DOCSLIB.ORG

Modified Variation of Parameters Method for Solving System of Second-Order Nonlinear Boundary Value Problem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Modified Variation of Parameters Method for Solving System of Second-Order Nonlinear Boundary Value Problem International Journal of the Physical Sciences Vol. 5(16), pp. 2426-2431, 4 December, 2010 Available online at http://www.academicjournals.org/IJPS ISSN 1992 - 1950 ©2010 Academic Journals Full Length Research Paper Modified variation of parameters method for solving system of second-order nonlinear boundary value problem Muhammad Aslam Noor1,2*, Khalida Inayat Noor1, Asif Waheed1 and Eisa-Al-Said2 1Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan. 2Department of Mathematics, College of Science, King Saud University, Riyadh, Saudi Arabia. Accepted 1 December, 2010 In this paper, we use the modified variation of parameters method, which is an elegant coupling of variation of parameters method and Adomian’s decomposition method, for solving the solution system of nonlinear boundary value problems associated with obstacle, contact and unilateral problems. The results are calculated in terms of series with easily computable components. The suggested method is applied without any discretization, transformation and restrictive assumptions. To illustrate the implementation and efficiency of the proposed method, an example is given. Key words: Variation of parameters method, Adomian’s polynomials, system of nonlinear boundary value problems. INTRODUCTION In this paper, we consider the following system of been studied extensively. To the best of our knowledge, second-order boundary value problem: problem (1) has not been investigated. Such type of problems arise in the study of obstacle, contact, unilateral Àf (x,u( x)) a≤x<c and equilibrium problems arising in economics, trans- Œ portation, nonlinear optimization, oceanography, ocean u′′=Ãf (x,u( x)) +u( x) g( x) +r c≤x<d (1) wave engineering, fluid flow through porous media, some Œ other branches of pure and applied sciences and ÕŒf (x,u( x)) d≤x≤b engineering (Al-Said et al.,1995, 2002, 2004). In this paper, we consider the case where f in the system (1) is highly nonlinear with arbitrary choices of with boundary conditions u a = α , u b = α and ( ) 1 ( ) 2 r a n d g ( x ) . continuity conditions of u ( x) and u′( x) at internal points Therefore, we have to use some powerful analytic or numerical technique for obtaining the approximate c and d of the interval [a,b]. Here r, α1 and α2 are solution of system (1). Several techniques have been real and finite constants and g x is a continuous used to solve system of boundary value problems ( ) associated with obstacle problems. Noor et al. (1986) f x,u x = f u applied finite difference method for unilateral problems; function and ( ( )) ( ) is a nonlinear function. Al-Said et al. (2004) used finite difference method for If f (u) = f , a linear function, then this problem had obstacle problems; Khalifa et al. (1990) applied quintic spline method for contact problems; Noor et al. (1994) applied quartic spline method for obstacle problems; Al- Said et al. (2002) used quartic spline method for obstacle *Corresponding author. E-mail: [email protected]. problems, and Momani et al. (2006) used decomposition Noor et al. 2427 method for system of forth-order obstacle problems. for differential equations, we consider the general differential These numerical methods provide discrete point solution. equation in operator form. The most noticeable fact which is necessary to mention here that all these methods are proposed to solve linear Lu ( x) + Ru ( x) + Nu ( x) = g ( x), (2) system of boundary value problems associated with obstacle, unilateral and contact problems but less where L is a higher order linear partial operator with respect to attention is drawn towards nonlinear system of boundary time , R is a linear partial operator of order less than L , N is a value problems. We use modified variation of parameters nonlinear partial operator and g is a source term. method to solve system of nonlinear boundary value By using the variation of parameters method as developed in problems associated with obstacle problems. Noor et al. Mohyud-Din et al. (2009, 2011), we have the following general (2008) have used variation of parameters method for solution of Equation (2) as solving a wide classes of higher orders initial and boundary value problems. Ma et al. (2004, 2008) have n−1 i x applied variation of parameters method for solving some Bix u(x)= + λ( x,s) −Nu(s) −Ru(s) +g(s) ds, (3) non homogenous partial differential equations. Ramos — ( ) i=0 i! 0 (2008) had used this technique to find the frequency of some nonlinear oscillators. Ramos (2008) has also shown