system. Distillation is energy intensive process and design optimization is essential for energy saving as well as consistent quality. Divided wall distillation system is an optimized distillation method which involves separation of multicomponent mixture in single unit where pre-fractionation and main column are separated by a wall which leads to saving in capital as well as operating cost. The divided wall distillation is nonlinear in nature so the model predictive controller is useful in which three component purities are controlled. In the proposed work, dynamic model of the system has been simulated in Matlab for the separation of (n-butanol)-(water)-(n-butyl levulinate). The effectiveness of the developed controller is checked for a step change in n-butanol purity.

Keywords:Divided wall distillation, globally linearizing controller, dynamic model, system of ordinary differential equations

AMS subject classiﬁcations : 34B60 (Under ODE: applications)

48 ASYMPTOTIC ANALYSIS OF BOUNDARY OPTIMAL CONTROLPROBLEM ON A GENERAL BRANCHED STRUCTURE S. AIYAPPAN,A. K. NANDAKUMARAN , AND ABU SUFIAN [email protected],[email protected],abusuﬁ[email protected]

We consider an optimal control problem posed on a domain with a highly oscillating smooth boundary where the controls are applied on the oscillating part of the boundary. There are many results on domains with oscillating boundaries where the oscillations are pillar-type (non-smooth). The literature on smooth oscillating boundary is very few and recently Aiyappan et. al have studied a homogenization problem on such oscillating domain by developing new unfolding operators. In this article, we use appropriate scaling on the controls acting on the oscillating boundary leading to different limit control problems; namely, boundary optimal control and interior optimal control problem. In the last part of the article, we visualize the domains as a branched structure and we introduce unfolding operators to get contributions from each level at every branch.

Keywords:Optimal control, Asymptotic analysis, Unfolding operator, Oscillating boundary domain, Homogenization

AMS subject classiﬁcations : 80M35, 80M40, 35B27, 49J20

Approximate controllability of multi-term time-fractional differential inclusions with nonlocal conditions Ashish Kumar, Dwijendra N. Pandey Indian Institute of Technology, Roorkee-247667 [email protected], [email protected]

A set of sufﬁcient conditions for the approximate controllability for a class of multi-term time fractional differential inclusions with the nonlocal condition of the form

n c 1+β X c γj D y(t) + αj D y(t) ∈ Ay(t) + Bu(t) + F (t, y(t)) t ∈ [0, b] j=1 y(0) + g(y) = σ, y0(0) + h(y) = χ. has been established in this article. Keywords:approximate controllability, multi-term time-fractional delay differential system, (β, γj)− resolvent family, ﬁxed point theorems, differential inclusions.

AMS subject classiﬁcations : 34B60 (Under ODE: applications)

49 Approximate Controllability of Semilinear Fractional Evolution Systems with multiple Delays in Control Abdul Haq, N. Sukavanam Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India, Pin-247667. [email protected] This work studies the controllability of a class of fractional evolution differential equations with Riemann- Liouville fractional derivatives and with multiple delays in control. We establish suitable assumptions to prove the existence and uniqueness of mild solutions. Approximate controllability of the system is shown using sequence method. Finally, an illustrative example has been provided.

Keywords: Riemann-Liouville fractional derivatives, fractional evolution systems, delay system, mild solution, approximate controllability. AMS subject classiﬁcations : 93B05

CONTROLLABILITY OF NONLOCAL FRACTIONAL ORDER INTEGRO-DIFFERENTIAL SYSTEMS WITH TIME VARYING DELAY Ajay Kumar, N. Sukavanam Department of Mathematics, Indian Institute of Technology Roorkee Roorkee-247667, India e-mail: [email protected] In this article, we established some new set of sufﬁcient conditions for the controllability of nonlocal fractional order integro-differential systems in Banach space with time varying delay under the assumption that semigroup operator is non-compact. The theory of fractional calculus, strongly continuous semigroup, and Nusbaum ﬁxed point theorem are the main tools used in this problem. An example is given to verify the application of our proposed results.

Keywords: Controllability, Fractional control delay systems, Semigroup Theory, Nusbaum Fixed Point Theorem. AMS subject classiﬁcations : 93B05, 93C10.

Development of Higher-order Implicit-Explicit Robert-Asselin Type Time Filters Praveen K. Maurya, Manoj K. Rajpoot Department of Mathematics Rajiv Gandhi Institute of Petroleum Technology Jais 229 304, U.P., India, e-mail: [email protected] A class of new hybrid implicit-explicit (IMEX)time filters based on three time- level leapfrog temporal integration scheme are developed. The developed filters are used for the dispersive and non-dispersive model systems by considering the space-time discretization together. As, the leapfrog time-stepping method is most commonly used in ocean and atmospheric modelling, however, it also admits a spurious mode in numerical computations. The developed hybrid IMEX time filters are based on the optimized values which are capable in suppress- ing the computational mode(s) more effectively without affecting the physical mode. Finally, the developed IMEX filters are tested for one-dimensional non-dispersive convection and two-dimensional dispersive rotating shallow water equation (LRSWE), and by solving the incompressible flow problem governed by Navier-Stokes equation at different Reynolds numbers. Keywords: IMEX time filters; Spurious dispersion; Grid staggering; Rotary shallow water equation; Navier-Stokes equation. AMS 2010 Classifications: Primary 65Mxx; Secondary 65Txx.

50 Modelling hydraulic characteristics of Gabion weir by soft computing techniques Siddharth Sonkar1, N.K.Tiwari2, and Subodh Ranjan Vajesnayee3 1,2,3M.Tech Civil Engineering, National Institute of Technology Kurukshetra, Haryana-136119, India. [email protected]

e-mail: [email protected] A conventional weir essentially consists of an impermeable body constructed by concrete and the main function of a weir is to raise the water level and efficiently regulate the ow. Gabion is a cage, cylinder, or box filled with rocks, concrete, or sometimes sand and soil for use in civil engineering and other purposes. In this paper,soft computing based modelling techniques were used to estimate the hydraulic characteristics of Gabion weir and results were compared with conventional solid broad crested weir. The modelling techniques employed for analysis are Artificial Neural Network(ANN) and Gaussian Process (GP). The parameters used as input are water head at upstream, discharge,mean size of gabion materials , porosity,etc. and output as coefficient of discharge. Keywords: Gabion weir,ANN, GP and Hydraulic Characteristics.

Effect of radiative heat transfer on the growth and decay of acceleration waves in non-ideal magnetogasdynamics Shobhit Kumar Srivastava Indian Institute of Technology (B.H.U.) Varanasi 221005, India. [email protected]

In the present paper, the evolutionary behavior of weak shock waves propagating in an unsteady one- dimensional ﬂow in non-ideal radiating gas under the effect of transverse magnetic ﬁeld is examined. For the effect of thermal radiation, the radiative transfer equations are approximated under the optically thin limit. It is observed that a linear solution in the characteristic plane may exhibit a non-linear behavior in the physical plane. The transport equation governing the evolution of weak shock waves is obtained which introduces the conditions for shock formation. The time for the rst breaking of the wave is determined. Also, the effect of radiative heat transfer on the growth of compressive waves and decay of expansive waves in ideal and non-ideal magnetogasdynamics regime is discussed.

Keywords: Thermal radiation; Non-ideal gas; Acceleration waves; Magnetic eld; Shock formation.

Wind turbine blade section optimization using a quantitative study M.Balachandar, B.U Raja Ramakrishnaa and N.Ramanan Sri Sai Ram Institute of Technology,Chennai,Tamilnadu,India , R&D, Synce engineering service,Chennai,Tamilnadu,India [email protected], inﬁ[email protected], [email protected]

Recently there has been an increase in the demand for the utilization of clean renewable energy sources. This is due to increase in the oil prices and increased awareness of human induced climate change. Wind energy has been shown to be one of the most promising sources of renewable energy. The variation has been made with the cross sections of the wind turbine blades. The chosen airfoil is NACA five digit series. Two different series were chosen (NACA 63412 and NACA 63415). These airfoils were studied using a simulation software. Then analyzed using CFD software. Various graphs depicting the lift, angle of attack and drag co-efficient were studied to conclude the air foil having the highest lift. Later the finalized airfoil shape was fitted as one single wind turbine using a simulation software. The modified air foil was man- ufactured from Nutmeg Hickory wood. The given material provides better stability and finish and hence

51 was chosen. The power driven was found using calculation. The observed results were veriﬁed using the computed results from the software. The results shows that for low speed range of 2.5m/s - 9m/s generates 1.28kW of power.

Keywords: CFD, NACA 63415, wind turbine, air foil, renewable energy,power

Hybrid impulsive effects on quasi-synchronization of neural networks with parameter mismatch and mixed time-varying delays Rakesh Kumar Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi-221005 Emails : [email protected]

This article is deeply concerned about the effects of hybrid impulses on quasi-synchronization of neural networks with mixed time-varying delays and parameter mismatches. The hybrid impulsive controller has been designed to deal with the difficulties in achieving the quasi-synchronization under the effects of hybrid impulses which occur all the time, but their density decreases gradually with time. In addition to hybrid impulses, the new concept of average impulsive interval and average impulsive gain have been applied to cope with the simultaneous existence of synchronizing and de-synchronizing impulses. Based on the Lyapunov method together with some mathematical techniques and the extended comparison principle combined with the formula of variation of parameters for mixed time-varying delayed impulsive system, the delay-dependent sufficient criteria of quasi-synchronization have been derived for two separate cases, viz., a Ta < ∞ and a Ta = ∞. Finally, the efficiency of the theoretical results has been illustrated by providing two numerical examples.

Keywords: Neural networks, Quasi synchronization, Hybrid impulses, Parameter mismatch, Mixed time-varying delays.

Novel divergence measure for reﬁned single valued neutrosophic sets and its utility in decision making Adeeba Umar, R. N. Saraswat Department of Mathematics and Statistics Manipal University Jaipur, Jaipur -303007, Rajasthan, India. [email protected], [email protected]

Neutrosophic set which is a branch of neutrosophy, studies the scope, characterization and origin of the neutralities and their interconnection with various ideational expansion. Neutrosophic set is an inﬂuential framework which has been proposed recently. In this communication, a novel divergence measure for re- ﬁned single valued neutrosophic sets is introduced with the proof of its validity. An application of novel divergence measure is shown with an illustration for decision making in medical investigation and project selection.

Keywords: Divergence measure, reﬁned single valued neutrosophic sets, medical diagnosis, decision making.

AMS subject classiﬁcations : 94A15, 62B86, 94A17

52 Analysis of Surface Seismic Waves in Piezomagnetic Layered Structure Suman Goyal, Sanjeev Anand Sahu Department of Applied Mathematics, IIT (ISM) Dhanbad (826004), Jharkhand, India. [email protected] , [email protected]

We propose an analytical solution for the transference of surface seismic waves in a piezo-composite smart structure. A model describing the propagation of the considered wave in the piezomagnetic layered structure is considered. The model is comprised of piezomagnetic layer lying on inhomogeneous elastic half-space. The inhomogeneity in the half-space is due to exponential variation of elastic constants with the depth. The governing equations are presented in form of direct Sturm-Liouville problem. Frequency equations in the form of determinant have been obtained for both the magnetically open and short cases. The inﬂuences of layer thickness, presence and absence of piezomagnetic coefﬁcient on the phase velocity of surface wave are depicted through graphs. The results may be useful in designing and optimization of Surface Acoustic Wave (SAW) devices.

Keywords:Love-type wave, inhomogeneity, Sturm-Liouville, Smart Material.

AMS subject classiﬁcations : 74J15, 74D05, 74G10, 74H10.

Velocity Proﬁle of Shear Horizontal (SH) surface waves in Bi-layered FGPM/Porous Piezoelectric Plate Shreeta Kumari, Sanjeev A. Sahu1, Kamlesh K. Pankaj Department of Applied Mathematics, IIT(ISM), Dhanbad-826004, Jharkhand, India. ofﬁ[email protected], [email protected], [email protected]

The present problem is confined to study the SH-wave propagation in a functionally graded piezoelectric plate fused together with porous piezoelectric plate. The functionally graded material is assumed to be varying quadratically. Separation of variable method is used to get the solution for the displacement and stress components. Solutions of the constitutive equations are obtained in terms of modified Bessels function of the first and second kind. Dispersion relation is obtained for both; electrically open and short conditions. Effects of thickness of the plates, gradient coefficient, dielectric coefficients and piezoelectric coefficients have been discussed and shown distinctly through graphs. Findings of the present investigation may lead to the theoretical foundation for designing the SAW devices of higher efficiency.

Keywords: Functionally graded piezoelectric material, Porous-Piezoelectric material, SH-wave, Dis- persion.

AMS subject classiﬁcations : 74J15, 74D05, 74G10, 74H10

53 Thermal analysis of convective-radiative pin ﬁn with MATLABs inbuilt tool Pdepe considering temperature dependent properties Sarvjeet Singh, Rohit K. Singla Department of Mechanical Engineering, Thapar Institute of Engineering and Technology, Patiala, India. [email protected], [email protected]

The present work investigates the performance of the nonlinear longitudinal circular pin ﬁn with all temperature dependent thermal parameters. The involved temperature dependency is either linear or nonlinear for different parameters of conduction, convection, and radiation. The solution of the problem is evaluated with various boundary condition from easier one, i.e., constant temperature, insulated, etc. to realistic, i.e., heat gain or lost by convection and radiation. The mathematical non-linear equations was solved with the MATLAB based code of Pdepe. The stability of solution has been veried. The numerical results of the present technique were validated with the literature and found to be very well in agreement with the available results in the literature. The study investigates the effect of various parameters on the temperature distribution of the nonlinear longitudinal circular pin ﬁn.

Keywords: Pin ﬁn, Conduction, Convection-Radiation, Pdepe tool.

Kumar.tex

Effect of Solar Flare on climate change by Solar Flare Wave Model and Its Application Sumit Bainjwan, Vishal Dhakane Pandit Deendayal Petroleum University, Gandhinagar Pandit Deendayal Petroleum University, Gandhinagar. [email protected], [email protected]

In the last few hundred years, the number of sunspots has gradually increasing leading to Global Warm- ing. Evidence suggests that solar activity has affected the global climate adversely. Sunspots are storms on the sun’s surface that are marked by intense magnetic activity and play host to solar flares and hot gassy ejections from the sun’s corona. Solar Flares consist of magnetized plasma flares which emerge from the sun and influences galactic rays that may in turn affect atmospheric phenomena on Earth, such as cloud cover. During solar bursts and associated solar flares and hot gassy ejections from the coronal mass excre- tion, the outer atmosphere of the earth is subjected to large amounts of energy. The heat wave produced by the flurry eruptions in the sun are energetic particles which fall in the upper layers of the atmosphere and convey energy at point of incidence. The action creates spectacular Aurora Borealis around the poles which heat the atmosphere significantly. The mean temperature of earth can be explained by variation of short wave arriving at surface of earth and influence of long term change in radiation, this paper focus on modeling and analysis of solar wave which could be used to analyze the global warming phenomenon. The model is subjected to observational data which could be examine methodically leading predictive control to analyze and take possible steps to mitigate the Global warming.

Keywords: Solar Flares, Global Warming, Non-Linear Wave Equation, Coronal Mass Ejections, Plas- mas, Radiation, Atmosphere, Energetic Particle.

Kumar.tex

54 Modelling Aeration Efﬁciency of Hydraulic Jump at Under Sluice Gate Nirali Vashishth, Subodh Ranjan Vajasneyee, and N K Tiwari Department of Civil Engineering, National Institute of Technology Kurukshetra, Haryana-136119, India. [email protected], [email protected], [email protected]

Hydraulic jump is the most frequently occurring hydraulic phenomena in the river observed when water at high velocity discharges into a zone of lower velocity which follows an abrupt rise creating turbulence in the water surface thereby dissolving and increasing the dissolved oxygen (DO) in the river. The amount of DO is a key indicator of quality of water. Higher is the DO better is the quality of water as it means minimal presence of organic matter. The aim of this paper is to compute the aeration efficiency of hydraulic jump at under sluice gate using AI-based modelling techniques namely SMO Regression, Meta and Trees for the values taken from the data set of published results. The techniques are compared based on statistical performance evaluation parameters and agreement diagrams and it was found that Meta gave better results than others. The findings of this paper will help in selecting the best fit technique for modelling of aeration efficiency of hydraulic jump at under sluice gate.

