Chen, Yi Tung; Li, Jichun Computational Partial Differential Equations Using MATLAB
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Chen, Yi Tung; Li, Jichun Computational Partial Differential Equations Using MATLAB


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Computational Partial 
Differential Equations 
Using MATLAB®
CHAPMAN & HALL/CRC APPLIED MATHEMATICS 
AND NONLINEAR SCIENCE SERIES
Series Editors Goong Chen and Thomas J. Bridges 
Published Titles
Computing with hp-ADAPTIVE FINITE ELEMENTS, Volume 1, One and Two Dimensional 
 Elliptic and Maxwell Problems, Leszek Demkowicz
Computing with hp-ADAPTIVE FINITE ELEMENTS, Volume 2, Frontiers: Three
 Dimensional Elliptic and Maxwell Problems with Applications, Leszek Demkowicz, 
 Jason Kurtz, David Pardo, Maciej Paszyn´ski, Waldemar Rachowicz, and Adam Zdunek
CRC Standard Curves and Surfaces with Mathematica®: Second Edition, 
 David H. von Seggern
Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in
 Mechanics and Physics, Victor A. Galaktionov and Sergey R. Svirshchevskii
Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications, 
 Victor A. Galaktionov 
Introduction to Fuzzy Systems, Guanrong Chen and Trung Tat Pham
Introduction to non-Kerr Law Optical Solitons, Anjan Biswas and Swapan Konar
Introduction to Partial Differential Equations with MATLAB®, Matthew P. Coleman
Introduction to Quantum Control and Dynamics, Domenico D\u2019Alessandro
Mathematical Methods in Physics and Engineering with Mathematica, Ferdinand F. Cap
Mathematical Theory of Quantum Computation, Goong Chen and Zijian Diao
Mathematics of Quantum Computation and Quantum Technology, Goong Chen, 
 Louis Kauffman, and Samuel J. Lomonaco
Mixed Boundary Value Problems, Dean G. Duffy
Multi-Resolution Methods for Modeling and Control of Dynamical Systems, 
 Puneet Singla and John L. Junkins 
Optimal Estimation of Dynamic Systems, John L. Crassidis and John L. Junkins
Quantum Computing Devices: Principles, Designs, and Analysis, Goong Chen, 
 David A. Church, Berthold-Georg Englert, Carsten Henkel, Bernd Rohwedder, 
 Marlan O. Scully, and M. Suhail Zubairy
Stochastic Partial Differential Equations, Pao-Liu Chow
Computational Partial 
Differential Equations 
Using MATLAB®
CHAPMAN & HALL/CRC APPLIED MATHEMATICS
AND NONLINEAR SCIENCE SERIES 
Jichun Li
University of Nevada
Las Vegas, NV, U.S.A.
Yi-Tung Chen
University of Nevada
Las Vegas, NV, U.S.A.
MATLAB® and Simulink® are trademarks of the Math Works, Inc. and are used with permission. The Math-
works does not warrant the accuracy of the text or exercises in this book. This book\u2019s use or discussion of 
MATLAB® and Simulink® software or related products does not constitute endorsement or sponsorship by 
the Math Works of a particular pedagogical approach or particular use of the MATLAB® and Simulink® 
software.
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© 2008 by Taylor & Francis Group, LLC
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Version Date: 20131121
International Standard Book Number-13: 978-1-4200-8905-9 (eBook - PDF)
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Contents
Preface xi
Acknowledgments xiii
1 Brief Overview of Partial Di\ufb00erential Equations 1
1.1 The parabolic equations . . . . . . . . . . . . . . . . . . . . . 1
1.2 The wave equations . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 The elliptic equations . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Di\ufb00erential equations in broader areas . . . . . . . . . . . . . 3
1.4.1 Electromagnetics . . . . . . . . . . . . . . . . . . . . . 3
1.4.2 Fluid mechanics . . . . . . . . . . . . . . . . . . . . . . 4
1.4.3 Ground water contamination . . . . . . . . . . . . . . 5
1.4.4 Petroleum reservoir simulation . . . . . . . . . . . . . 6
1.4.5 Finance modeling . . . . . . . . . . . . . . . . . . . . . 7
1.4.6 Image processing . . . . . . . . . . . . . . . . . . . . . 7
1.5 A quick review of numerical methods for PDEs . . . . . . . . 8
References 10
2 Finite Di\ufb00erence Methods for Parabolic Equations 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Theoretical issues: stability, consistence, and convergence . . 15
2.3 1-D parabolic equations . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 The \u3b8-method . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.2 Some extensions . . . . . . . . . . . . . . . . . . . . . 19
2.4 2-D and 3-D parabolic equations . . . . . . . . . . . . . . . . 23
2.4.1 Standard explicit and implicit methods . . . . . . . . . 23
2.4.2 The ADI methods for 2-D problems . . . . . . . . . . 25
2.4.3 The ADI methods for 3-D problems . . . . . . . . . . 28
2.5 Numerical examples with MATLAB codes . . . . . . . . . . . 30
2.6 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . 33
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
References 36
v
vi Computational Partial Di\ufb00erential Equations Using MATLAB
3 Finite Di\ufb00erence Methods for Hyperbolic Equations 39
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Some basic di\ufb00erence schemes . . . . . . . . . . . . . . . . . . 40
3.3 Dissipation and dispersion errors . . . . . . . . . . . . . . . . 42
3.4 Extensions to conservation laws . . . . . . . . . . . . . . . . . 44
3.5 The second-order hyperbolic PDEs . . . . . . . . . . . . . . . 45
3.6 Numerical examples with MATLAB codes . . . . . . . . . . . 49
3.7 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . 52
3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
References 54
4 Finite Di\ufb00erence Methods for Elliptic Equations 57
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Numerical solution of linear systems . . . . . . . . . . . . . . 59
4.2.1 Direct methods . . . . . . . . . . . . . . . . . . . . . . 59
4.2.2 Simple iterative methods . . . . . . . . . . . . . . . . . 61
4.2.3 Modern iterative methods . . . . . . . . . . . . . . . . 64
4.3 Error analysis with a maximum principle . . . . . . . . . . . . 66
4.4 Some extensions . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4.1 Mixed boundary conditions . . . . . . . . . . . . . . . 68
4.4.2