the equivalence of this technique with variational where n is a order of given differential equation and Bi 's are iteration method. It is further investigated by Noor et al. unknowns which are further determined by initial/boundary (2008) that proposed technique is totally different from conditions. λ ( x, s) is multiplier, it is obtained with the help of variation iteration method in many aspects. The multiplier Wronskian technique used in this method. This multiplier removes used in variation of parameters method is obtained by the successive application of integrals in iterative scheme and it Wronskian technique and is totally different from depends upon the order of equation formula. Mohyud-Din et al. Lagrange multiplier of variation of parameters method. (2009, 2011), have obtained the following for finding the multiplier The variation of parameters method had reduced lot of λ x, s as computational work involved due to this term as ( ) compared to some other existing techniques using this i−1 term which is clear advantage of proposed technique n si−1xn−i (−1) over them. λ ( x, s) = ƒ . (4) We now consider the modified variation of parameters i=1 (i −1)!(n − i)! method, which is an elegant combination of variation of parameters method and Adomian’s decomposition For different choices of n, one can obtain the following values of method. It turned out that the modified variation of λ parameters method is very flexible. We would like to mention that the modified variation of parameters method n =1, λ x, s =1, has the efficiency of both the techniques. The use of ( ) multiplier and Adomian’s polynomial together in the n = 2, λ ( x, s) = x − s, modified variation of parameters method increases the x2 s2 rate of convergence by reducing the number of iterations n = 3, λ ( x, s) = − sx + , and successive application of integral operators. This 2! 2! technique makes the solution procedure simple while still x3 sx2 s2 x s3 maintaining the higher level of accuracy. In the present n = 4, λ ( x, s) = − + − , study, we implement this technique for solving a system 3! 2! 2! 3! of second-order nonlinear boundary value problems associated with obstacle, unilateral and contact From (3), we have the following iterative scheme for solving the problems. It is well-known that the obstacle problems can Equation (2). be studied via the variational inequalities (Noor, 1994, x 2004, 2009, 2009a; Noor et al., 1993). It is an interesting 2 (5) uk+1 ( x) =uk ( x) +—λ( x,s)(−Nuk (s) −Ruk (s) +g(s)) ds k =0,1,2, . and open problem to extend this technique for solving the 0 variational inequalities. This may lead to further research in this field and related optimization problems. The It is observed that the fix value of initial guess in each iteration interested readers are advised to discover novel and provides the better approximation, that is 3 innovative applications for this technique. uk ( x) = u0 ( x), for k =1,2, .. However, we can modify the initial guess by dividing u0 ( x) in two parts and using one of them MODIFIED VARIATION OF PARAMETERS METHOD as initial guess. It is a more convenient way in case of more than two terms in u x . In modified variation of parameters method, To convey the basic concept of the variation of parameter method 0 ( ) 2428 Int. J. Phys. Sci. À x we define the solution u(x) by the following series 1 1 Œuk (x) +—λ( x,s)( Ak +uk +1) ds, for−1≤ x < − and ≤ x ≤1 Œ 0 2 2 ∞ Œ uk (x) = à u(x)=uk (x), and the nonlinear terms are decomposed by +1 Œ x k=0 1 1 Œu (x) + λ x,s A +2u ds, for − ≤ x < , infinite number of polynomials as follows Œ k — ( )( k k ) Õ 0 2 2 ∞ Using λ ( x, s) = x − s, since the governing equation is of N(u)=Ak (u0,u1,u2,...,ui), k=0 second-order, À x where u is a function of x and Ak are the so-called Adomian’s 1 1 Œuk (x) +—( x−s)( Ak +uk +1)ds, for−1≤ x<− and ≤x≤1 polynomials. These polynomials can be generated for various Œ 0 2 2 classes of nonlinearities by specific algorithm developed in Wazwaz Œ (2000) as follows uk+1(x)= à Œ x Œ 1 1 k n uk (x) +—( x−s)( Ak +2uk ) ds, for − ≤x< . ≈1 ’≈ d ’ ’ Œ 2 2 i 3 Õ 0 Ak = ∆ k ÷N ∆(λ ui ) ÷ , k = 0,1,2, . k! dλ i=0 λ=0 1 Case 1: −1≤ x < − . 2 Hence, we have the following iterative scheme for finding the In this case, we implement the modified variation of approximate solution of (2) as parameters method as follows by taking the initial value: u = c x + c . and obtain further iterations as follows: x 0 1 2 uk+1(x) =uk (x) +—λ(x,s)(−Ak −Ruk (s)+ g(s))ds. (6) x 0 u1 ( x) = c2 + — ( x − s)( A0 + u0 +1)ds, 0 We would like to emphasize that the modified variation of parameters method (MVPM) for solving system of second-order nonlinear boundary value problems may be viewed as an important 1 2 1 3 1 4 2 1 5 3 and significant refinement of the previous known methods.