Keywords: hydraulic jump, aeration, dissolved oxygen, sluice gate.

Interference of Closely Placed Bridge Piers On Local Scour Anuj Kataria1 and Baldev Setia2 1M.Tech Civil Engineering, National Institute of Technology Kurukshetra, Haryana-136119, India, e-mail: [email protected] 2Professor, Department of Civil Engineering, National Institute of Technology Kurukshetra, India. e-mail: [email protected]

Bridge is a communication provided when the road is obstructed on account of a river, water bodies etc. Most of the bridge failed due to the hydraulic related issues in general scouring around bridge elements. Scouring is a natural phenomenon that occurs due high velocity resulting in removal of sediment particles from the bed, bank of streams and bridge piers. Extensive work on scour around isolated pier is available in literature but relatively lesser work is available on scour around closely placed bridge piers. The present study is concerned with the experimental investigation of interference of bridge piers on local scour by varying pier spacing. The experimental work was conducted in a ume of length 15 m, width 0.61 and depth 0.70 m in the Water Resources Engineering laboratory at National Institute of Technology, Kurukshetra, In- dia. A set of 20 experiments have been conducted under clear water condition. There are two pier arrangements, namely Tandem and Side by side arrangement that were tried during the experimental study. A single experiment was also conducted to measure the scour depth at upstream face of isolated pier, which formed the basis of reference for the studies on group of piers. Scour depth in tandem arrangement for two piers touching each other was found to be 5.17 percent more than that at the isolated pier. At a clear spacing of 16 times the diameter of pier, maximum scour depth at any of the two piers was found to be same as that at isolated pier. In case of two piers in side by side arrangement, scour depth was found to be 62 percent more than that at isolated pier at no clear spacing between the two equal piers.

Keywords: Local scour, Bridge pier, Arrangements, Interference, Scour depth.

55 The inﬂuence of vegetation type and cover on rain garden hydrological performance Anuj kumar1, Krishna Kumar Singh2 Department of Civil Engineering, National Institute of Technology Kurukshetra, Haryana-136119, India. [email protected]

Rain gardens are the best storm water management tool these days and frequently used all over the globe especially in urban areas to reduce urban storm water impacts. Infiltration characteristics of three small rain gardens (1 m2) constructed in the hydraulics lab of NIT Kurukshetra campus with different slope profiles were monitored for nearly 9 months, covering 5 observations over the whole period. And water applied to the Gardens through a constant volume cylindrical tank. The final result indicates that the rain garden with the flat profile having the highest infiltration rate. This year few modifications were made in these rain gardens, like the size of the two gardens get doubled and the vegetation cover also changed and the observation with different type of vegetation had taken. In the present study, three vegetation types were considered: native scutch grass (cynodon dactylon) and chandni flower plants as well as daisy flower plants. Overall, results indicate that the rain garden with the higher density of scutch grass vegetation cover infiltrate the water most expeditiously. And the effect of size on the hydrological performance of the rain garden is very minor or it can be neglected.

Keywords: Rain garden, Infiltration rate, flat profile

Non-linear Deformation of Thin Elastic Model Membrane Driven by Electrostatic Forces Amar Shrivastava and Paritosh Mahata Department of Mechanical Engineering, Birla Institute of Technology, Mesra, Ranchi - 835215, Jharkhand, India. [email protected]

Deformation of thin elastic membrane driven by electrostatic forces has important implications in the field of engineering and biological sciences. For example, in cell-biological systems, peripheral proteins bend the cell membrane into curved structure due to electrostatic interactions between them. This is an important phenomenon for several cell-biological processes like endocytosis and exocytosis. Malfunction- ing of these processes produces diseases like cancer and Parkinson’s. To analyze the binding mechanism between the protein and cell membrane, the out of plane deformation of cell membrane is analyzed while treating it as an elastic membrane. In addition to this, the membrane deformation, driven by electrostatic forces plays an important role in many industrial applications like electrophotography, powder technology, semiconductor and pharmaceutical industries. In this work, we analyze large and non-linear deformation behavior of a thin elastic membrane sheet interacting electrostatically with a rigid curved domain in the presence of dielectric fluid. The membrane has negative charge distribution on its upper surface. A rigid curved domain with uniform positive charge distribution at its concave face interacts with the membrane electrostatically and deform it. Mechanical deformation of the membrane is coupled with the electrostatic interaction in its equilibrium configuration. Large and non-linear mechanical deformation of the membrane is predicted by the Neo-Hookean strain energy function. The electrostatic forces acting between membrane and curved domain are calculated by using Debye-Huckel equation. The non-linear ordinary differential equations obtained from the equilibrium configurations of the membrane are solved numerically to calculate the deformations of membrane. It is observed that the membrane deformation is proportional with the increasing charge density of the rigid curved domain. Increase in inverse Debye length (which signifies the strength of electrostatic field) of the dielectric fluid decreases the membrane deformation. Though the present study is focused on the elastic

56 deformation of thin membrane, but it can also form a basis for understanding the deformation of complex biological lipid membrane.

Keywords: Non-linear deformation, thin membrane, electrostatic interaction

Integral Equation technique for solution of diffraction of obliquely incident water waves by rectangular asymmetric trench 1Amandeep Kaur, 1S. C. Martha, 2A. Chakrabarti 1Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar, Punjab 140001, India, 2Department of Mathematics, Indian Institute of Science, Bangalore 560012, India, Email: [email protected]

The problem involving diffraction of obliquely incident water waves by a rectangular trench is examined for its solution with the aid of a system of integral equations of ﬁrst kind. The resulting integral equations are solved by using suitably designed polynomial approximations of the unknown functions. The numerical values of physical quantities associated with the water wave problem are found to be in excellent agreement with the known results where a Galerkin type of approximation has been used to obtain the solutions.

Life Cycle Analysis (LCA) of Low Volume Rural Hill Roads Akhilesh Nautiyala, Sunil Sharmab PhD Scholar, Civil Engineering Department, NIT Hamirpur, 177005 India, Assistant Professor, Civil Engineering Department, NIT Hamirpur, 177005 India. [email protected], [email protected]

Road maintenance projects involve limited amount of funds to be allocated judiciously, for this precise Life Cycle Assessment (LCA) of pavements is crucial. A precise LCA of can save huge amount of money and time by applying satisfactory maintenance and rehabilitation techniques at the suitable time to the most suitable roads. This paper present an approach to forecast behaviour of pavements in rural hill roads in its overall life cycle and then predict its performance throughout its service life. Pavement condition rat- ing and traffic volume are two major factors considered in this study to determine the overall performance of pavement. Rutting, Ravelling, Cracking, Patching and Pothole are five major defects taken to evalu- ate overall condition of pavement. Linear regression analysis was used to develop a relationship between pavement age and pavement condition index (PCI), which gives a scientific relationship to predict remaining service life of pavement’s in rural roads in hilly regions. Linear regression analysis was performed to develop correlation between PCI and pavement age which gives relationship in the form of an equation: y = −25.62ln(x) + 116.16, where y is the pavement age and x is the current PCI. Correlation coefficient of 0.8899 was obtained for this equation which shows that the curve is a good fit. The average service life of rural roads in the study area was obtained as 21.65 years.

Keywords: Life Cycle Assessment, Remaining service life, Pavement condition index, Rural hill roads, Liner regression analysis

57 Flow Characteristics at the Conﬂuence of WJC and SYL Canals Pradeep kumar1 and Baldev Setia2 1M.Tech Civil Engineering, National Institute of Technology Kurukshetra, Haryana-136119, India, e-mail: [email protected] 2Professor, Department of Civil Engineering, National Institute of Technology Kurukshetra, India. [email protected] Presenter

Haryana is among major agricultural states of India with approximately 2.9mha of cultivable land under surface irrigation aided mainly by a network of canal system. Development of water for irrigation can be cited as one of major contrib- utors to Haryana’s agriculture success. At one of the con uences in this network, two canals Western Jamuna Canal(WJC) and Satluj Yamuna Canal(SYL) meet near Karnal a city in Haryana at 29.692171N, 76.956648E. While WJC with a bed width 76.25m carries 320.24cumec, the SYL with bed width of 16.104m carries 175.52cumec. Owing to geometric and discharge differences, the con uence is not smooth and results in formation of eddies resulting in scouring of bed and banks at the specific con uence point. This follows as a damaging the right bank of canal. The Present study analyses the causes of the scouring at the confluence with the help of momentum equation with different magnitudes of Froude Number, Discharge ratio, Depth ratio, Confluence angle, etc. The momentum equation has been developed analytically and solved using MATLAB. Results show that there is decrease in depth ratio with an increase in discharge ratio at a given Froude Number. Hence with the understanding of field data and Flow characteristics gives a reasonable solution regarding erosion, scour and sedimentation in the downstream con uence canal. Being a field problem there are some limitations of the work like, the availability of eld data from not more than one source at confluence point. Keywords: Canal con uence, Momentum equation, MATLAB, Depth ratio, Discharge ratio.

TREND ANALYSIS OF HYDROLOGICAL PARAMETERS OF TWO AGRARIAN DISTRICTS OF HARYANA, INDIA Mridula Sharma1, Arun Goel Department of Civil Engineering, National Institute of Technology Kurukshetra, Haryana-136119, India. [email protected], [email protected]

Hydro-geological parameters and land-use pattern vary spatio-temporally and have a profound effect on groundwater level and quality of any region. Kurukshetra and Kaithal are two prominent agrarian districts in Haryana, India which have suffered sharp decline in ground water levels below ground in freshwater aquifer areas and rise in water levels in saline-water aquifer areas which is detrimental for agriculture, economy and environment. Therefore, in order to ascertain sustainability of irrigated agriculture, in the present study, the ground water levels of these districts of Haryana, including areas with both saline-water and fresh-water aquifers are studied in co-relation with other hydrological parameters data of last 37 years (1981-2017) including rainfall, temperature, wind speed etc. The statistical analyses of past trends has been done annually and seasonally using Mann-Kendall test and Sens slope estimation technique which is vital for sustainable planning of water-resources and cropping pattern of the area.

Keywords: Rainfall, Hydrology, Ground water, Mann-Kendall test, Sens slope.

AMS subject classiﬁcations : 315 (Data Analysis)

58 GEOMETRIC PROPERTIES OF THE EXTENDED τ GAUSS HYPERGEOMETRIC FUNCTION R. ROY AND R. K. JANA Department of Applied Mathematics and Humanities, S. V. National Institute of Technology, Surat, 395007, India. e-mail : [email protected]

In this paper, we have considered the normalized form of extended τ Gauss hypergeometric function, 2 2 τ Γ(c) P∞ (a;p)nΓ(b+τn) zn which is deﬁned as R1(z) = R1 ((a; p), b; c, z) = Γ(b) 0 Γ(c+τn) n! , where p ≥ 0, τ > 0, |z| < 1; R(c) > R(b) > 0, when p = 0. We obtained several conditions so that the extended τ Gauss hypergeometric function has some geometric properties including univalency, starlikeness and convexity inside the unit disk |z| < 1. Keywords: Analytic function, Univalent function, Close-to-convex function, Starlike function. AMS subject classiﬁcations : 33E12, 30C45.

Theoretical Investigation of Networks of Interacting Exclusion Processes Tripti Midha, Arvind Kumar Gupta Department of Mathematics, Indian Institute of Technology Ropar, Punjab. [email protected]

Motivated by the biological and vehicular transport processes, we investigate a network consisting of a vertex V from which several lattices, undergoing the totally asymmetric simple exclusion process with interactions, converge and diverge together. The vertex V act as an exit and entry reservoir, respectively, for the segments arriving and leaving from it. The interactions in the bulk of every lattice are theoretically handled with the two-cluster meanfield framework. We calculate the effective entrance and exit rates, respectively, for the outgoing and incoming segments to the vertex V, as a function of its average particle density, by ignoring the nearest-neighbor correlations for the sites at the boundaries. The theoretical calculations for the particle density, correlation profiles and phase diagrams for the entire network match efficiently with the extensively performed computer Monte Carlo simulations. We theoretically compute the general existence conditions for the various phases in a phase diagram and found that their existence highly depends on the total number of outgoing, incoming segments and the interaction energy among the particles. We find that the correlations weaken in a network with more number of incoming or outgoing segments. Keywords: Exclusion processes, microtubule network, cluster mean-field theory, Monte Carlo simulations

59 60 CNA

On solvability of some nonlinear functional-integral equations with applications Amar Deep and Deepmal PDPM-Indian Institute of Information Technology, Design and Manufacturing, Jabalpur - 482005 (MP), India. [email protected]

Using the concept of measure of noncompactness in Banach algebra, we establish some existence results for a generalized nonlinear functional-integral equation, which contains several known functional- integral equations as a particular case. Our results unify and improve some known results in the recent literature. For application, an example of a functional-integral equation is also provided to illustrate our main result. Keywords: Measure of noncompactness, Fixed point, Functional-integral equation, Banach algebra. 2010 Mathematics Subject Classiﬁcation. 90C39, 47H10.

Keywords: Solar Flares, Global Warming, Non-Linear Wave Equation, Coronal Mass Ejections, Plas- mas, Radiation, Atmosphere, Energetic Particle.

61 Entropy Generation in the Flow of Sisko Nanoﬂuids over a Stretching Sheet Ankita Bisht and Rajesh Sharma Department of Mathematics, National Institute of Technology, Hamirpur, Hamirpur 177 005, Himachal Pradesh, India. [email protected], [email protected]

The main motive of the present article is to investigate the entropy generation in MHD Sisko nanofluid flow towards linear stretching of the sheet. The governing Sisko nanofluid flow equations comprise momentum, energy and nanoparticle volume fraction are reduced to nonlinear ordinary differential equations by using suitable similarity variables. The coupled nonlinear ordinary differential equations are then solved numerically by using finite difference method in MATLAB software. The impact of different physical parameters on entropy generation number, velocity, Bejan number, and temperature are presented graphi- ∗ cally. The obtained results indicate that the entropy generation number (NG) increases with increases in the Brinkman number (Br∗) and magnetic parameter (M ∗) while decreases with the material parameter for Sisko fluid (A∗) whereas Bejan number (Be∗) enhances for large values of the material parameter for Sisko fluid (A∗) while diminishes for the magnetic parameter (M ∗) and Brinkman number (Br∗). Moreover, ∗ ∗ ∗ increasing trends are observed for both (NG) and (Be ) for higher values of diffusion parameter (γ ). Ve- locity profile increases while the temperature profile decreases for large values of the magnetic parameter. It can be observed from the present analysis that the obtained results are beneficial in understanding the entropy generation (or irreversibility) in non-Newtonian fluid flows.

Keywords: Sisko nanoﬂuid, Entropy generation, Finite difference method, Numerical solution.

AMS Subject Classiﬁcation: 76A05, 76M20, 76N20, 76W05, 35G30

Adomian Decomposition Method for the solution of a Parial Differential Equations of Fractional Order Pratibha Verma and Manoj Kumar Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Allahabad, Prayagraj211004, Uttar Pradesh, India. [email protected] and [email protected].

In this paper, we investigate the method for solving linear and non-linear partial differential equations of fractional order. The Adomian Decomposition Method is one of the reliable and popular method for solving linear and non-linear differential equations and provides solutions in the form of series. In this study, our main purpose is to achieve more accurate and fast convergent solution with less iterations. Here we adopt Two-Step Adomian Decomposition Method (TSADM) for solving linear and non-linear partial differential equations of fractional order and compared with Standard Adomian decomposition method and Modiﬁed Adomain Decomposition Method. It is successfully applied to autonomous linear and non-linear partial differential equations with variable coefﬁcients. It is shown that TSADM is more effective and promising method with one iteration and provides exact solution of both the problems without discretization and lin- earisation.

