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Solitary and Periodic Traveling Wave Solutions of Nonlinear Partial Differential Equations in Mathematical Physics

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Solitary and Periodic Traveling Wave Solutions of Nonlinear Partial Differential Equations in Mathematical Physics Solitary and Periodic Traveling Wave Solutions of Nonlinear Partial Differential Equations in Mathematical Physics By ZAHIDUL ISLAM Student No.: 112703P Registration No.: 04652, Session: 2011-2012 MASTER OF PHILOSOPHY IN MATHEMATICS Department of Mathematics DHAKA UNIVERSITY OF ENGINEERING & TECHNOLOGY, GAZIPUR Solitary and Periodic Traveling Wave Solutions of Nonlinear Partial Differential Equations in Mathematical Physics The thesis submitted to the Department of Mathematics, Dhaka University of Engineering & Technology, Gazipur in partial fulfillment of the requirement for the award of the degree of MASTER OF PHILOSOPHY IN MATHEMATICS By ZAHIDUL ISLAM Student No.: 112703P Registration No.: 04652, Session: 2011-2012 Supervisor Prof. Dr. Md. Abu Naim Sheikh Department of Mathematics Dhaka University of Engineering & Technology, Gazipur ii The thesis entitled Solitary and Periodic Traveling Wave Solutions of Nonlinear Partial Differential Equations in Mathematical Physics Submitted by ZAHIDUL ISLAM Student No.: 112703P, Registration No.: 04652, Session: 2011-2012 a student of M.Phil. (Mathematics) has been accepted as satisfactory in partial fulfillment for the degree of Master of Philosophy in Mathematics On February 2017, BOARD OF EXAMINERS 1. Prof. Dr. Md. Abu Naim Sheikh Supervisor Dept. of Mathematics Chairman Dhaka University of Engineering & Technology, Gazipur 2. Head Member Dept. of Mathematics (Ex-Office) Dhaka University of Engineering & Technology, Gazipur 3. Prof. Dr. Md. Mahmud Alam Member Dept. of Mathematics Dhaka University of Engineering & Technology, Gazipur 4. Prof. Dr. Md. Shirazul Hoque Mollah Member Dept. of Mathematics Dhaka University of Engineering & Technology, Gazipur 5. Dr. Harun-Or-Roshid Member Associate Professor, Dept. of Mathematics (External) Pabna University of Science and Technology, Pabna-6600 iii DEDICATED To My Parents iv Abstract The nonlinear partial differential equations are very significant due to their wide-ranging of applications. These are frequently used to describe many physical problems of plasma physics, diffusion process, geo-chemistry, protein chemistry, chemically reactive materials, mathematical biology, ecology (models of population growth), physics of the heat flow, solid state physics, biology, meteorology, electricity and optical fibers etc. In the modern days of science the nonlinear phenomena is one of the most active fields of research seems to be occur in numerous branches of science and engineering. The main ingredient of it is the nonlinear partial differential equation. The nonlinear partial differential equations are habitually used to reveal the motion of isolated waves. By using the improved (G/ G) -expansion method, we obtained some travelling wave solutions of well-known nonlinear Sobolev type partial differential equations, namely, the Benney- Luke equation. We show that the improved (G/ G) -expansion method is a useful, reliable and concise method to solve these types of equations. Finally, we have investigated the traveling wave solutions of the nonlinear partial differential equations, namely, combined KdV-mKdV equation via modified rational exponential method. We have shown that this method is concise, reliable and efficient to solve these types of nonlinear partial differential equations. v Author’s Declaration This is to certify that the work presented in this thesis is the outcome of the investigation carried out by the author under the supervision of Prof. Dr. Md. Abu Naim Sheikh, Department of Mathematics, Dhaka University of Engineering & Technology, Gazipur in partial fulfillment of the requirements of the degree of Master of Philosophy in Mathematics at Dhaka University of Engineering & Technology, Gazipur and that it has not been submitted anywhere for the award of any degree or diploma. Author vi CERTIFICATE I have the pleasure to certify that the Master of Philosophy thesis entitled “Solitary and Periodic Traveling Wave Solutions of Nonlinear Partial Differential Equations in Mathematical Physics” submitted by Mr. Zahidul Islam in partial fulfillment of the requirement of the degree of Master of Philosophy in Mathematics, Dhaka University of Engineering & Technology, Gazipur, Bangladesh has been completed under my supervision. I believe that the research work is an original one and it has not been submitted elsewhere for any kind of degree or diploma. I wish him a bright future and every success in life. ( Prof. Dr. Md. Abu Naim Sheikh ) Supervisor Department of Mathematics Dhaka University of Engineering & Technology, Gazipur vii Acknowledgements I am most indebted