Load more

Recommended publications
  • Finite Difference Methods for Solving Differential
    FINITE DIFFERENCE METHODS FOR SOLVING DIFFERENTIAL EQUATIONS I-Liang Chern Department of Mathematics National Taiwan University May 16, 2013 2 Contents 1 Introduction 3 1.1 Finite Difference Approximation . ........ 3 1.2 Basic Numerical Methods for Ordinary Differential Equations ........... 5 1.3 Runge-Kuttamethods..... ..... ...... ..... ...... ... ... 8 1.4 Multistepmethods ................................ 10 1.5 Linear difference equation . ...... 14 1.6 Stabilityanalysis ............................... 17 1.6.1 ZeroStability................................. 18 2 Finite Difference Methods for Linear Parabolic Equations 23 2.1 Finite Difference Methods for the Heat Equation . ........... 23 2.1.1 Some discretization methods . 23 2.1.2 Stability and Convergence for the Forward Euler method.......... 25 2.2 L2 Stability – von Neumann Analysis . 26 2.3 Energymethod .................................... 28 2.4 Stability Analysis for Montone Operators– Entropy Estimates ........... 29 2.5 Entropy estimate for backward Euler method . ......... 30 2.6 ExistenceTheory ................................. 32 2.6.1 Existence via forward Euler method . ..... 32 2.6.2 A Sharper Energy Estimate for backward Euler method . ......... 33 2.7 Relaxationoferrors.............................. 34 2.8 BoundaryConditions ..... ..... ...... ..... ...... ... 36 2.8.1 Dirichlet boundary condition . ..... 36 2.8.2 Neumann boundary condition . 37 2.9 The discrete Laplacian and its inversion . .......... 38 2.9.1 Dirichlet boundary condition . ..... 38 3 Finite Difference Methods for Linear elliptic Equations 41 3.1 Discrete Laplacian in two dimensions . ........ 41 3.1.1 Discretization methods . 41 3.1.2 The 9-point discrete Laplacian . ..... 42 3.2 Stability of the discrete Laplacian . ......... 43 3 4 CONTENTS 3.2.1 Fouriermethod ................................ 43 3.2.2 Energymethod ................................ 44 4 Finite Difference Theory For Linear Hyperbolic Equations 47 4.1 A review of smooth theory of linear hyperbolic equations .............