Keywords: Adomian decomposition method, Modiﬁed Adomain Decomposition Method , Two Step Adomain Decomposition Method, Fractional Partial Differential Equation.

62 A new efﬁcient numerical scheme for variable order fractional sub-diffusion equation Sarita Nandal, Dwijendra Narain Pandey Depaertment of Mathematics, IIT Roorkee, Roorkee-247667, India, [email protected], [email protected]

In this paper, we propose to construct a new efficient finite difference scheme for variable order fractional sub-diffusion wave equation. Fractional derivative will be considered in the sense of Caputo and approximated using L2 − 1σ formula which gives second order convergence for α(t) ∈ (0, 1). For spatial dimensions, we will consider a compact difference operator which improves the convergence order to O(h4), where h is spatial mesh size. Next, we will prove the stability, solvability, and convergence of our constructed scheme using discrete energy method with help of L2-norm. Our proposed scheme is new and efficient in terms of convergence orders in both time and spatial dimensions. Then, a few examples will be provided to demonstrate the accuracy and efficiency of the proposed scheme. This scheme can further be implemented easily to non-linear and time delay oriented problems.

AMS subject classiﬁcation: Analytical and numerical methods for ODEs and PDE.

Key words: Variable order fractional derivative, L2-1σ formula, Compact difference scheme, Stability, Convergence.

Transverse Hydromagnetic and Media Permeability Effect on Mixed Convective Flow in a Channel Filled by Porous Medium with asymmetric wall heating condition Km. Renu, Ashok Kumar Hamvati Gandan Bahugana Garhwal, India. [email protected], [email protected]

An analytical as well as numerical study of steady, fully developed unidirectional non-Darcy mixed convective flow in a vertical channel with transverse hydromagnetic effect is reported in this article. The fluid in channel is assumed electrically conducted and flow is due buoyancy forces and an external pressure gradient. The non-Darcy Brinkman-Forchheimer extended model is considered to characterized the flow in porous media. Chebyshev spectral collocation method is used to solved the governing equations and get the magnificent agreement with the analytic solution for the special case. The governing parameters for this problem are media permeability (Darcy number,Da, Forchheimer number, F ∗), Hartmann number Gr (M), Heat generating/absorption parameter (H), and mixed convective parameter ( Re ). From the numerical investigation, it is found an unnatural deviation in the velocity, temperature and Nusselt number after the threshold value of positive H (heat generating case). The Nusselt number is a linear function of negative H. Further, on the basis of other fixed controlling parameters, the velocity, temperature and Nusselt number are increasing on increasing M up to certain value and beyond it reducing asymptotically. The flow strength and heat transfer rate decreases on decreasing media permeability on reducing Da and increasing F ∗.

Keywords - Magnetohydrodynamics, Mixed-convection, Non-Darcy Brinkman-Forchheimer extended model, Chebyshev Spectral collocation method, Gauss-Chebyshev quadrature formula.

Classiﬁcation - 76Dxx, 76Fxx, 76Sxx, 76Wxx, 76Rxx

63 Numerical study of entropy generation in porous medium vertical channel subjected to mixed convection Paresh Vyas, Kusum Yadav UOR, Jaipur, Rajasthan, India. [email protected], [email protected]

This paper is aimed to analyze entropy generation in a vertical plate channel filled with porous medium. The flow is modified by buoyancy force, viscous and ohmic dissipations. One wall of the channel is subjected to convective flux whereas other wall is kept at uniform temperature. The governing partial differential equations are reduced to ordinary differential equations by using similarity transformations and the resulting boundary value problem is solved numerically by finite difference scheme. The effects of pertinent parameters on the entropy generation number, global entropy and Bejan number have been reported graphically and discussed.

Keywords: entropy, buoyancy force, porous medium, mixed convection, vertical channel.

Curvelet optimized method for solving partial differential equations on general manifolds Deepika Sharma, Kavita Goyal School of Mathematics, Thapar Institute of Engineering and Technology, Patiala-147004, India Emails: [email protected] [email protected]

In this work, a dynamically adaptive curvelet method has been developed for solving partial differential equations (PDEs) on the general manifolds. Approximation formulae for Laplacian-Beltrami (2) operator is derived using the closest point method. The grid on which the equation is solved is obtained using curvelets. The CPU time taken by the proposed method is compared with the CPU time taken by the closest point method and it is observed that the proposed method performs better. To best of our knowledge the useful properties of curvelet is for the ﬁrst time exploited to solve PDEs on general manifolds. The method is tested on three test problems namely reaction-diffusion equation on a sphere, Schnakenberg model evolving on the surface of ellipsoid and the well-known Fitzhugh-Nagumo equations. The numerical results show that the method can accurately capture the emergence of the localized patterns at all the scales and the node arrangement is accordingly adapted. The convergence of the method has also been veriﬁed.

Keywords; Multiresolution analysis (MRA); Adaptive node arrangement; Curvelet; Numerical method; Partial differential equations.

AMS subject classiﬁcation : 65M99, 35J25

64 Solution of inverse fractional Fisher’s equation by differential quadrature method G. Arora, Pratiksha Lovely Professional University, Punjab, India. [email protected], [email protected] This work is an attempt to solve the inverse problem on fractional Fisher equation. A method is proposed to find the numerical solution of the problem, comprising of Lubich’s approach to discretize the time fractional derivative and differential quadrature method with modified B-spline basis function to approximate the space derivatives A stable numerical solution is obtained for this problem and then a comparison is made with the existing results. The obtained results are presented in form of tables and figures.

keyword: Fractional Fisher’s equation, differential quadrature method, discretization, fractional differential equation, inverse problem.

MSC [2010] : 26A33, 34A08, 35R11, 65L20

Uncertainty propagation using Wiener-Bspline wavelet expansion Navjot Kaur and Kavita Goyal School of Mathematics, Thapar Institute of Engineering and Technology, Patiala-147004, India Emails: nkaur [email protected] [email protected] In this paper, we have constructed a scheme combining generalized Polynomial Chaos (gPC) expansion and B-spline wavelets. To begin with, semi-orthogonal compactly supported B-spline wavelets are constructed for the bounded interval [0, 1] which is used as a PC expansion for possible stochastic processes. To compute the deterministic coefﬁcients of expansion, we have applied both semiorthogonal Galerkin projection and pseudo-spectral projection of uncertain data and the solution variables. Then, to determine the behaviour of the stochastic process, the system of equations obtained from projection are integrated using fourth order Runge-Kutta method. The scheme is illustrated through model problems of real life importance. It has been observed that the wavelet function based expansion shows superior results as compared to scaling function based expansion.

Keywords : Uncertainty quantiﬁcation; B-spline wavelets; generalized Polynomial Chaos (gPC); Stochastic differential equation.

AMS subject classiﬁcation : 65M99, 35J25.

Entropy Generation Analysis For A Micropolar Fluid Flow Due To A Moving Surface Paresh Vyas, Manvi Adha University of Rajasthan, Jaipur, Rajasthan, India. [email protected], [email protected] In this paper, an analytical solution has been developed for micropolar fluid flow in porous medium. The flow is caused by convection due to moving surface and buoyancy. The governing equations are solved by perturbation method. The velocity and thermal fields are used to compute entropy generation. Profiles for entropy generation and bejan number are portrayed and discussed.

Keywords: micropolar ﬂuid ﬂow, porous medium, entropy

65 Mathematical analysis of surface wave propagation in Functionally Graded Material using WKB approximation Sonali Mondal, Sanjeev A. Sahu Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad-826004, India. [email protected], [email protected]

This paper aims to study the Love-type surface waves propagation in a composite structure comprising of functionally graded material (FGM) layer and porous piezoelectric (PP) substrate. The FGM layer is supposed to be under constant tensile initial stress. Wentzel Kramers Brillouin (WKB) approximation technique and variable separable method are used for the wave solutions. Dispersion relations are obtained using suitable boundary conditions for the two cases i.e. electrically open and electrically short case. Numerical example is given in support of findings. Effects of gradient coefficient of the FGM layer, initial tensile stress as well as the width of the FGM layer have been shown graphically. The present study contributes towards designing and optimization of underwater acoustic devices. Mathematics Subject Classification (2010). 74B10 74J15.

Keywords : Functionally Graded Material, Surface waves, WKB approximation, Porous piezoelectricity.

Data bounded WENO reconstructions of high order schemes Ritesh Kumar Dubey, Sabana Parvin Research Institute & Department of Mathematics, SRM Institute of Science and Technology, Chennai, India. [email protected] [email protected]

High order accurate shock capturing Weno schemes are mainly based on high order WENO reconstructions. These reconstructions lack signiﬁcantly in terms of rigor mathematical results except in some particular cases.The main aim of this work is to establish estimates on stability mainly data boundedness of such reconstructions. More precisely, conditions on non-linear WENO weights are provided on the region of a smoothness measurement which is again a function of consecutive gradient ratio, so as to make WENO reconstruction data bounded in that regions. Data boundedness i.e Boundedness of the polynomial reconstruction by its adjacent values, will helps in preservation of extrema in WENO polynomial solution. Numerical results for various test problems are given and compared.

Keywords: Hyperbolic conservation laws, WENO reconstructions, Non-linear weights, Data dependent stability, Data-bounded polynomials.

AMS subject classiﬁcations : 65M06, 65D05, 35L65.

66 Solution of singular fractional Lane-Emden type equations by an analytical technique Anoop Kumar, Seema Department of Mathematics and Statistics School of Basic and Applied Sciences Central University of Punjab, Bathinda, Punjab-151 001, India. Email: [email protected], [email protected].

This paper deals with the approximate solution for the singular fractional Lane-Emden equations using Variational iteration method(VIM). The fractional derivative are described in Caputos sense. The obtained results shows that VIM is effective and convenient.

Keywords: Variational iteration method, Lane- Emden equations, Lagrange multiplier.

AMS subject classiﬁcations : 30C70, 35N99, 35J05.

NUMERICAL APPROACH FOR A COUPLED SYSTEM OF SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEM WITH NON-SMOOTH DATA Aarthika K, V. Shanthi Department of Mathematics, National Institute of Technology, Tiruchirappalli-620 015, Tamilnadu, India. [email protected], [email protected].

The time-dependent weakly coupled linear system of singularly perturbed one-dimensional parabolic convection-diffusion partial differential equation with a discontinuous source term is analyzed in this paper. We build a numerical technique by handling an efﬁcient ﬁnite difference scheme to achieve a reliable approximation of the solution. This method involves an appropriate piecewise-uniform mesh to generate uniformly convergent numerical estimates to the solution. The execution of the linear system successfully experimented which validates the analytical results.

Keywords: convection-diffusion, non-smooth data, piecewise-uniform mesh, singular perturbation, weakly coupled system, uniform convergence

AMS subject classiﬁcations : 35B25, 35R05, 65N06

THRESHOLDING FUNCTIONS INVOLVED IN THE WAVELET BASED DENOISING METHOD Princess Raina and Zaheer Abbas Department of Mathematical Sciences, Baba Ghulam Shah Badshah University Rajouri [email protected]

Every scientist or engineer dealing with the real world data knows very well that, in general, signals do not exist without noise - Gaussian Noise, Impulsive Noise(salt-and- pepper noise or spike noise), Speckle Noise. So to obtain the correct information, the signals hve to be denoised for which there are abundance of methods available in the literature. Among these methods, the wavelet based methods are very effective and have been very successfully applied in many areas of science and technology. The wavelet method involves three steps - a linear forward wavelet transformation, a nonlinear thresholding step and a linear inverse wavelet transform. The second step uses a function, called as thresholding function. Donoho and Johnstone

67 proposed two types of thresholding functions - hard and the soft thresholding functions. However, these thresholding techniques suffer from certain drawbacks and hence a number of thresholding functions have been introduced by the researchers since then. In this paper, we shall present a detailed report on the various threshholding functions available in the literature after presenting various types of noises, wavelet denoising method and a brief introduction to wavelets.

One-dimensional solute migration model with ﬁrst-order production term in semi-inﬁnite porous media Affreen Akhter, Mritunjay Kumar Singh Department of Applied Mathematics, Indian Institute of Technology, Indian School of Mines), Dhanbad 826004, Jharkhand, India. [email protected], [email protected]

In the present study developed an analytical solution for one-dimensional contaminant migration model in homogeneous as well as heterogeneous semi-inﬁnite porous media. The effect of linear isotherm, sink/source term, porosity, density and ﬁrst-order production term are considered into account for obtained the exact analytical solution of contaminant migration. Initially, a not solute-free domain is assumed as a linear combination of the initial source concentration with the effect of zero order production term with space dependent is considered. Temporally dependent linear function is taken in one end of the boundary and other end of the boundary concentration gradient is assumed to be zero. Laplace Integral Transform Technique (LITT) is used to demonstrate the analytical solution. An asymptotically and exponentially decreasing function is considered to illustrate the proposed solutions.

Keywords: Advection, Dispersion, Linear Isotherm, Solute, Analytical Solution.

STUDY OF FRACTIONAL THERMOELASTIC PROBLEM WITH MOVING HEAT SOURCE Jaya Bikram, G.D. Kedar 1Department of Mathematics, RTM Nagpur University, Nagpur, India. [email protected]

In this paper, we have studied one dimensional problem of thermoelastic rod with moving heat source. The heat conduction equation is taken in space-time fractional derivative form. Finite Riez fractional and Caputo derivative are considered in heat conduction. The integral transforms are used to determine the temperature distribution. The numerical results are presented and illustrated graphically.

Keywords: Fractional heat conduction equation, generalized Mittage Lefﬂer function, thermal stresses, fractional thermoelasticity. AMS subject classiﬁcations :

68 Numerical Solution of Fractional Order Non-Conservative Advection-Diffusion Equation Anup Singh and S. Das Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi-221005, India [email protected]

In this article, the Laplace transform method is used to solve the advection- diffusion equation having source or sink term with initial and boundary conditions. The solution profile of normalized field variable for both conservative and non-conservative systems are calculated numerically using the Bellman method and the results are presented through graphs for different particular cases. A comparison of the numerical solution with the existing analytical solution for standard order conservative system clearly exhibits that the method is effective and reliable. The important part of the study is the graphical presentations of the effect of the reaction term on the solution profile for the non-conservative case in the fractional order as well as standard order system. The salient feature of the article is the exhibition of stochastic nature of the considered fractional order model.

Keywords: Advection; Diffusion; Laplace transformation; Non-conservative system

AMS subject classiﬁcations : 35Q35, 35B35, 35B30

Finite element analysis of semilinear time-fractional diffusion equation Dileep Kumar∗a, Sudhakar Chaudharyb, V.V.K Srinivas Kumar∗ ∗ Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India, [email protected] b Department of Mathematics, Institute of Infrastructure Technology Research and Management, Ahmedabad, Gujarat, India, [email protected]

This work is devoted to the finite element analysis of semilinear time-fractional diffusion equation. Grünwald-Letnikov approximation is used to approximate time-fractional derivative. We propose fully- discrete scheme to solve semilinear time-fractional diffusion equation and discuss its existence-uniqueness of solution. For linearizing the nonlinear fully-discrete problem, Newton’s method is used. We derive a priori error estimate for the fully-discrete solution in L2(Ω) norm. Numerical results are presented which confirms the theoretical findings.