to the grace of “Allah” who guides entire humanity awards knowledge, truth and eternal joys. This is the best opportunity to thanks my supervisor Prof. Dr. Md. Abu Naim Sheikh, Department of Mathematics, Dhaka University of Engineering & Technology, Gazipur and to express my utter gratitude to him. He inspired me to take the present topic and then, guided me at every step of the research. My special thanks are due to Prof. Dr. Md. Azmal Hossain, Prof. Dr. Md. Mahmud Alam, Prof. Dr. Md. Shirazul Hoque Mollah and Prof. Dr. Most. Nasrin Akhter, Department of Mathematics, Dhaka University of Engineering & Technology, Gazipur for providing me all sorts of cooperation and assistance. I would also like to extend my thanks to all the faculty members of the Department of Mathematics, Dhaka University of Engineering & Technology, Gazipur for their help and encouragement. Finally, I wish to express my deep regards to my parents and all other family members and friends for their constant cooperation and motivation. Their sincerest wishes for me have played a very important role in the study. ( Zahidul Islam ) viii CONTENTS Page No. Board of Examiners iii Abstract v Author’s Declaration vi Acknowledgements viii List of Figures xii-xiv Chapter 1: Introduction 1 – 4 1.1 Overview 1 1.2 Historical Background 1 1.3 Literature Review 2 1.4 Objectives 4 1.5 The Proposal 4 Chapter 2: Basic Definitions and Discussion 5 – 40 2.1 Introduction 5 2.2 Differential Equation 5 2.3 History of Differential Equation 5 2.4 Ordinary differential equations 7 2.5 Partial differential equations 7 2.6 Linear differential equations 8 2.7 Non-linear differential equations 8 2.8 Quasi-linear differential equation 9 2.9 Applications of Differential Equations 9 2.10 Some applied Field of Differential Equations 10 2.10.1 Physics 10 2.10.2 Classical mechanics 11 2.10.3 Electrodynamics 11 ix 2.10.4 General relativity 11 2.10.5 Quantum mechanics 11 2.10.6 Biology 12 2.10.7 Predator-prey equations 12 2.10.8 Chemistry 12 2.10.9 Economics 12 2.11 Wave 12 2.12 Formation of Wave 13 2.13 Particle Interaction of Wave 14 2.14 Mechanism of Wave Transports Energy and Not 15 Matter 2.15 Wave equation 17 2.16 Wave forms 18 2.17 Amplitude and modulation of Wave 19 2.18 Wind wave 20 2.19 Seismic wave 21 2.20 Shock wave 22 2.21 Electromagnetic waves 23 2.22 Gravity wave 24 2.23 The generation of ocean waves by wind 25 2.24 Traveling Wave 27 2.25 A solitary wave is shallow water 28 2.26 John Scott Russell and the solitary wave 29 x 2.27 Rogue wave 32 2.28 Completely Integrable Shallow Water Wave 34 Equations 2.29 Soliton 35 2.29.1 Explanation of Soliton 36 2.29.2 Classification of solitons 38 2.30 The solitary wave menagerie 38 Chapter 3: Exact Traveling Wave Solutions to Benney-Luke 41 – 56 Equation 3.1 Introduction 41 3.2 Description of the improved (G′/G)-expansion method 42 3.3 Traveling Wave Solution of Benney-Luke Equation 44 3.4 Result and discussion 49 3.5 Conclusion 56 Chapter 4: Modified Rational Exponential Method for Exact 57- 69 Traveling Wave Solutions of the Combined KdV- mKdV Equation 4.1 Introduction 57 4.2 Direct Modified Exponential Function method 58 4.3 Traveling wave solution of the combined KdV-mKdV 59 equation 4.4 Results and Discussion 63 4.5 Conclusion 69 Reference: 70- 73 xi List of Figures Page No. Fig.-2.1 Formation of Wave in a medium 13 Fig.-2.2 Wave Passing in a medium 14 Fig.-2.3 Wave with Wavelength λ, can be measured between any two 17 corresponding points on a waveform Fig.-2.4 Graph for 2 wavelength, green wave traverse to the right while 18 blue wave transverse left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves.note that f(x,t) + g(x,t) = u(x,t) Fig.-2.5 Several Shape of Wave 19 Fig.-2.6 Amplitude and modulation of Wave 20 Fig.-2.7 Wind Wave 21 Fig.-2.8 Seismic wave 22 Fig.-2.9 Shock waves 23 Fig.-2.10 Electromagnetic waves 24 Fig.-2.11 Atmospheric gravity waves as seen from space 25 Fig.-2.12 Graph of ocean waves generated by wind 26 Fig.-2.13 Physical Model of Travelling Wave 27 Fig.-2.14 John Scott Russell. 30 Fig.-2.15 Recreation of a solitary wave on the Scott Russell Aqueduct on 31 the Union Canal. Photograph courtesy of Heriot-Watt University. Fig.-2.16 Rogue wave 33 Fig.-2.17 Coordinate frame and periodic wave on the surface of water. 34 Fig.-2.18 Solitary wave in a laboratory wave channel 35 xii Fig.-2.19 A hyperbolic secant (sech) envelope soliton for water waves. 36 The blue line is the carrier signal, while the red line is the envelope soliton. Fig.-3.1(a) 3D Shape of equation (3.3.20) 51 Fig.-3.1(b) 2D Shape of equation (3.3.20) for the same values of the 51 parameters of Fig.- 3.1(a) Fig.- 3.2(a) 3D Shape of equation (3.3.21) 52 Fig.- 3.2(b) 2D Shape of equation (3.3.21) for the same values of the 52 parameters of Fig.- 3.2(a) Fig.- 3.3(a) 3D Shape of equation (3.3.24) 53 Fig.- 3.3(b) 2D Shape of equation (3.3.24) for the same values
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