    [Show full text]
  • Exercises from Finite Difference Methods for Ordinary and Partial
    Exercises from Finite Difference Methods for Ordinary and Partial Differential Equations by Randall J. LeVeque SIAM, Philadelphia, 2007 http://www.amath.washington.edu/ rjl/fdmbook ∼ Under construction | more to appear. Contents Chapter 1 4 Exercise 1.1 (derivation of finite difference formula) . 4 Exercise 1.2 (use of fdstencil) . 4 Chapter 2 5 Exercise 2.1 (inverse matrix and Green's functions) . 5 Exercise 2.2 (Green's function with Neumann boundary conditions) . 5 Exercise 2.3 (solvability condition for Neumann problem) . 5 Exercise 2.4 (boundary conditions in bvp codes) . 5 Exercise 2.5 (accuracy on nonuniform grids) . 6 Exercise 2.6 (ill-posed boundary value problem) . 6 Exercise 2.7 (nonlinear pendulum) . 7 Chapter 3 8 Exercise 3.1 (code for Poisson problem) . 8 Exercise 3.2 (9-point Laplacian) . 8 Chapter 4 9 Exercise 4.1 (Convergence of SOR) . 9 Exercise 4.2 (Forward vs. backward Gauss-Seidel) . 9 Chapter 5 11 Exercise 5.1 (Uniqueness for an ODE) . 11 Exercise 5.2 (Lipschitz constant for an ODE) . 11 Exercise 5.3 (Lipschitz constant for a system of ODEs) . 11 Exercise 5.4 (Duhamel's principle) . 11 Exercise 5.5 (matrix exponential form of solution) . 11 Exercise 5.6 (matrix exponential form of solution) . 12 Exercise 5.7 (matrix exponential for a defective matrix) . 12 Exercise 5.8 (Use of ode113 and ode45) . 12 Exercise 5.9 (truncation errors) . 13 Exercise 5.10 (Derivation of Adams-Moulton) . 13 Exercise 5.11 (Characteristic polynomials) . 14 Exercise 5.12 (predictor-corrector methods) . 14 Exercise 5.13 (Order of accuracy of Runge-Kutta methods) .
    [Show full text]
  • Finite Difference and Discontinuous Galerkin Methods for Wave Equations
    Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1522 Finite Difference and Discontinuous Galerkin Methods for Wave Equations SIYANG WANG ACTA UNIVERSITATIS UPSALIENSIS ISSN 1651-6214 ISBN 978-91-554-9927-3 UPPSALA urn:nbn:se:uu:diva-320614 2017 Dissertation presented at Uppsala University to be publicly examined in Room 2446, Polacksbacken, Lägerhyddsvägen 2, Uppsala, Tuesday, 13 June 2017 at 10:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Professor Thomas Hagstrom (Department of Mathematics, Southern Methodist University). Abstract Wang, S. 2017. Finite Difference and Discontinuous Galerkin Methods for Wave Equations. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1522. 53 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-9927-3. Wave propagation problems can be modeled by partial differential equations. In this thesis, we study wave propagation in fluids and in solids, modeled by the acoustic wave equation and the elastic wave equation, respectively. In real-world applications, waves often propagate in heterogeneous media with complex geometries, which makes it impossible to derive exact solutions to the governing equations. Alternatively, we seek approximated solutions by constructing numerical methods and implementing on modern computers. An efficient numerical method produces accurate approximations at low computational cost. There are many choices of numerical methods for solving partial differential equations. Which method is more efficient than the others depends on the particular problem we consider. In this thesis, we study two numerical methods: the finite difference method and the discontinuous Galerkin method.
    [Show full text]
  • Modifications of the Method of Variation of Parameters
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector International Joumal Available online at www.sciencedirect.com computers & oc,-.c~ ~°..-cT- mathematics with applications Computers and Mathematics with Applications 51 (2006) 451-466 www.elsevier.com/locate/camwa Modifications of the Method of Variation of Parameters ROBERTO BARRIO* GME, Depto. Matem~tica Aplicada Facultad de Ciencias Universidad de Zaragoza E-50009 Zaragoza, Spain rbarrio@unizar, es SERGIO SERRANO GME, Depto. MatemAtica Aplicada Centro Politdcnico Superior Universidad de Zaragoza Maria de Luna 3, E-50015 Zaragoza, Spain sserrano©unizar, es (Received February PO05; revised and accepted October 2005) Abstract--ln spite of being a classical method for solving differential equations, the method of variation of parameters continues having a great interest in theoretical and practical applications, as in astrodynamics. In this paper we analyse this method providing some modifications and generalised theoretical results. Finally, we present an application to the determination of the ephemeris of an artificial satellite, showing the benefits of the method of variation of parameters for this kind of problems. ~) 2006 Elsevier Ltd. All rights reserved. Keywords--Variation of parameters, Lagrange equations, Gauss equations, Satellite orbits, Dif- ferential equations. 1. INTRODUCTION The method of variation of parameters was developed by Leonhard Euler in the middle of XVIII century to describe the mutual perturbations of Jupiter and Saturn. However, his results were not absolutely correct because he did not consider the orbital elements varying simultaneously. In 1766, Lagrange improved the method developed by Euler, although he kept considering some of the orbital elements as constants, which caused some of his equations to be incorrect.