Keywords: Fractional diffusion equation, ﬁnite element methods, Newton’s method, Gr¨unwald-Letnikov approximation, Error estimates

AMS subject classiﬁcations : 65N12, 65N30, 35K61

69 Effect of Aspect Ratio on Natural Convective ﬂow in a Rectangular Enclosure Occupied by Anisotropic Porous Medium Ashok Kumar, Ajay Kumar∗1, Km. Renu and M. S. Rawat 1Department of Mathematics, H N B Garhwal University (A Central University) Srinagar -246174, India [email protected], [email protected], [email protected] and [email protected]

A compreherensive numerical study of natural convective flow in rectangular enclosure occupied by anisotropic porous medium using spectral element methods is presented . In this study the main emphasize is given on the effect of aspect ratio on fluid flow and heat transfer rate. The flow is induced by the non- uniform partial heating at the bottom wall and cooling at vertical walls anlong with the adiabatic top wall. The non-uniform partial heating is given on the one third part in middle of the bottom wall and the rest part of the bottom wall is adiabatic. The flow in porous medium is characterized by the non-Darcy model in which media is taken hydrodynamically anisotropic and thermaly isotropic. The numerical results are presented in terms of stream function, isotherm, local as well as average heat transfer rate. The numerical results presented here shows that the heat transfer rate as well as flow streanth is reducing on increasing the aspect ratio. The maximum values stream fuction is also reducing on enhancing the Aspect ratio.

Keywords: Spectral Element Method, Natural Convection, Porous Media, Partial Heating

AMS subject classiﬁcations : Numerical Method in Fluid Mechanics 76Z99, Fluid Mechanics 76D33 & Flows in Porous Media 78A99

NUMERICAL SOLUTION OF DAMPED FORCED OSCILLATOR PROBLEMS BY OPERATIONAL MATRIX OF INTEGRATION Dr. Mithilesh Singh1, Seema Sharma2, Sunil Rawan3 1Rajkiya Engineering College, Churk, Sonbhadra, Uttar Pradesh 2Kanya Gurukul Campus, Haridwar, Uttarakhand 3Gurukula Kangri Vishwavidyalaya, Haridwar,Uttarakhand [email protected],[email protected],[email protected]

In this paper, the operational matrix of integration Bernoulli orthonormal polynomials has been used to determine the numerical solution of damped forced oscillator problem. The integral operator on Bernoulli orthonormal polynomials has been applied to determine the operational matrix of integration. Numeri- cal examples of two different problems of spring are given to illustrate the efﬁciency and accuracy of the method and compared with exact solution.

Keywords: Orthonormal Bernoulli polynomials; Operational matrix; Damped forced oscillator

AMS subject classiﬁcations : 65L05,34B60

70 Flow of a Hydromagnetic Fluid through a porous medium between permeable beds with damping effects. Ravi Kumar and B.G. Prasad Post Graduate Department of Mathematics, Patna University, Patna-800005, India [email protected]; [email protected]

The Navier-Stoke’s equations and Darcy’s law have been used to study the flow of a hydromagnetic fluid under a uniform magnetic field, with suction and injection through the lower and the upper permeable −(n1+in2)t beds. The pressure gradient is taken as e , where n1 and n2 being positive constants. Analytical solutions for the velocity field and the volume flux have been obtained. For n2 = 0 the problem reduces to that discussed by Prasad and Kumar (2011) and for n1 = 0 and porosity of medium tending to zero, the problem reduces to that discussed by Malathy and Srinivasan (2008).

Finite element solution of a problem on coupled thermoelasticity for functionally graded material by two different approaches for time domain Om Namha Shivay∗ and Santwana Mukhopadhyay Department of Mathematical Sciences, IIT (BHU), Varanasi-221005 ∗Corresponding author; [email protected]

The present work is concerned with the modified Green-Lindsay thermoelasticity theory involving strain-rate. This theory has been proposed very recently to modify the Green-Lindsay model of thermoelasticity and overcome its drawback by introducing both temperature and strain-rate terms in the constitutive relations of coupled thermos-mechanics. We consider a problem involving coupled thermo-mechanical interactions inside a functionally graded hollow disk due to a thermal shock applied at the inner boundary which is assumed to be stress free. The material properties of the disk are assumed to change along the radial direction according to a volume fraction rule with a power of non-homogeneity index term. We for- mulate the problem by considering the basic governing equations of GL and modified GL thermoelasticity theories in a unified form to derive a non-linear system of coupled partial differential equations. We solve this system by applying a Galerkin’s approach of FEM for the space domain and derived the time differential system of equations. We apply two different methods; namely, the FE approach and Newmark method to obtain the solution time domain. Further, to show the advantages of the present methods over transFE methods, the CPU time to obtain the solutions by the present formulations are compared with the CPU time to obtain the solution by trans-finite element method. The variation of different physical field variables with space and time have been discussed for different values of the nonhomogeneity index by highlighting the difference in the results under GL model and modified GL model.

Keywords: Coupled Thermoelasticity Theory; Temperature-rate Dependent theory; Functionally Graded Materials (FGM); Finite Element Method; Trans-ﬁnite Element Method

AMS subject classiﬁcations : 74A60

71 Simulation of Heat transfer of Ferro ﬂuid in Cylindrical Micro-Channel Ramesh kumar1∗, Harry Garg2, S.K Dhiman1 1Department of Mechanical Engineering, Birla Institute of Technology Mesra, Ranchi, India, 835215 2Central Scientiﬁc Instruments Organization (CSIR-CSIO), Chandigarh, India, 160030 1,2∗[email protected]; [email protected]; [email protected] ∗CORRESPONDING AUTHOR

In the present work the computational fluid dynamics and differential method is used to study the flow behaviour and heat transfer rate of kerosene based Ferro fluid. The Ferro fluid is mixture of magnetic particles and based fluid therefore treated as mixture phase. The numerical simulation results are obtained in the form of different parameter e.g. Nusselt Number, Reynolds number and heat transfer coefficient.

Keywords: Heat transfer, Ferro ﬂuid, Magnetization, Kelvin body forces

Computational Analysis of Bed of a Mobile Channel Atul Ailawadhi1 Baldev Setia2, 1M.Tech. Student, Department of Civil Engineering, National Institute of Technology, Kurukshetra, Haryana 2Professor, Department of Civil Engineering, National Institute of Technology, Kurukshetra, Haryana [email protected], [email protected]

The process of sediment transport being a topic of utmost importance has been emphasized on, since long. Understanding of the process has undergone significant transformations through laboratory experi- mentations and field observations. However, its solutions through the use of computational tools, though having a wide scope, have not been fully developed. Some significant work in this direction has been reported in the works of Wen Xiong (2017), Shi Liu (2017), ZHU Zhi-wen (2012), etc. In the present paper the transport process has been modelled with the help of a three-dimensional simulation model of a flume with mobile sand bed using the principles of CFD. The data used for simulation was obtained from the experimental works carried out in another study in the laboratory of NIT Kurukshetra. The flow has been simulated by employing the k-? turbulence model and the sand-water coupling being modelled by Euler- Euler model with two-phase flow. Navier-Stokes Equation was solved independently for both the phases and all three- mass, momentum and energy conservation laws were applied. The volume fraction method helped in determining the flux exchange of the sand phase from the bed and has been followed by updating the bed boundary using dynamic meshing technique. The bed levels at various time steps have been obtained using the threshold value of the critical bed shear stresses. The computed values have been compared with the laboratory results and satisfactorily good co-relation has been observed in terms of bed levels and incipience velocity for the fine sand particles as well.

Keywords: CFD-Computational Fluid Dynamics; critical bed shear stress; incipience velocity.

AMS subject classiﬁcations : 35Q35, 35B35, 35B30

72 Numerical study on MHD ﬂow of nanoﬂuid due to a rotating disk with heat generation and partial slip effect V. K. Chaurasiya, Ramayan Singh and Rajat Tripathi Department of Mathematics, National Institute of Technology, Jamshedpur-831014, Jharkhand, India. [email protected]

In the present study, our aim is to study the heat and mass transfer mechanism in Von Krmn swirling flow problem of nanofluid due to a rotating disc with the consideration of magnetic field and heat generation effect. The well-known Buongiorno model exhibits the characteristics of Brownian motion and thermophoresis. Using Karman similarity transformations, the governing partial differential equations (PDEs) are transformed into the non-linear and coupled ordinary differential equations (ODEs) which are then solved by a numerical method for the broad range of the involved parameters. The effects of the magnetic parameter, heat generation parameter and slip coefficients on the heat and mass transfer are discussed. The effect of relavent parameters on the velocities (radial and azimuthal) and temperature distributions is represented by plotting graphs. Expressions of wall skin friction and rate of heat transfer at the surface of the disk are calculated and are presented in the tabular form. It is noted that the velocities (radial and azimuthal) are reduced on increasing the magnetic field and slip coefficient. The temperature of fluid is enhanced for heat generation parameter.

Fibonacci collocation method to solve nonlinear space-time fractional order advection-reaction-diffusion equation Kushal Dhar Dwivedi Department of Mathematical Sciences IIT(BHU), Varanasi-221005, India. [email protected]

In this article, a new algorithm is proposed to solve the fractional order nonlinear one-dimensional solute transport system. The spectral collocation technique is considered with the Fibonacci polynomial as a basic function for approximation. The Fibonacci polynomial is used to obtain derivative in terms of an operational matrix. The proposed algorithm is based on the fact that the terms of the considered problem are approximated through a series expansion of double Fibonacci polynomials and then collocated those on speciﬁc points, which provide a system of non-linear algebraic equations which can be solved using New- tons method. To validate the accuracy of the proposed method, it is applied to solve three different problems having analytical solutions. The comparison of the results through error analysis depicted through tables clearly shows the higher accuracy of order of convergence of the proposed method in less CPU time. The salient feature of the article is the graphical exhibition of the movement of solute concentration for different particular cases due to presence and absence of reaction term when the proposed method is applied to the considered fractional order nonlinear space-time advection-reaction-diffusion model.

Keywords: Fibonacci polynomial, Spectral method, Fractional-order, Diffusion equation, Operational matrix

AMS subject classiﬁcations : 35Q35, 35B35, 35B30

73 Abhishek Verma,N K Singh 1 1Department of Mechanical Enginee Abhishek Verma,N K Singh Department of Mechanical Engineering, NIT Kurukshetra, India. [email protected]

Structures experience various forces exerted over them in routine. Thus they are designed in order to bear them all in certain limits modified with some factors of safety. Wind forces are one of the most important of such forces especially in high rise buildings and tall structures like towers, etc. The same has been simulated through computational technique using CFD based software package Ansys Fluent available in CFD lab of NIT Kurukshetra. In the same process, k-omega SSD model has been used as a standard flow turbulent model. The wind forces have been tracked in the vicinity of the structure at NUMERICAL SIM- ULATION OF FLOW AROUND VARIOUS STRUCTURAL SHAPES the faces treated as walls for the fluid-structure interaction. The study involved comparison of the drag force magnitude and the vortices for various structural shapes feasible for construction. The observation of vortices in the nearby areas and the magnitude of forces has been compared with various other studies and has been found in good accordance with each other.

Keywords: CFD- Computational Fluid Dynamics; Ansys Fluent; Aerodynamics Drag and Lift; Turbulent Flow

Numerical study of inclined stretchable partially heated enclosure ﬁlled with nanoﬂuid Pentyala Srinivasa Rao, Anil Kumar Department of Applied Mathematics, Indian Institute of Technology (ISM) Dhanbad 826004, India, e-mail: [email protected]

In this article, we analysed numerically the heat transfer behaviour of nanofluid due to buoyancy force inside a two-dimensional inclined partially heated stretchable square cavity filled with nanofluid. This model is used to test the performance of stretchable walls and inclination on stream function and temperature field inside the cavity. The proposed governing equations are numerically solved by Implicit alternate direction finite difference method for modified Rayleigh number (103 ≤ Ra ≤ 105), Prandtl number(Pr = 0.1; 0.7; 3), volume fraction(0 ≤ φ ≤ 0.3), stretching parameter (0.01 ≤ τ ≤ 1.2) and inclination. It is found that the convective heat transfer and mean Nusselt number both are effected with variation of stretching parameter and inclination. Therefore, the details of results are graphically presented to show that the effect of stretching parameter on flow strength, heat transfer and mean Nusselt number for laminar flow in the cavity.

Keywords: Nanoﬂuid, Natural convection, Stretching sheet, Moving boundaries, Enclosure. AMS subject classiﬁcations : 76DXX; 76RXX; 80AXX.

74 CP

Keywords:Divided wall distillation, globally linearizing controller, dynamic model, system of ordinary differential equations

AMS subject classiﬁcations : 34B60 (Under ODE: applications)

75 76 DE

Existence of mild solutions for neutral fractional functional integro-differential equations with non instantaneous impulses of order α ∈ (1, 2) Pallavi Bedi, Anoop Kumar Department of Mathematics and Statistics, School of Basic and Applied Sciences, Central University of Punjab, Bathinda, Punjab-151 001, India. [email protected], [email protected].

In this manuscript we establish the existence of mild solutions for neutral fractional functional integro- differential equations with non instantaneous impulses and state dependent delay of order α ∈ (1, 2) of the form. Z t α−1 c α (t − s) Dt u(t) + g s, uρ(s,us) ds = Au(t) + f t, uρ(t,ut), Bu(t) , (0.0.4) 0 Γ(α)

t ∈ (si, ti+1] ⊂ J = [0,T ], i = 0, 1, 2, ..., N.

0 u(t) = φi(t, u(t)); u (t) = ψi(t, u(t)), t ∈ (ti, si], i = 1, 2, ..., N. (0.0.5)

0 u(t) = q1(t); u (t) = q2(t), t ∈ [−d, 0]. (0.3)

c α 0 where Dt denotes the Caputo s fractional derivative of order α ∈ (1, 2) and A : D(A) ⊂ X −→ X is the sectorial operator defined on a complex Banach space X, 0 = t0 = s0 < t1 ≤ t2 < ... < tN ≤ SN ≤ tN+1 = T are prefixed numbers and φi, ψi ∈ C (ti, si] × X; X ∀i = 1, 2, 3..., N. The functions f : J × PCo × X −→ X; g : J × PCo −→ X and ρ : J × PCo −→ [−d, T ] are continuous functions. The history function ut ∈ PCo = C [−d, 0],X and is defined as ut(θ) = u(t + θ), θ ∈ [−d, 0]. The maps 0 0 0 q1(t), q2(t) ∈ PCo and u (t) denotes the derivative of u(t)w.r.t to t and φi, ψi ∈ C (ti, si] × X; X ∀i = 1, 2, 3..., N. R t + The term B(x)(t) = 0 K(t, s)x(s)ds where K ∈ C(D, R ) is the set of all positive functions which are continuous on D = {(t, s) ∈ R2 : 0 ≤ s ≤ t < T }. The existence results are proved with the help of analytic operator functions and fixed point theorems. An example is offered to demonstrate the application of results obtained.

Keywords: Fractional differential equations; non instantaneous impulse conditions; ﬁxed point theorems; state-dependent delay; sectorial operator; analtyic operator functions.

AMS subject classiﬁcations : 34A08, 34K40, 34A37, 26A33, 34A12.

77 CHARACTERIZATION OF POLARIZED SHEAR WAVES IN POROUS-PIEZOELECTRIC MEDIUM OVER A HETEROGENEOUS ELASTIC SUBSTRATE CONTAINING POINT SOURCE SUBHASHIS KARMAKAR*, SANJEEV A. SAHU Dept. of Applied Mathematics, IIT(ISM), Dhanbad, 826004 E-mails: [email protected], [email protected]

In the present study a model consisting of porous piezoelectric layer lying over a heterogeneous half- space has been considered. An analytical approach has been adopted by using Greens function method to solve the inhomogeneous linear differential equations for horizontally polarized shear wave (SH-wave) propagation. Point source, situated at the interface of two media has been considered to inﬂuence the propagation of SH-waves. It is found that the angular frequency of SH wave depends considerably on the material properties (thickness of the layer, elastic parameter, piezoelectric constant, heterogeneity, and porosity). Obtained results are presented graphically to exhibit the inﬂuence of parameters on the phase and group velocity of considered wave. A special case is also presented by reducing the upper layer to a piezoelectric layer. Numerical example has been illustrated by taking Lead Zirconate Titanate (PZT-1, PZT-5H and PZT-7)as the upper layer. Results may be useful to investigate the dispersion characteristics of surface seismic waves in smart materials.

Keywords: Greens function, Porous piezoelectric layer, Point source, SH-wave

AMS subject classiﬁcations : 34L10, 35Q74, 74J15.

Smooth stable manifold for delay equations with arbitrary growth rates Lokesh Singh*, Dhirendra Bahuguna Indian Institute of Technology, Kanpur [email protected]

Invariant manifolds play crucial role in understanding the dynamics of differential equations. Perron and Lyapunov developed a method to get the stable invariant manifold for hyperbolic trajectories with the assumption that solution operator satisfies [uniform] exponential dichotomy. Later on, in 2005 L. Barreira and C. Valls generalized the notion of uniform exponential dichotomy to nonuniform exponential dichotomy for nonhyperbolic trajectories and proved the existence of stable invariant manifold for nonautonomous differential equations. In this talk, first I will give the generalization of nonuniform exponential dichotomy. Later, assuming the generalized nonuniform exponential dichotomy, I will prove the existence of Smooth stable invariant manifold for semiflows generated by a nonautonomous differential equations with infinite delay of the form given by

0 x = Ax(t) + Lxt + f(t, xt), xs = φ, on some Banach space X.

Keywords: Stable Invariant Manifold, Nonautonomous system, Nonuniform Exponential dichotomy

78 Approximation of ﬁxed points and the solution of delay differential equation via new iterative scheme Javid Ali, Faeem Ali Department of Mathematics, Aligarh Muslim University, Aligarh- 202002, India [email protected]

The purpose of this paper is to introduce a new iterative scheme, called F iterative scheme for approximating ﬁxed points of contraction mappings in an arbitrary Banach space. We also prove that F iterative scheme is T -stable and converges faster than the iterative schemes S, Picard-S, Vatan, Thakur-New, M ∗,M and many more. As an application, we approximate the solution of a delay differential equation by using F iterative scheme. Also, we give two numerical examples to support our analytic proofs and illustrate the efﬁciency of purposed iterative scheme.

Keywords: F iterative scheme, contraction mappings, ﬁxed points, convergence results, Banach spaces, delay differential equations

AMS subject classiﬁcations : 47H09, 47H10.

Classiﬁcation of Some Functional Differential Equations with Constant Coefﬁcients to Solvable Lie Algebras Jervin Zen Lobo, Y.S Valaulikar Department of Mathematics, St. Xaviers College, Mapusa, Goa - 403507, India Department of Mathematics, Goa University, Talegao Plateau, Goa - 403206, India [email protected], [email protected]

In this paper we shall apply symmetry analysis to some functional differential equations with constant coefficients. The determining equations of the admitted Lie group are constructed in a manner different from that of the existing literature for delay differential equations. We define the standard Lie bracket and make a complete classification of linear functional differential equations with constant coefficients, to solvable Lie algebras. We also classify some non-linear functional differential equations with constant coefficients, to solvable Lie algebras.

Keywords: Delay differential equations, determining equations, group analysis, neutral differential equations, solvable lie algebras.

Mathematics Subject Classiﬁcation (2010): 17B30, 34K06, 76M60, 58J70.

79 80 MD

Speech analysis based emotion-aware healthcare system Akshita Abrol, Praveen K. Lehana Department of Electronics Jammu University Jammu, J& K, India. Email: [email protected]

One of the most important use of speech communication in health care would be to detect an illness through the analysis of speech of a patient. Glottal analysis of speech constitutes an integral part of clinical interpretation. It helps in determination of emotional expression and its relation to the overall state of the speaker. Emotional or mood disorders such as clinical depression constitute potentially crippling illnesses. Depressive disorders are among the leading causes of disability in the world and have been linked to high rates of suicides. This paper discusses the relation of emotional impairment in speech with the clinical state of the speaker. With the advent of big data oriented wireless technologies including Internet of Things in the next generation networks, emotional health care monitoring speciﬁcally for children and elderly and mentally ill people is receiving considerable attention. Emotional impairment is considered to be one of the hallmark features of Schizophrenic speech which has further been investigated in this paper. It has been concluded that objective classiﬁcation of emotional speech to detect an illness would contribute to person- alized and seamless state-of-the-art health care technologies in near future.

Keywords: Emotional impairment, depression, speech, healthcare, next generation networks.

Analysing the efﬁciency of hexagonal microﬂuidic fuel cell Jyoti Lalotra, Praveen K. Lehana Department of Electronics Jammu University Jammu, J& K, India, Email: [email protected]

The greed of power and energy has blindfolded the human race by triggering the never ending process of depletion of energy resources on earth. It has alarmed to search for alternative sources of energy as a consequence of deteriorating climatic changes and rapid depletion of fossil fuels. It is time to shift to green energy otherwise with the pace of polluting and utilizing natural resources the earth will not sustain more. Besides this, combustion of fossil fuels is slowly changing the composition of atmosphere and it may lead to climatic disasters. Developing alternative renewable and safe resources is a goal of energy engineering to safeguard the future of coming generations. Further, major part environmental pollution is contributed by using conventional techniques for generating electricity. A recent device for tackling these problems is the fuel cell. It is like a battery which generates energy from an electrochemical reaction. Both batteries and fuel cells convert chemical energy into electrical energy. Battery supplies energy only for some specified time, fuel cell has the efficiency to supply energy indefinitely as long as the supply of hydrogen and oxygen is maintained. Fuel cells have higher efficiency than other conventional alternatives. Proton exchange membrane (PEM) based fuel cells use Hydrogen and Oxygen for generating electric voltage. The only by-product of this fuel is the water. In this paper, different microfluidic structures have been investigated for enhancing the efficiency. The investigations have shown that hexagonal structure is a better option for

81 supplying gases and placement fuel cell as arrays.

Keywords: Efﬁciency, PEM, Fuel cell.

Varying Trends of Smart Bandaging Priti Rajput, Praveen K. Lehana Department of Electronics Jammu University Jammu, J& K, India. [email protected]

Living tissue inheritably has the ability of self-healing on getting wounded or misaligned. The healing mechanism starts within seconds of getting injured. The damaged blood vessels send a signal to the blood cell platelets that race together and form a clot to prevent further blood loss. The clots turn to scab with time. The blood vessels open to allow fresh nutrients and oxygen into the wound that speeds healing and white blood vessels joins in to fight infection. Oxygen rich red blood cells arrives at the wound site to build up the damaged tissue. With time the tissue heals completely and is same as it was before. Bodys mechanism takes a fixed time to heal the wound depending upon the conditioning of the wound and immune system. Sometimes it takes more than a year to heal the wound and in that case manmade modes are required to heal the wound at a pace. A wound heals on its own in living tissues unless it becomes chronic. Bandages are a mean to aid healing. The traditional mechanisms of healing the wounds involve examining it by removing the bandages regularly that hinders the healing mechanism. With advancement in technology, the traditional techniques are being replaced by smart methods of wound management. Researchers throughout the world are working on smart bandage technology. The current fast and technological world demands a bandage that needs to be monitored at home for spread of infection or restorative process. Different bandages have been under development in various research institutes across the globe. The smart bandages react to the varying level of oxygen, pH, moisture, or infection at the wounded area. Its great relief to the wounded soldiers in the battle field as it helps to monitor the wound in the absence of doctors. The paper presents a review of various smart bandages developed by different researchers and the prototype of one developed by us using herbal ingredients.

Keywords: Bandages, infection, microﬂuidics, moisture, wounds.

Varying Trends of Smart Bandaging Verasis Kour, Praveen K. Lehana Department of Electronics Jammu University Jammu, J& K, India. [email protected]

Internet of Things (IoT), a novel technology that connects day-to-day objects with the internet, for leveraging the usefulness of data, has started to find its applications in numerous domains including automobile, aviation, healthcare, logistics, manufacturing, industries, transport etc. Additionally, the concept of machine health and status monitoring is being incorporated widely in industrial as well as domestic environment owing to its various advantages such as detection of anomalies, reducing the maintenance costs, eliminating the machinery breakdown, performance optimization, fault detection, and machinery condition monitoring. Techniques such as analysis of machines electromagnetic field signature, noise and vibration signature, current signature, machinerys temperature profile analysis, etc are usually used for monitoring the health of a machine. Integration of machinery maintenance with IoT, would serve as a powerful tool, leading to enhanced capabilities and increased potential, offering wider opportunities in terms of machinery healthcare. Data collection in real time through various sensors and actuators, transfer of this data to the

82 internet, its analysis, extraction of information, exchange of this information among wider section of machinery via the internet as well as machine-to-machine communication opens doors for investigating novel techniques for efﬁcient machinery healthcare using IoT.

Keywords: Internet of Things, machinery health monitoring, condition monitoring, status monitoring, machine learning.

Impact of rigid surface on dispersion and damping characteristic of Love-type wave propagating in an initially stressed piezoelectric layer Shishir Gupta, Rachaita Dutta, Soumik Das Department of Applied Mathematics IIT(ISM), Dhanbad, India. shishir [email protected], [email protected] [email protected] In this article, an investigation regarding Love-type wave propagation through initially stressed piezoelectric layer deals with the effect of overlying fractured porous medium saturated with liquid and an underlying ﬁber-reinforced half-space with initial stress. Two individual cases for the interposed piezoelectric layer have been analysed; one is electrically open circuit and other one is electrically short circuit. More- over, behaviour of Love-type wave propagation is determined in presence of rigid surface. Displacement components are acquired by using method of separation of variables. Frequency equations in both cases appear to be complex which are dissociated into real and imaginary expressions indicating dispersion and attenuation properties of Love-type wave respectively. Impact of total porosity, volume fraction of fractures, initial stress parameters, dielectric constants, piezoelectricity parameter and ﬁber-reinforcement parameters are portrayed through various graphical implementations. In every graph of dispersion and attenuation, both cases of piezoelectricity have been compared by taking variation in above discussed parameters.

Keywords: Love-type wave, fractured porosity, initial stress, piezoelectricity, ﬁber-reinforcement.

AMS subject classiﬁcations : 74J15.

Discharge characteristic of multi-cycle triangular labyrinth weir Subhankar Das, Parthajit Roy M.Tech Student, National Institute of Technology, Silchar, India, Assistant Professor, National Institute of Technology, Silchar, India . [email protected], parthajit [email protected] In rivers and canals, weirs are the most common and useful water retaining structure and it allows ow of excess water over its crest. The flow over a weir is largely governed by its length and head over the weir crest. Labyrinth weirs are more efficient than conventional weirs due to increased crest length. In this study, ANSYS Fluent software was used to investigate the effect of included angle and number of cycles on the performance of the labyrinth weir. The grid-independent study was done to check the sensitivity of the mesh in the numerical results and a fine mesh (10mm) was selected for numerical study with 1.874% discretization error. Standard k- turbulence model was used to consider the turbulence effects. The volume of fluid (VOF) model was used for tracking the free surface. The discharge performance obtained from the computational fluid dynamic (CFD) analysis shows a good agreement with the results obtained through laboratory experiments. Comparison of the results shows that single cycle labyrinth weirs are more efficient than multi-cycle labyrinth weir. The results encouraged to use of ANSYS Fluent and computational fluid dynamics methods for further analysis.

Keywords: Labyrinth weir, Computational ﬂuid dynamics, Free surface ﬂow

83 Geometrical changes in journal bearing due to piezoelectric actuators Aayush Trivedia, Wolfgang Seemannb, and Mohammad Talhac a,cSchool of Engineering, Indian Institute of Technology Mandi, HP, 175005, India bInstitut ﬁir Technische Mechanik, Karlsruher Institut ﬁir Technologie, Karlsruhe, Germany. [email protected]

In this study, an analytical solution to ﬁnd the deformation in journal bearing under the action of piezoelectric actuators is obtained. Here, steel AISI 4340 is taken as bearing material and PZT-4 is taken as material for piezoelectric actuators. The differential equation relating the bending moment and displacement of thin curved rods are coupled with the stress-charge form of reverse piezoelectric effect. The two- dimensional plane stress condition is used for additional simplifcation of the model. The present results are validated with the ﬁnite element method-based package COMSOL multiphysics. It has been observed that the results tend to converge at higher values of radius to thickness ratios of the bearing, and lower breadth to height ratios of the piezoelectric actuators. Keywords: Piezoelectric actuators, journal bearing, FEM, deformation analysis, thin curved rods.

Studies of Hyperloop Vehicle for Transportation: A Review V.K. Srivastav1, Aditya Priyanka1, Abhishek Kumar1, Shudhanshu Kumar1, Anand Raj1 1 Mothari College of Engineering Mothari, Bihar

This paper presents a critical review of hyperloop vehicle that is useful for next generation transportation. The Hyperloop concept is proposed as quicker, cheaper alternative to high speed rail.It is seen from literature that computational simulation play an important role to simulate the air flow around the capsule considering the effect of the axial-flow air compressor and the levitation cushion on the air flow. In this paper, provide all the boundary conditions in tabular from that was used in both computational as well as experimental papers. The present work will also compare different hyperloop models used by the .

Keywords: Hyperloop, Computational Fluid Dynamics(CFD), Tube transport system, Tube Vehi- cle.

Studies of Hyperloop Vehicle for Transportation: A Review V.K. Srivastav1, Aditya Priyanka1, Abhishek Kumar1, Shudhanshu Kumar1, Anand Raj1 1 Motihari College of Engineering Motihari, Bihar. [email protected]

Keywords: Hyperloop, Computational Fluid Dynamics(CFD), Tube transport system, Tube Vehi- cle.

84 Bivariate Bernstein-Schurer-Stancu type GBS operators based on (p, q)-integers Mohd. Ahasan and M. Mursaleen Department of mathematics, Aligarh Muslim University, Aligarh-202002, India. [email protected]

The purpose of this paper is to construct (p, q)-analogue of Bernstein-Schurer-Stancu type GBS operators for approximating B-continuous functions and establish the uniform convergence theorem for these new operators. In terms of mixed modulus of continuity, the rate of convergence for these new operators is determined. Keywords: (p, q)-integers, Bernstein-Schurer-Stancu type operators, GBS operators, Positive linear operators, B-continuous functions and Mixed modulus of continuity. AMS subject classiﬁcations : 41A10, 41A25 and 41A36

Uncertain eigenvalue analysis of ﬁnite element modelled functionally gradient arches Mohammad Amir#, Mohammad Talha∗ School of Engineering, Indian Institute of Technology Mandi, HP, 175005, India. [email protected]

Uncertain eigenvalue analysis of the functionally gradient arches using finite element method is presented in this study. The system parameters, like material properties of each constituent’s material, and volume fraction index are taken as uncorrelated random input variables. These random input variables are modelled using first order perturbation technique (FOPT). An efficient stochastic finite element model based on FOPT is developed and employed to examine the second-order statistics (mean and standard deviation (SD)) of the vibration response of the functionally graded curved beams. The efficacy and accuracy of the present formulation is ensured by comparing the present results with results obtained by Monte Carlo sampling (MSC) method. Few new numerical results are calculated which can be used as benchmark for further studies.

Keywords:First order perturbation technique (FOPT), Stochastic ﬁnite element method (SFEM), Monte Carlo sampling (MSC) method, Uncertainty quantiﬁcation, FG curved beams etc.

Studies of Hyperloop Vehicle for Transportation: A Review V.K. Srivastav1, Aditya Priyanka1, Abhishek Kumar1, Shudhanshu Kumar1, Anand Raj1 1 Motihari College of Engineering Motihari, Bihar

Keywords: Hyperloop, Computational Fluid Dynamics(CFD), Tube transport system, Tube Vehi- cle.