    [Show full text]
  • A Quasi-Static Particle-In-Cell Algorithm Based on an Azimuthal Fourier Decomposition for Highly Efficient Simulations of Plasma-Based Acceleration
    A quasi-static particle-in-cell algorithm based on an azimuthal Fourier decomposition for highly efficient simulations of plasma-based acceleration: QPAD Fei Lia,c, Weiming And,∗, Viktor K. Decykb, Xinlu Xuc, Mark J. Hoganc, Warren B. Moria,b aDepartment of Electrical Engineering, University of California Los Angeles, Los Angeles, CA 90095, USA bDepartment of Physics and Astronomy, University of California Los Angeles, Los Angeles, CA 90095, USA cSLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA dDepartment of Astronomy, Beijing Normal University, Beijing 100875, China Abstract The three-dimensional (3D) quasi-static particle-in-cell (PIC) algorithm is a very effi- cient method for modeling short-pulse laser or relativistic charged particle beam-plasma interactions. In this algorithm, the plasma response, i.e., plasma wave wake, to a non- evolving laser or particle beam is calculated using a set of Maxwell's equations based on the quasi-static approximate equations that exclude radiation. The plasma fields are then used to advance the laser or beam forward using a large time step. The algorithm is many orders of magnitude faster than a 3D fully explicit relativistic electromagnetic PIC algorithm. It has been shown to be capable to accurately model the evolution of lasers and particle beams in a variety of scenarios. At the same time, an algorithm in which the fields, currents and Maxwell equations are decomposed into azimuthal harmonics has been shown to reduce the complexity of a 3D explicit PIC algorithm to that of a 2D algorithm when the expansion is truncated while maintaining accuracy for problems with near azimuthal symmetry.
    [Show full text]
  • A Numerical Method of Characteristics for Solving Hyperbolic Partial Differential Equations David Lenz Simpson Iowa State University
    Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1967 A numerical method of characteristics for solving hyperbolic partial differential equations David Lenz Simpson Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mathematics Commons Recommended Citation Simpson, David Lenz, "A numerical method of characteristics for solving hyperbolic partial differential equations " (1967). Retrospective Theses and Dissertations. 3430. https://lib.dr.iastate.edu/rtd/3430 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. This dissertation has been microfilmed exactly as received 68-2862 SIMPSON, David Lenz, 1938- A NUMERICAL METHOD OF CHARACTERISTICS FOR SOLVING HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS. Iowa State University, PluD., 1967 Mathematics University Microfilms, Inc., Ann Arbor, Michigan A NUMERICAL METHOD OF CHARACTERISTICS FOR SOLVING HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS by David Lenz Simpson A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject: Mathematics Approved: Signature was redacted for privacy. In Cijarge of Major Work Signature was redacted for privacy. Headead of^ajorof Major DepartmentDepartmen Signature was redacted for privacy. Dean Graduate Co11^^ Iowa State University of Science and Technology Ames, Iowa 1967 ii TABLE OF CONTENTS Page I. INTRODUCTION 1 II. DISCUSSION OF THE ALGORITHM 3 III. DISCUSSION OF LOCAL ERROR, EXISTENCE AND UNIQUENESS THEOREMS 10 IV.