85 Second-order statistics of the elastic buckling of skewed functionally gradient plates with material uncertainties Mohammed Shakir1∗ and Mohammad Talha 2 1,2School of Engineering, Indian Institute of Technology, Mandi-175005, India. [email protected]

The manufacturing of functionally gradient materials (FGM) are highly sophisticated due to large number of design parameters are associated in the fabrication processes and led uncertainties in their material properties. These materials offer immense capability and excellent performance in a vast range of engineering applications, such as in Aerospace, Mechanical and Marine engineering. In the present work, second-order statistics of the elastic buckling of skewed FGM plates has been studied subjected to uniaxial and biaxial loadings. The effective material properties of the gradient plates are customized in the transverse direction only according to the power-law distribution of the volume fractions of the constituents. Youngs moduli, Poissons ratios and power law index are considered as random system parameters. The plate kine- matics is based on Reddys higher order shear deformation theory. An efficient C0 stochastic finite element based on the first-order perturbation technique (FOPT) is proposed to access second-order statistics of the buckling load. Convergence and comparison studies have been performed to assess the adequacy of the present mathematical model. The numerical results have been presented with different system parameters, and its dispersion with respect to various random variables. Keywords: Functionally gradient skewed plate, material uncertainty, finite element method, first- order perturbation technique.

Estimation of mean rainfalls using Geostatistical techniques in Kabul River Basin, Afghanistan Shamsullah Sultani1 1∗, Arun Goel 2 1.∗ Master of technology student, Civil Engineering Department, National Institute of Technology Kurukshetra, Haryana-136119, India email: [email protected] 2. Professor, Civil Engineering Department, National Institute of Technology, Kurukshetra, Haryana-136119, India. drarun [email protected]

Rainfall is part of Hydrologic cycle, it has much spatial variability over the basin. The amount of rainfall received at periodic intervals like weeks, months, seasons and years indicates its distribution. Rainfall magnitude varies in space and time so it needs suitable implements to anticipate accurate mean values in space as well as in time. Many rainfall estimation methods are available in which interpolation is very helpful. In this study two geostatistical interpolation techniques (kriging, cokriging) and one deterministic technique IDW (Inverse Distance Weighted) have been applied to estimate monthly (January-December), seasonal (spring, summer, autumn, winter) and annual rainfalls in Kabul River Basin, Afghanistan. It is highly mountainous region ranging from 400-6000 meters lies in the northeast quarter of the country and includes eight provinces and the capital Kabul city. Interpolation has been conducted for a period of 1958- 1978 by using ground data obtained from 15 rain gauge stations. Among all three methods, kriging and cokriging produced good results with nearly zero Mean Error (ME) and least Root Mean Square Error (RMSE) where IDW produced results out of limits. Elevation was used as co-variant in cokriging method to improve the results but due to lack of density of stations and correlation between two variables (rainfall and elevation) the outcome has not been improved considerably. Geographic Information System (GIS) with geostatistical extension has been used. Keywords: Kabul River Basin, Afghanistan, Kriging, Cokriging, IDW, Rainfall

86 COMPUTATIONAL AND EXPERIMENTAL STUDIES OF FLY ASH BRICK V. K. Srivastav1 R. K. Singh1, R. Kumar1, A Kumar1, A. K. Chhotu1, A. R. Paul2 1Motihari College of Engineering, Motihari, Bihar 2Motilal Nehru National Institute of Technology Allahabad, Prayagraj, Uttar Pradesh. [email protected]

This paper is studied compressive strength of fly ash bricks both computationally as well as experimentally. The experiment is performed in the Department of Civil Engineering, MCE Motihari. However, computational work is done using Ansys Software. In India, It is seen that more than 50the electricity is produced through coal fired thermal power plant and in this process about 65of the total coal produced in India. Electricity formed through coal is cheap, reliable and most widely used in India but it has much bad effect on human health because of disposal interns of fly ash. Fly ash are deposited in big pond which creates a bad impact on environment as well as human health so recycling of fly ash is the biggest challenge for researchers. The fly ash bricks are reasonably lighter in weight and stronger as comparison of clay bricks. Therefore, the present work is focused on to compute the strength of fly ash bricks. The computed result shows good agreement with the experimental results. Keywords : Fly ash brick, Finite Element Method, Strength of Fly Ash Bricks.

3D Stress Analysis of Reinforced Concrete Sleeper Tejas M. Gondhalekar1 and S. K. Panigrahi2 1PG student, Defence Institute of Advanced Technology (Deemed University), Pune, India 2Professor, Defence Institute of Advanced Technology (Deemed University), Pune, India. [email protected]

Railway is a significant and widespread means of conveyance for people and cargo. One of the most vital part of railway system is sleeper which is rested beneath the rails and supports the track. Its function is to transfer and mete out the loads coming from the above to the ballast and later to ground and transversely secure the rails to maintain the correct gauge-width. Till this time many materials have been used in making of sleeper. But the problems such as biodegradation and splitting in timber, corrosion and electrical conductivity for steel and excessive brittleness for cast iron constricted the use of these materials and shifted the research towards the concrete sleeper. In the railway track structure the strength of sleeper determines the type of locomotive to be used over that path. In near future due to time constraints the railway transportation is going to increase. This condition necessitates the appropriate use of exhaustive analysis for the design analysis of railway sleeper. In the present work, a three dimensional stress analysis has been carried out with the help of ANSYS under the vertical static loading finite element based model. Also it is shown that the loading capacity of the sleeper is strongly affected by the longitudinal reinforcements. This paper also presents the study of dynamic behaviour of railway sleeper subjected to transient concentrated load. This load also takes into account the dynamic load factor for the analysis. Numerical finite element solutions to this fundamental transient dynamic load problem are obtained.

Keywords: Concrete sleeper, ﬁnite element method, Stress correlation, longitudinal reinforcement, transient load, impact factor.

87 Triple-diffusive convection with more realistic two temperature model for nanoﬂuids Urvashi Gupta Dr. S.S. Bhatnagar University Institute of Chemical Engineering and Technology, Panjab University, Chandigarh-160014, India. dr urvashi [email protected].

Triple-diffusive convection for nanofluids in which driving density differences are caused due to the variation of three diffusive components nanoparticles, heat and solute; using more realistic two temperature model i.e. separate thermal energy equations for fluid phase and nanoparticle phase has been investigated by making use of the method of superposition of basic possible modes and single term Galerkin method. The system of partial differential equations are converted to ordinary differential equations which are solved using Normal mode technique and single term Galerkin method. A complex system with local thermal nonequilibrium (LTNE) effects among the fluid and particle phases along with the Brownian motion and thermophoresis to account for nanoparticles has been considered. Numerical computations are carried out using the software Mathematica for permissible range of values of various parameters under consideration. An additional solute concentration equation supplements the conservation equations due to the existence of solute which introduces two additional non-dimensional parameters whereas three additional parameters came into existence due to the consideration of LTNE effects. The critical Rayleigh number remains constant for smaller values of interphase heat transfer parameter, decreases for intermediate values and again reaches to constant value for higher values of the parameter. It implies that stabilizing and destabilizing effects are observed only for the intermediate values of Nield parameter. Further Lewis Number, solute Rayleigh number, concentration Rayleigh number and modified diffusivity ratio tend to enhance the impact of destabilization while modified thermal diffusivity ratio inhibit the destabilizing impact though very slightly. Keywords: Nanofluids; Binary convection; Thermophoresis; Rayleigh Bnard problem; Local thermal non-equilibrium.

Design of wind turbine for the application of desalination process S.Ramachandran1 ,S.Vasanth1∗,S.Devarajan1 1Sri Sai Ram Engineering college,Chennai,Tamilnadu, India. [email protected]

In this research work , design, development and performance analysis of a cluster of wind energy based desalination units are proposed. The design shall be made for small scale of desalinated water per day per unit depending on the requirement. Various blade profiles shall be used to optimize the profles. NACA 0015 airfoil profile is used for the windmill which rotates at a very low speed of 1m/s. A high pressure reciprocating pump is used for pumping seawater through high pressure RO Membrane to get the desalinated water. Glass Fiber Reinforced Polypropylene material is used for the fabrication of the blades of the windmill. This would facilitate the studies on the performance various NACA profiles for optimized for various units of desalination depending on the sites. Also suitable materials for the fabrication of the turbine blades would be suggested based on Carbon Fiber Reinforced Plastics (CFRP) blades. CFD was used for the analysis of air ow and the results are presented. Keywords: CFD, NACA 0015, wind turbine, turbine blades, desalination

88 Enhancing software reliability through the generalized inﬂection S-shaped failure rate with testing effort in growth model Vishal Pradhan1, Ajay Kumar1 and Joydip Dhar1 1Department of Applied Sciences, ABV-Indian Institute of Information Technology and Management Gwalior-474015 India. [email protected]

This paper presents a methodology for estimating software reliability incorporating generalized inflection S-shaped failure rate using the testing effort function. Here, testing effort function is involved in failure rate function. Further, we consider both the perfect and imperfect debugging environment. In the reliability growth model, we also develop for both detection and correction process. A software project manager has to conduct testing activities economically and efficiently with limited testing resources due to the limited delivery, cost, and testing effort. Therefore, we reckon the testing effort function (TEF) that classified as detection and correction effort. Here, we estimate the values of unknown parameters of TEF, detection model and correction model separately by least square estimation. All the measures of performance are calculated using real data-sets and comparison made between the existing model and the proposed models. It is observed that the performance of the proposed model is better. Release time of software with a trade-off between reliability and cost is also calculated. Keywords: Software reliability growth model (SRGM), Testing effort function, Optimal release, Generalized Inflection S-shaped.

Experimental and numerical study of blood ow characteristics in human coronary artery with plaque Wasim Saliha,Pradeep Kumarb, Parmod Kumarc, Mohammad Talha ∗ School of Engineering, Indian Institute of Technology Mandi, HP, 175005, India. [email protected]

The present study investigates the characteristics of blood ow in human coronary artery with plaque using combined experimental and numerical simulations. A computational model has been developed using uid ow solver AN- SYS Fluent. In the present framework blood is considered as Newtonian uid and the corresponding governing equations are discretised using Eulerian approach based ﬁnite volume method. Two different set of plaque con gurations with concentrated, and uniform distribution along the circumference have been simulated. The computational results have been veriﬁed with the corresponding experiments. It has been observed that central line velocity in axial direction takes sudden rise at the throat of uniformly distributed plaque. Similar rise of central line velocity is observed in concentrated plaque also, however it has shown localised uctuation in velocity magnitude at other locations as well. The detailed study will include the comparison of velocity and pressure distributions for wide range of ow conditions. These analysis have great importance in designing of cardiovascular devices in order to circumvent heart related problems. Keywords: Coronary Artery, Plaque, Blood ow, Numerical simulations, Expermental Setup

89 Numerical simulation for radial stress and displacement of jet engine Exhaust pipe Y.K.Singh1 and S. K. Panigrahi2 1 PG student, Defence Institute of Advanced Technology (Deemed University), Pune, India 2 Professor, Defence Institute of Advanced Technology (Deemed University), Pune, India.

Exhaust pipe is used in exhaust section of jet engine. Exhaust pipe is subjected to combined mechanical and thermal loads like internal pressure and high temperature of exhaust gases and the ambient conditions. Exhaust pipe is actually thick cylindrical angular pipe cross-section. In this research a basic 3D model that can be used to study the effects of temperature and internal pressure of exhaust gases on the radial stress distributions and radial displacement ﬁelds in circular angled cylinder. A linear distribution of temperature through the thickness is considered for thick cylindrical exhaust pipe. The analytical model will be studied on the thick walled cylinders theory using FEA. Results of this analytical study will be used to study the critical cross-section of Exhaust pipe. To sustain very temperature of exhaust gases the material should have good thermal resistance properties, the common alloy used for Exhaust pipe is stainless steel or INCONEL 718. The alloy with good results will suggested for manufacturing of exhaust pipe.

Keywords: Exhaust pipe, Jet engine, Thermal and Mechanical load 3D-Finite Element Analysis (FEA), Steel and Nickel Alloys.

90 MM

Cross-diffusion induced Turing and non-Turing patterns in Rosenzweig-MacArthur model Nayana Mukherjee Department of Mathematics & Statistics, IIT Kanpur, Kanpur, India. [email protected]

Pattern formation is widely studied in spatio-temporal prey-predator models with only selfdiffusion terms. Models with cross-diffusion, in addition to self-diffusion terms, take care of the situation in which presence, absence, abundance or scarcity of one species affect the movement of a population of another species in a given domain. In this talk, I consider cross-diffusion induced pattern formation in a prey- predator model with Rosenzweig-MacArthur type reaction kinetics in a one-dimensional spatial domain. Spatio-temporal prey-predator model with Rosenzweig-MacArthur type reaction kinetics and self-diffusion is unable to generate Turing patterns rather produces traveling wave, periodic traveling wave, modulated periodic traveling wave and spatio-temporal chaotic patterns. However, addition of density dependent cross- diffusion term leads to satisfaction of Turing instability conditions and generation of stationary Turing patterns. Also the dynamics of the patterns generated by the self-diffusion model is preserved. Further, cross-diffusion term affects the speed of traveling waves produced in the self-diffusion model. The focus of my talk is to investigate the bifurcation of traveling wave solution into Turing patterns and transition of one pattern into another in the presence of cross-diffusion.

Keywords: cross-diffusion; Turing instability; traveling wave; stationary pattern; spatiotemporal chaos; bifurcation.

AMS subject classiﬁcations : 35B36, 37L10, 34D30

POPULATION DYNAMIC CONSEQUENCES OF FEARFUL PREY IN A SPATIOTEMPORAL PREDATOR-PREY SYSTEM Ranjit Kumar Upadhyay, Swati Mishra Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad, Jharkhand-826004, India. [email protected]

of responses spanning the physiology, morphology, ontogeny and the behavior of scared organisms. To explore the effect of fear and its dynamic consequences, we have formulated a predator-prey model with the cost of fear in prey reproduction term. Spatial movement of species in one and two dimensions have been considered for the better understanding of the model system dynamics. Stability analysis, Hopf-bifurcation, direction and stability of bifurcating periodic solutions have been studied. Conditions for Turing pattern formation have been established through diffusion-driven instability. The existence of both supercritical and subcritical Hopf-bifurcations have been investigated by numerical simulations. Various Turing patterns are

91 presented and found that the change in the level of fear and diffusion coefﬁcients alter these structures signiﬁcantly. Holes and holes-stripes mixed type of ecologically realistic patterns are observed for small values of fear and relative increase in the level of fear may reduce the overall population size.

Keywords: Predator-prey interactions, Fear effect, Anti-predator response, Stability, Hopf-bifurcation, Pattern formation.

AMS subject classiﬁcations : 92D25, 92D40, 34A34, 35B36.

A Mathematical model for Human Papillomavirus and its impact on cervical cancer in India R Praveen Kumar, K Murugesana aDepartment of Mathematics National Institute of Technology, Tiruchirappalli-620015, Tamil Nadu, India. [email protected], nand [email protected], [email protected]

In this article, We develop a mathematical model for Human Papillomavirus transmission and its impact of cervical cancer in India. The existence and stability of the system are discussed. Using next-generation matrix method we obtain the basic reproduction number and show that the disease-free equilibrium becomes asymptotically stable under certain conditions. Finally, numerical simulation is applied to study the system.

Keywords:

Predator-prey system: Herd behavior and anti-predator traits contribute in enriching the evolution of stronger prey defence Rajat Kaushik, Sandip Banerjee Indian Institute of Technology Roorkee, Roorkee-247447, Uttrakhand (India). [email protected], [email protected]

In this paper, we focus on a stage-structured predator-prey model with herd behavior and anti-predator phenomenon. It is our purpose in this paper to illustrate the combined effects of the herd behavior and anti-predator strategy on the predator-prey ecology. By constructing the model equations, we present an analytical study of the system and the occurrence of Hopf and transcritical bifurcation for the anti-predator rate as a bifurcation parameter. Finally, the extinction of predators is explored via numerical simulations, that is the prey population survives successfully in the habitat due to their defense mechanism and anti- predator behavior.

Keywords: Herd behavior; Anti-predator strategy; Asymptotically stable; Hopf bifurcation.