    [Show full text]
  • Analysis of Finite Difference Discretization Schemes For
    Computers and Chemical Engineering 71 (2014) 241–252 Contents lists available at ScienceDirect Computers and Chemical Engineering j ournal homepage: www.elsevier.com/locate/compchemeng Analysis of finite difference discretization schemes for diffusion in spheres with variable diffusivity ∗ Ashlee N. Ford Versypt, Richard D. Braatz Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA a r t i c l e i n f o a b s t r a c t Article history: Two finite difference discretization schemes for approximating the spatial derivatives in the diffusion Received 28 April 2014 equation in spherical coordinates with variable diffusivity are presented and analyzed. The numerical Accepted 27 May 2014 solutions obtained by the discretization schemes are compared for five cases of the functional form Available online 27 August 2014 for the variable diffusivity: (I) constant diffusivity, (II) temporally dependent diffusivity, (III) spatially dependent diffusivity, (IV) concentration-dependent diffusivity, and (V) implicitly defined, temporally Keywords: and spatially dependent diffusivity. Although the schemes have similar agreement to known analytical Finite difference method or semi-analytical solutions in the first four cases, in the fifth case for the variable diffusivity, one scheme Variable coefficient Diffusion produces a stable, physically reasonable solution, while the other diverges. We recommend the adoption of the more accurate and stable of these finite difference discretization schemes to numerically approxi- Spherical geometry Method of lines mate the spatial derivatives of the diffusion equation in spherical coordinates for any functional form of variable diffusivity, especially cases where the diffusivity is a function of position. © 2014 Elsevier Ltd.
    [Show full text]
  • MTHE / MATH 237 Differential Equations for Engineering Science
    i Queen’s University Mathematics and Engineering and Mathematics and Statistics MTHE / MATH 237 Differential Equations for Engineering Science Supplemental Course Notes Serdar Y¨uksel November 26, 2015 ii This document is a collection of supplemental lecture notes used for MTHE / MATH 237: Differential Equations for Engineering Science. Serdar Y¨uksel Contents 1 Introduction to Differential Equations ................................................... .......... 1 1.1 Introduction.................................... ............................................ 1 1.2 Classification of DifferentialEquations ............ ............................................. 1 1.2.1 OrdinaryDifferentialEquations ................. ........................................ 2 1.2.2 PartialDifferentialEquations .................. ......................................... 2 1.2.3 HomogeneousDifferentialEquations .............. ...................................... 2 1.2.4 N-thorderDifferentialEquations................ ........................................ 2 1.2.5 LinearDifferentialEquations ................... ........................................ 2 1.3 SolutionsofDifferentialequations ................ ............................................. 3 1.4 DirectionFields................................. ............................................ 3 1.5 Fundamental Questions on First-Order Differential Equations ...................................... 4 2 First-Order Ordinary Differential Equations ..................................................
    [Show full text]
  • Domain Decomposition Methods for Parallel Solution of Shape Sensitivity Analysis Problems
    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng. 44, 281—303 (1999) DOMAIN DECOMPOSITION METHODS FOR PARALLEL SOLUTION OF SHAPE SENSITIVITY ANALYSIS PROBLEMS MANOLIS PAPADRAKAKIS* AND YIANNIS TSOMPANAKIS Institute of Structural Analysis and Seismic Research, National Technical University of Athens, Athens 15773, Greece ABSTRACT This paper presents the implementation of advanced domain decomposition techniques for parallel solution of large-scale shape sensitivity analysis problems. The methods presented in this study are based on the FETI method proposed by Farhat and Roux which is a dual domain decomposition implementation. Two variants of the basic FETI method have been implemented in this study: (i) FETI-1 where the rigid-body modes of the floating subdomains are computed explicitly. (ii) FETI-2 where the local problem at each subdomain is solved by the PCG method and the rigid-body modes are computed explicitly. A two-level iterative method is proposed particularly tailored to solve re-analysis type of problems, where the dual domain decomposition method is incorporated in the preconditioning step of a subdomain global PCG implementation. The superiority of this two-level iterative solver is demonstrated with a number of numerical tests in serial as well as in parallel computing environments. Copyright ( 1999 John Wiley & Sons, Ltd. KEY WORDS: shape sensitivity analysis; domain decomposition methods; preconditioned conjugate gradient; multiple right-hand sides; reanalysis problems INTRODUCTION Shape optimization aims to improve the shape of a structure, defined with a number of parameters called design variables, by minimizing an objective function subject to certain constraint functions. The shape optimization algorithm proceeds with the following steps: (i) a finite element mesh is generated, (ii) displacements, stresses, frequencies, etc.