92 Effect of porosities with and without fractures on propagation of SH wave: case wise study Shishir Gupta, Soumik Das, Rachaita Dutta [email protected]

A mathematical model is presented in this article for a comparative study between SH wave propagation in corrugated double porous stratum and that in corrugated single porous medium. Among the two cases, the first one depicts propagation of SH wave through fluid saturated creased porous layer having fractures and resting over a half-space consisting of void pores, whereas in the second case, only liquid saturated porous medium with corrugated boundaries has been considered for SH wave propagation while the half-space remains unchanged. Effects of heterogeneity and anisotropy on wave propagation have been analysed in both cases. Method of separation of variable is used to attain the complex frequency equation according as both cases and those complex dispersion relations are separated into two equations that illustrate dispersion and attenuation properties of SH wave. In every graphical execution, double porosity and single porosity have been compared in terms of wave velocity and attenuation coefficient by using variation of several parameters such as inhomogeneity parameter, position parameter, fluctuation parameter, flatness parameter, total porosity and ratio of rigidity modulus.

Dynamical behaviour of predator-prey systems with variation of Allee function in predator’s growth: Structural sensitivity analysis Deeptajyoti Sen Department of Mathematics and Statistics, IIT Kanpur, Kanpur, India. [email protected]

Prey-predator model with Allee effect in predator’s growth is one of the important research areas in Mathematical biology now a days as it capture some realistic features . This due to the fact that the complex behaviour exhibited by such model also agrees with the experiment held in laboratory. Allee effect occurs in predator’s growth due to various reasons such as lack of consumption of prey, unable to ﬁnd suitable mate etc. Now Allee functions, to model Allee effect on predator growth, have certain properties. In this talk, I wish to compare the dynamics of the system with genreal form of Allee function in predator’s growth. The ay a results will be elaborated with some base functions (such as b+y ,, b (1 − exp(−by)) etc) which has same charecteristic as that of Allee function and perturb them slightly by variation of the parameters. Finally I will present the comparative dynamics resulting from the variation of Allee effect function using structural sensitivity analysis . Keywords: Allee effect, functional response, Bifurcation,Stability MSC: 37L10, 34D30

93 MODELLING OF SEDIMENT EXCLUDER Dibyendu Das1, N.K.Tiwari2,Subodh Ranjan vajesnayee3 1,2,3Department of Civil Engineering,National Institute of Technology Kurukshetra, Haryana. e-mail: [email protected]

A tunnel-type sediment excluder is commonly used at the headwork of a canal for preventing excess sediment from entering the off-taking canal. In such excluders the sediment-laden water, which flows mainly near the bed, is made to flow through the tunnels provided at the canal bed. It may be then discharged back into the river downstream through the undersluice bays. Comparatively sediment-free water in the top layers is allowed to enter the canal. Presently, the only hydraulic principle utilized in its design is that energy loss is kept to a minimum and a minimum velocity of flow is ensured through the tunnel for the nondeposition of the coarse material.This paper attempts to investigate the efficiency of Sediment Excluder at the headwork of canal by decreasing the number of tunnels,change the sediment size and concentrations. By using the AI based modelling techniques,It has been predicted the efficiency of Sediment Excluder by changing number of tunnels,sediment size and concentrations. In this study the output value of efficiency of Sediment Excluder due to decreasing number of tunnels were predicted using ANN and Gaussian Process by taking Parameters like Geometric mean,discharge,concentrations,efficiency of Sediment Excluder. Keywords: ANN, Gaussian process, Geometric mean, concentrations, Efficiency of Sediment Ex- cluder.

FEM Analysis of Adhesively Bonded Composite Patches V.Divakar1 and S. K. PanigrahiR2 1,2Defence Institute of Advanced Technology (Deemed University), Pune, India. [email protected]

Structural adhesives as a medium of bonding as opposed to other securing strategies such as screws and rivets presents various favorable circumstances like lightweight, larger area of contact. This paper deals with the numerical simulation of these adhesively bonded joints in Aluminum 2024 T3 Aircraft Skin having corrosion grind out at center with pessimistic load range. The FEA model generated is validated by comparing the stresses with the results available in the literature. Variation due to change in Material Anisotropy, Patch shape, Ply Sequences and its impact on stresses are studied in this paper. The Maximum Principal stress in Aluminium Skin, Critical Failure Indices in Adhesive as well as Composite Doubler due to out-of plane bending moments are determined for different ﬁbres made of Graphite and Boron in Epoxy matrix with varied laminate stacking sequence viz. unidirectional [0]13, cross-ply [(0/90)3(0)(90/0)3] and angle ply [(+45/45)3(0)(45/45)3]. Results show that judicious choice of the above mentioned parameters in Composite Doubler Patch allows for a noteworthy increment of the static strength compared to previously corrosion damaged aluminium plate and facilitates a cost-effective method to broaden the lives of the part. Keywords: Composite materials; Composite repair; Patch repair; adhesively bonded patch; Car- bon ﬁbre reinforced polymer; Failure indices.

94 Solving a variational inclusion problem with its corresponding resolvent equation problem involving XOR-Operation 1Rais Ahmad,2Javid Iqbal,2Shakeel Ahmed,1Saddam Husain 1Department of Mathematics Aligarh Muslim University, Aligarh 202002, India. 2Department of Mathematical Sciences Baba Ghulam Shah Badshah University Rajouri,Jammu and Kashmir-185234, India. [email protected]

In this paper, we consider a variational inclusion problem involving XOR-operation with its resolvent equation problem involving XOR-operation. We suggest separate iterative algorithms for solving both the problems. The existence and convergence results are proved for variational inclusion problem and for corresponding resolvent equation problem in ordered Hilbert spaces. We claim that results of this paper are new and reﬁnement of previously known results Keywords: Resolvent, Existence, Convergence, XOR-Operation, Equation. AMS Subject Classiﬁcation: 49J40, 47H06, 49J53, 47J20.

Modeling and Simulation of Nylon Liner Shaped Charge Jets Yadav Ombir Singh∗, Nimje S.V and Choudha P.K. [email protected]

Warhead an integral part of ammunition, which causes desired damage to the target by converting the energy released by rapid decomposition (detonation) of explosives into desired forms like temperature, blast, K.E. to projectiles etc. by a suitable warhead mechanism. Warheads can be designed for omni- directional effects or for a particular direction i.e. directed energy warheads. Shaped charges belong to class of directed energy warheads and are usually employed in defeating armoured protected targets. A shaped charge is a cylinder of explosives with an empty cavity with a liner (mostly of metals) on one side and a detonator at the other end of it. When the explosive is detonated, a high pressure is generated which causes the solid liner to collapse and form a jet with velocities in the range of 10 km/s. When this high energy jet strikes a metal plate, a deep cavity is formed and that is more effective as compared to other conventional ammunitions against an armoured target. But the conventional shaped charges have some disadvantages i.e. they are easily detectable by metal detectors and activate the modern Explosive Reactive Armours (ERA) resulting into failure of its intended task. Also they are very heavy, thus are not suitable for hand held ﬁred ammunitions. Hence, in the current research nylon is used as a substitute for the traditional metal liners to achieve short comings of the traditional metal liner shaped charges. Exhaustive numerical simulations have been carried out for nylon liner shaped charge using commercial software Autodyn. The objective of research work is to assess the properties of the nylon jet and to create an empirical relationship to forecast the jet length, by changing the crucial dimensional parameters such as length-to-diameter ratio, charge diameter, liner thickness, cone angle and stand-off-distance. Results indicate that maximum jet tip velocity can be achieved for 80 cone angle and liner thickness (2-2.5 mm). Keywords: Warhead, shaped charge, detonation, ERA.

95 Modeling the impact of sanitation and awareness on the spread of infectious diseases Rajanish Kumar Raia, A.K. Misraa∗, Yasuhiro takeuchib aDepartment of Mathematics, Institute of Science, Banaras Hindu University, Varanasi - 221005, India. bCollege of Science and Engineering, Department of Physics and Mathematics, Aoyama Gakuin University, Kanagawa 252-5258, Japan. [email protected]

Sanitation and awareness programs play a fundamental role and are much effective public health interventions to control the spread of infectious diseases. In this paper, a nonlinear mathematical model for the control of infectious diseases, such as typhoid fever is proposed and analyzed by considering budget required for sanitation and awareness programs as a dynamic variable. It is assumed that the budget allocation regarding the protection against the disease to warn people and for sanitation increases logistically and its per-capita growth rate increases with the increase in number of infected individuals. In the model formulation, it is assumed that the susceptible individuals contract infection through the direct contact with infected individuals as well as indirectly through bacteria shed in the environment. It is further assumed that a fraction of budget is used to warn people via propagating awareness whereas the remaining part is used for sanitation to reduce the density of bacteria. The condition when budget should spend on sanitation/awareness to reduce the number of infected individuals is obtained. Model analysis reveals that the sanitation and awareness programs have capability to reduce the epidemic threshold and thus control the spread of infection. However, delay in providing funds destabilizes the system and may cause stability switches through Hopf-bifurcation. Numerical simulations are also carried out to support analytical ﬁnd- ings. Keywords: Budget allocation, Delay, Sanitation, Awareness, Hopf-bifurcation, Stability switch. MSC (2010): 34A34, 34D20, 34C23, 92D30.

Modelling of scour around Cylindrical Piers M Vignesh1, Subodh Ranjan Vajesnayee2, and N K Tiwari3 1,2,3Department of Civil Engineering, National Institute of Technology Kurukshetra, India, e-mail: [email protected]

Sediment erosion in the region of bridge piers is a potential hazard to the safety of bridges. The presence of various roughness heights on the surface of bridge piers was explored as a scour countermeasure. Labo- ratory experiments under clear-water conditions are to be conducted using different roughness heights with different pier diameters under different ow water depths. The test program included a case of a smooth pier to determine the performance of the roughened piers and to benchmark some existing formulas of local scour around bridge piers. This paper attempts to investigate the scouring around the piers by increasing the roughness. By using the AI- based modelling technique, It has been predicted the scouring around the cylindrical piers by changing the roughness and dimension of piers. In this study, the output value of scour around piers due to increasing its surface roughness were predicted using ANN and Gaussian Process by taking parameters like mean size(d50), velocity, etc. Keywords: ANN, Gaussian process, scouring, piers, roughness, D50.

96 NA

Continuous wavelet transform of Schwartz tempered distributions in S0(Rn) Jay Singh Maurya Department of Mathematical Sciences Indian Institute of Technology (B.H.U.) Varanasi 221005, India. Email : [email protected]

In this paper we deﬁne a continuous wavelet transform of a Schwartz tempered distribution f ∈ S0(Rn) with wavelet kernel ψ ∈ S(Rn) and derive the corresponding wavelet inversion formula interpreting convergence in the weak topology of S0(Rn) It turns out that the wavelet transform of a constant distribution is zero and our wavelet inversion formula is not true for constant distribution, but it is true for a non-constant distribution which is not equal to the sum of a non-constant distribution with a non-zero constant distribution.

Keywords: Distributions; Generalized functions; Distribution space; Wavelet transform of generalized functions.

Continuous fractional wave packet system in Sobolev type space Manab Kundu and Akhilesh Prasad Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad-826004, India. [email protected], apr [email protected]

In this paper we introduce continuous fractional wave packet system in Sobolev type space and investigate their orthogonal properties. The sufﬁcient condition for such system to be orthogonal in Sobolev type space is investigated. Mathematics Subject Classiﬁcation (2010). 42C40 42C15.

Keywords: Wave packet transform, Fractional Fourier transform, Orthogonal wave packet, Frame.

97 Cechˇ proximity relation and Rough Set Theory Pankaj Kumar Singh, Surabhi Tiwari Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Prayagraj211004, India. [email protected], [email protected]

Zdzisław Pawlak used equivalence relation on a non-empty set to introduce approximation spaces during the early 1980s to classify objects employing attributes of information systems. It is an extension of set theory for the study of intelligent systems characterized by insufficient and incomplete information. Topol- ogy is a strong root of constructs that can be helpful to enrich the original model of approximation spaces. The study of proximal relation between two sets with respect to their approximations and its application in image analysis has been discussed in [James Peters et al., 2013], which motivates us to establish proximal relation between two sets using their respective upper approximations in approximation space. This paper introduces Cechˇ closure spaces on rough sets by preferring the approach of proximity relation on approximation spaces and studies its compactification. Some results have been proved on this advanced nearness structure named Cechˇ rough proximity. Examples well support the study. Since this approach identifies the nearness between sets in approximation space, therefore it may prove to be a better tool for such studies in the field of information science, artificial intelligence, computer science, pattern recognition, etc. 2010 AMS subject classification: 54A05, 54A10, 54C60, 54E05, 54E17.

Keywords: Rough sets, proximity spaces, closure space.

A Comparative Study of Monitor function in Mesh Reconstruction Prabhat Mishra and Ritesh Kumar Dubey SRM IST, Kattankulathur. [email protected], [email protected]

In this work new monitor functions are proposed using exponential function and compared with the existing prevailing monitor function in mesh adaptation algorithms for discontinuous flow problems. Dif- ferent time integration methods are used to analyze the non-oscillatory property of the adapted mesh. It is found that monitor functions designed by exponential function is efficient and yields the solution in less number iterations of the mesh adaptation algorithm. Mathematics Subject Classification(2010): 35L65, 65M50.

Keywords: Adaptive Mesh Reconstruction, Monitor Functions, Conservative scheme.

98 Studies of Hyperloop Vehicle for Transportation: A Review V.K. Srivastav1, Aditya Priyanka1, Abhishek Kumar1, Shudhanshu Kumar1, Anand Raj1 1 Motihari College of Engineering Motihari, Bihar. [email protected]

Keywords: Hyperloop, Computational Fluid Dynamics(CFD), Tube transport system, Tube Vehi- cle.

MESHLESS METHOD FOR THE NUMERICAL SOLUTION OF SPACE AND TIME FRACTIONAL WAVE EQUATION Hitesh Bansu ,Sushil Kumar S. V. National Institute of Technology, India. [email protected]

In this article, we have considered a space and time fractional wave equation (STFWE), that is obtained from the classical wave equation by replacing the space and time derivatives with a generalized (Caputo) derivative of fractional order. Furthermore, we have proposed novel and independent discretization for space and time fractional wave equations (STFWE) using two different basis functions namely, radial basis functions(in space) and Chebyshev polynomials(in time). The proposed numerical scheme is truly meshless and therefore is capable to manage both space and time fractional derivatives concurrently with appropriate initial and boundary conditions. Finally, we have included few numerical examples to conﬁrm accuracy and performance of the proposed scheme. Keywords: Radial basis function , Chebyshev Polynomial , Meshless method , Kronecker product , Fractional Wave equation AMS subject classiﬁcations : 26A33 65M70 65N35 35R11 65D25 35L05

Semi-Analytical Solution of Fractional Convection-Dispersion Equation by using Conformable Derivative Approach and Homotopy Analysis Method Manish Chaudhary∗, Rohit Kumar and Mritunjay Kumar Singh Department of Applied Mathematics Indian Institute of Technology (Indian School of Mines) Dhanbad - 826004 (Jharkhand). [email protected]

In this present study, semi-analytical solutions for one-dimensional fractional convection dispersion equation (FCDE) were developed. The concept of conformable fractional derivative of order α ∈ (0, 1] is considered along with Ho- motopy analysis method (HAM) for prediction of pollutant contamination in groundwater in different porous structures. Initially, the domain is considered to be pollutant free. The effect of retardation factor, seepage velocity and generalized dispersion coefﬁcient are included in the solution of convection dispersion model. In this work, problem is formulated for various types of spatial

99 dependency of velocity and dispersion coefﬁcient, that dealt with more realistic phenomenon of the pollutant transport in groundwater. FCDE is transformed into the standard convection-dispersion equation with conformal fractional derivative of order α = 1 and compared with the corresponding numerical solution obtained by Crank-Nicolson Scheme. It shows a good agreement in both solutions. It is observed that the pollutant concentration shows variation with conformable derivative of order α. Hence, this investigation helps to interpret the accurate description of time dependent behaviour of contaminant transport in porous structure.