    [Show full text]
  • Math 6730 : Asymptotic and Perturbation Methods Hyunjoong
    Math 6730 : Asymptotic and Perturbation Methods Hyunjoong Kim and Chee-Han Tan Last modified : January 13, 2018 2 Contents Preface 5 1 Introduction to Asymptotic Approximation7 1.1 Asymptotic Expansion................................8 1.1.1 Order symbols.................................9 1.1.2 Accuracy vs convergence........................... 10 1.1.3 Manipulating asymptotic expansions.................... 10 1.2 Algebraic and Transcendental Equations...................... 11 1.2.1 Singular quadratic equation......................... 11 1.2.2 Exponential equation............................. 14 1.2.3 Trigonometric equation............................ 15 1.3 Differential Equations: Regular Perturbation Theory............... 16 1.3.1 Projectile motion............................... 16 1.3.2 Nonlinear potential problem......................... 17 1.3.3 Fredholm alternative............................. 19 1.4 Problems........................................ 20 2 Matched Asymptotic Expansions 31 2.1 Introductory example................................. 31 2.1.1 Outer solution by regular perturbation................... 31 2.1.2 Boundary layer................................ 32 2.1.3 Matching................................... 33 2.1.4 Composite expression............................. 33 2.2 Extensions: multiple boundary layers, etc...................... 34 2.2.1 Multiple boundary layers........................... 34 2.2.2 Interior layers................................. 35 2.3 Partial differential equations............................. 36 2.4 Strongly
    [Show full text]
  • Solving the Schrödinger Equation Using the Finite Difference Time Domain Method
    Home Search Collections Journals About Contact us My IOPscience Solving the Schrödinger equation using the finite difference time domain method This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2007 J. Phys. A: Math. Theor. 40 1885 (http://iopscience.iop.org/1751-8121/40/8/013) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 148.224.96.21 The article was downloaded on 18/05/2010 at 20:10 Please note that terms and conditions apply. IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL J. Phys. A: Math. Theor. 40 (2007) 1885–1896 doi:10.1088/1751-8113/40/8/013 Solving the Schrodinger¨ equation using the finite difference time domain method I Wayan Sudiarta1 and D J Wallace Geldart1,2 1 Department of Physics and Atmospheric Science, Dalhousie University, Halifax, NS B3H 3J5, Canada 2 School of Physics, University of New South Wales, Sydney, NSW 2052, Australia E-mail: [email protected] Received 5 September 2006, in final form 7 January 2007 Published 6 February 2007 Online at stacks.iop.org/JPhysA/40/1885 Abstract In this paper, we solve the Schrodinger¨ equation using the finite difference time domain (FDTD) method to determine energies and eigenfunctions. In order to apply the FDTD method, the Schrodinger¨ equation is first transformed into a diffusion equation by the imaginary time transformation. The resulting time- domain diffusion equation is then solved numerically by the FDTD method. The theory and an algorithm are provided for the procedure.
    [Show full text]
  • Finite Difference Method for Numerical Computation of Discontinuous Solutions of the Equations of Fluid Dynamics Sergei Godunov, I
    Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics Sergei Godunov, I. Bohachevsky To cite this version: Sergei Godunov, I. Bohachevsky. Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics. Matematičeskij sbornik, Steklov Mathematical Institute of Russian Academy of Sciences, 1959, 47(89) (3), pp.271-306. hal-01620642 HAL Id: hal-01620642 https://hal.archives-ouvertes.fr/hal-01620642 Submitted on 25 Jul 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. MATEMA TICHESKII SBORNIK Finite Difference Method for Numerical Computation of Discontinuous Solutiona of the Equations of Fluid Dynamic• 1959 v. 47 (89) No.3 p. 271 S, K. Godunov Translated by 1. Bohacheveky Introduction The method of characteristics used for nume_rical computation of solutions of fluid dynamical equations is characterized by a large degree of nonetandard­ _ nesa and therefore il not suitable for _automatic computation on electronic computing machines t especially for problems with a large number of shock waves and contact discontinuities. In 1950 v. Neumann and Richtmyer (1) prOp<!ted to use, for the solution of fluid dynamics equations, difference equationa into which viscosity was introduced artificially; this has the effect of smearing out the shock wave over aeveral mesh points.
    [Show full text]