Keywords: Pollutant transport, convection, retardation factor, conformable Mathematics subject classiﬁcation: 76Rxx, 76Sxx

Convergence Analysis of New Hybrid Scheme for Singularly Perturbed Parabolic Problems with Interior Layers Mr. Narendra Singh Yadav1 Dr. Kaushik Mukherjee Department of Mathematics, Indian Institute of Space Science and Technology, Trivandrum-695547, India. [email protected]

This article deals with a class of singularly perturbed parabolic convection- diffusion problems with discontinuous convection coefﬁcient. To obtain better numerical approximation for solving this class of problems, the time derivative is discretized by the backward-Euler method on uniform mesh and for the spatial discretization, a new hybrid ﬁnite difference scheme is proposed utilizing a layer resolving piecewise-uniform Shishkin mesh. We prove that the proposed scheme is uniformly convergent in the discrete supremum norm; and almost second-order accurate in space, regardless of the larger and the smaller values of the parameter ”. Finally, extensive numerical experiments are conducted to verify the theoretical results and also to demonstrate the accuracy of the proposed scheme. Keywords: singularly perturbed parabolic problem, interior layer, numerical scheme, piecewise- uniform Shishkin mesh, uniform convergence

Fractional calculus for k-Mittag-Lefﬂer function of two variables with the kernel F3 Owais Khan Department of Applied Mathematics, A.M.U., Aligarh. [email protected]

In this paper, we introduced k-Mittag-Lefﬂer function of two variables and investigated its fractional integrals and derivatives using Marichev-Saigo-Maeda operators with the kernel F 3. Besides, we obtain certain interesting particular cases of derived results considering speciﬁc values of the parameters.

100 Approximation By Modiﬁed Lupas Operators Based On (p, q)Integers M. Mursaleena, Zaheer Abbasb and Mohd Qasimb aDepartment of Mathematics, Aligarh Muslim University, Aligarh- 202002, India bDepartment of Mathematics, Baba Ghulam Shah Badshah University, Rajouri-185234, JK,India. [email protected]

The purpose of this paper is to introduce a new modification of Lupas operators in the frame of post quantum setting and to investigate their approximation properties. First using the relations between q- calculus and post quantum calculus, the post quantum analogue of operators constructed will be linear and positive but will not follow Korovkins theorem. Hence a new modification of q-Lupas operators is constructed which will preserve test functions. Based on these modification of operators, approximation properties have been investigated. Further, the rate of convergence of operators by mean of modulus of continuity and functions belonging to the Lipschitz class as well as Peetres K-functional are studied.

Keywords: Lupas operators; Post quantum analogue; q analogue; Peetres K-functional; Ko- rovkins type theorem; Convergence theorems. AMS subject classiﬁcations :47H09, 47H10.

HURWITZ-LERCH ZETA FUNCTION AND SOLUTION OF FRACTIONAL KINETIC EQUATION Raghib Nadeem Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh202002, India. [email protected]

In the present article, we aim to extend the Hurwitz-Lerch Zeta function φδ,µ;ν(ξ, s, v; p) in terms of extended Beta function. We also study basic properties of this extended Hurwitz-Lerch Zeta function which comprises various integral formulas, derivative formula, Mellin transform, generating relation. Fractional kinetic equation for extended Hurwitz-Lerch Zeta function are also obtained from application perspec- tive. Further, we obtain certain interesting relations in the form of particular cases.

Computation of fractional integrals and derivatives for the product of Mathieu-type series and generalized Mittag-Lefﬂer function Mohd Saif Department of Applied Mathematics, A.M.U., Aligarh. [email protected]

Fractional calculus, in allowing integrals and derivatives of any positive order (the term ’fractional’ kept only for historical reasons), can be considered a branch of mathematical physics which mainly deals with integro-differential equations, where integrals are of convolution form with weakly singular kernels of power law type. In recent decades fractional calculus has won more and more interest in applications in several ﬁelds of applied sciences. In this line, our main object to investigate image formulas of generalized fractional hypergeometric operators involving the product of Mathieu type series and generalized Mittag- Lefﬂer function. We also consider some interesting special cases of derived results by specializing suitable value of the parameters..

101 Turing patterns in a Prey-Predator Model with Hassell-Varley Functional Response Vikas Kumar and Nitu Kumari School of Basic Sciences, Indian Institute of Technology Mandi, India. Emails: [email protected], [email protected]

In this article, we have studied a Prey Predator system with Hassell-Varley Functional Response. We assume fear in prey due to predator, the birth rate of prey population decreases. Further, we study the spatially extended model system to understand the impact of fear on the dynamics. Pattern formation phenomenon is illustrated via numerical simulation, which shows that different diffusive coefﬁcients give differ Turing patterns. Further, we observe the impact of fear in both temporal and spatiotemporal model systems.

Keywords: Hassell-Varley fuctional response, fear, local stability, Turing instability, Turing patterns.

AMS subject classiﬁcations : 92C15, 92B05, 35B36.

102 Invited Talks

103 104 Author Index

A Kumar1, 87 Anil Kumar , 74 A. Chakrabarti, 57 Anita Devi Thakur , 25 A. K. Chhotu1, 87 Anjali Jaiswal, 41 A. R. Paul2, 87 Ankit Parwaliya and Ajay kumar, 39 A.K. Misraa∗, 96 Ankita R Devdhara, 46 A.S.V. Ravi Kanth, 33 Anoop Kumar, 67, 77 Aarthika K, 67 Anuj Kataria1 and Baldev Setia2 , 55 Aayush Trivedia, 84 Anuj kumar1, 56 Abdul Haq, 50 Anup Singh, 69 ABDUR RAHEEM, 22 Anuraj Singh, 39 Abhilasha, 31 Anuraj Singh , 14 ABHISHAKE; GILLES BLANCHARD AND Aradhana Bandekar, 41 PETER MATHE, 22 Arun Goel, 58 Abhishek Kumar1, 84, 85, 99 Arun Goel 2 , 86 Abhishek Verma, 74 Arvind Kumar Gupta, 13, 59 ABU SUFIAN, 49 Ashish Kumar, 49 Adeeba Umar1, 52 Ashok Kumar, 63 Aditya Priyanka1, 84, 85, 99 ATIYA PERVEEN, 46 Affreen Akhter, 68 Atul Ailawadhi, 72 Aftab Alam, 45 B S Lakshmi and S S Phulsagar, 25 Ajay Kumar, 50, 70 B.U Raja Ramakrishnaa and N.Ramanan, 51 Ajay Kumar1 and Joydip Dhar1, 89 B.V. Rathish Kumar, 3, 36 Ajit Kumar, 48, 75 Baljinder Kour, 42 Akanksha Verma, 33 Bidhan Chandra Sardar, 15 Akash Bhavsar, 29 Akhilesh Nautiyal, 57 C. S. Mathpati, 48, 75 Akmal Raza, 43 Akshita Abrol, 81 D. Bahuguna, 41 Amandeep Kaur, 57 Deepak Goyal, 39 Amar Deep and Deepmala, 61 Deepika Sharma, 64 Amar Shrivastava and Paritosh Mahata, 56 Deeptajyoti Sen, 93 Amit kumar, 38 Devendra Kumar, 12, 24 1 Amit Parmar, 31 Dibyendu Das , 94 Anand Raj1, 84, 85, 99 Dileep Kumar, 69 ANANTA K. MAJEE, 20 Dr. RAMANABABU KALIGATLA, 35 and Mohammad Talhac, 84 Dwijendra Narain Pandey, 63 3 and N K Tiwari , 96 Faeem Ali, 79 and Nand kumar Tiwari, 38 and Subodh Ranjan Vajesnayee3, 51 G. Arora, 65 Anil , 14 G. Nath, 39 Anil Kumar, 39, 48 G.D. Kedar, 68

105 G.D. Veerappa Gowda , 2 Manish Chaudhary and Mritunjay Kumar Singh, Gaurav Saxena, 21 44 Ghanshyam G. Tejani, 26 Manish Chaudhary∗, 99 Giriraj Methi And Anil Kumar, 27 MANISHA, 35 Gnana Bhaskar Tenali , 1 Manisha Chowdhury, 36 Gomathi Bhavani, 21 Manoj Kumar, 33 Manvi Adha, 65 Harendra Kumar and Santwana Mukhopadhyay, Md Ataur Rahman Khan, 16 40 Md Hasanuzzaman and Waleed M. Alfaqih, 47 Harish Kumar , 8 Md Mansur Alam, 43 Harshita Tiwari, 44 Md Tarikul Islam, 16 Himanshu Rathore, 18 Md. Mehedi Hasan Bhuiyan., 16 Hira Fatima, 46 Md. Moniruzzaman Bhuyan, 16 Hitesh Bansu , 99 Mohammad Amir#, 85 Hitesh K. Singh and Dwijendra N. Pandey, 40 Mohammad Anwar Hossain, 16 Mohammad Arif and Mohammad Imdad, 45 Indresh Yadav, 17 Mohammad Imdad , 47 Jagdev Singh, 24 Mohammad Sajid, 2 ∗ JAIME H. ORTEGA, 5 Mohammad Talha , 85 1∗ 2 Jay Singh Maurya, 97 Mohammed Shakir and Mohammad Talha , Jaya Bikram, 68 86 Jervin Zen Lobo, 79 Mohd Saif , 101 Jyoti Lalotra, 81 Mohd. Ahasan and M. Mursaleen , 85 Jyoti Sharma, 19 Mr. Narendra Singh Yadav1 Dr. Kaushik Mukherjee , 100 K Murugesana, 92 Mridula Sharma1, 58 K. Balaje , 16 Mritunjay Kumar Singh, 68 K. Singh, 42 Kamaljeet, 21 N K Singh , 74 Kamlesh K. Pankaj, 53 N. Sukavanam , 50 Kapil Kumar Sharma, 4 N.K.Tiwari2, 51, 94 Kavita Goyal, 64 Nand Kumar Tiwari, 37, 54, 61 KM. Luxmi, 37, 54, 61 Navjot Kaur and Kavita Goyal, 65 Km. Renu, 63 Navnit Jha, 9 Krishna Kumar Singh2 , 56 Nayana Mukherjee, 91 Kunwer Singh Mathur, 30 Neetu Garg, 33 Kushal Dhar Dwivedi, 73 Nilesh Kumar Thakur and Archana Ojha , 21 Kusum Yadav, 64 Nimje S.V and Choudha P.K., 95 Nityananda Roy, 1 Lokesh Singh, 78 Om Namha Shivay, 71 1 M Vignesh , 96 Owais Khan , 100 M. Balachandar, 24 M. Krishna Prasad, 18 P. Dhanumjaya, 16 M. Mursaleena, 101 Pallavi Bedi, 77 M. Senthilvelan, 10 Pankaj Kumar Singh, 98 M. Tanveer , 16 Pankaj Kumar Tiwari, 27 M.Balachandar, 51 Paresh Vyas, 64, 65 Mahima Poonia, 42 Parmod Kumarc, 89 Manab Kundu and Akhilesh Prasad, 97 Partha Sarathi Mandal, 11 Manali Kokare, 48, 75 Parthajit Roy, 83

106 Pentyala Srinivasa Rao, 74 Roshan Lal, 5 Pooja Gupta, 35 S. C. Martha, 57 Poornesh Kumar Koorata1, 12 2 Prabhat Mishra and Ritesh Kumar Dubey, 98 S. Kumar , 7 1 2 S. S. Jogwar, 48, 75 Pradeep kumar and Baldev Setia , 58 1 Pradeep Kumarb, 89 S.C.S. Rao , 7 S.Devarajan1, 88 Pragya Shukla, 45 1 Prashant Pandey, 34 S.Ramachandran , 88 S.Sundar, 1 Pratibha Verma and Manoj Kumar, 62 ∗ Pratiksha, 65 S.Vasanth1 , 88 Praveen K. Lehana, 81, 82 Sabana Parvin, 66 Praveen K. Maurya Manoj K. Rajpoot , 50 Sachin Kumar, 36, 42 Prawal Sinha, 4 Sandip Banerjee, 92 Preeti, 34 Sandip Rakshit, 32 Prerna Singh, 47 Sangeeta Kumari, 38 Princess Raina and Zaheer Abbas, 67 Sanjeev A. Sahu, 37, 66 Priti Rajput, 82 Sanjeev A. Sahu1, 53 Prof. A. K. Nandkumaran, 1 Sanjeev Anand Sahu, 53 Sanket Tikare, 29 R Praveen Kumar, 92 Sapna Sharma, 11 R. Kumar1, 87 Sarika, 17 R. N. Saraswat, 52 Sarita Nandal, 63 R. P. Gupta , 8 Saroj Kumar Sahani, 9 R. ROY AND R. K. JANA , 59 Sarvjeet Singh, 54 Rachaita Dutta, 83, 93 Seema, 67 RAGHIB NADEEM , 101 Shahna, 17 Rahul Kumar Chaturvedi, 41 Shamsullah Sultani1 1∗, 86 Rais Ahmad , 95 Sheetal Dharmatti , 32 Rajanish Kumar Raia, 96 Shilpee S. Saxena, 21 Rajat Kaushik, 92 Shilpee Srivastava, 15 Rajeev, 7 Shishir Gupta, 83, 93 Rajesh Pandey , 8 Shobhit Kumar Srivastava, 51 Rajib Haloi and Subha Pal, 30 Shreeta Kumari, 53 Raju K. George, 3 Shudhanshu Kumar1, 84, 85, 99 Rakesh Choudhary and Shalini Jain, 26 Siddharth Sonkar1, 51 Rakesh Kumar, 52 Somveer Singh, 44 Ramakanta Meher , 45 Sonal Nirwal, 37 Ramesh kumar, 72 Sonali Mondal, 66 Randhir Singh Baghel, 28 Soumik Das, 83, 93 Ranjit Kumar, 47 Srinivasa Rao Nadiminti1; N. Gayathri Devi1 Ranjit Kumar Upadhyay, 91 and A.Kandasamy, 10 Ravi Kumar, 71 Subhankar Das, 83 RENU CHAUDHARY, 20 SUBHASHIS KARMAKAR, 78 Renuka Rai, 23 Subodh Ranjan Vajesnayee, 37, 54, 61 Rishabh Daal , 7 Subodh ranjan vajesnayee, 38 Rishikesh Yadav , 45 Subodh Ranjan Vajesnayee2, 96 Ritesh Kumar Dubey, 66 Subodh Ranjan vajesnayee3, 94 Rohit K. Singla, 54 Sultan Singh, 39 Rohit Kumar and Mritunjay Kumar Singh , 99 Suman Goyal, 53 Rohit Tripathi; G. N. Tiwari; Deepak Sharma, 28 Sumeeta Singh, 39 RohitKumar, 44 Sumit Kumar Vishwakarma, 13

107 Sunil Rawan, 70 27 Sunil Sharma, 57 V. R. Lakshmi Gorty, 19 Surabhi Tiwari, 98 V. Shanthi, 67 Sushil Kumar, 99 V.Divakar1 and S. K. PanigrahiR2, 94 Swati Mishra, 91 V.K. Srivastav1, 84, 85, 99 Syed Veena Sharma , 32 Abbas, 62 Verasis Kour, 82 Vikas Kumar and Nitu Kumari , 102 Talat Sultana and Arshad Khan, 17 Vishal Pradhan1, 89 1 2 Tejas M. Gondhalekar and S. K. Panigrahi , 87 Vishnu Narayan Mishra, 45, 46 Thomas Goetz , 1 Tripti Midha, 59 Wasim Saliha, 89 Wolfgang Seemannb, 84 UMESH, 35 URVASHI ARORA AND N.SUKAVANAM, 23 Y. S. Valaulikar , 41 Urvashi Gupta , 88 Y.K.Singh1 and S. K. Panigrahi2 , 90 Yadav Ombir Singh∗, 95 V. K. Chaurasiya, 73 Yasuhiro takeuchib, 96 V. K. Srivastav1 R. K. Singh1, 87 V. K. Srivastav; Aditya Priyanka; Abhishek ZAHEER ABBAS, 23 Kumar; Shudhanshu Kumar; Anand Raj., Zaheer Abbasb and Mohd Qasimb, 101

108