3rd Edition
An Introduction to Partial Differential Equations with MATLAB
The first and second editions of “An Introduction to Partial Differential Equation with MATLAB®” gained popularity among instructors and students at various universities throughout the world. Plain mathematical language is used in a friendly manner to provide a basic introduction to partial differential equations focusing on Fourier series and integrals.
Suitable for a one- or two-semester introduction to PDEs and Fourier series, the book offers equations based on method of solution and provides both physical and mathematical motivation as much as possible.
This third edition changes the book structure by lifting the role of the computational part much closer to the revised analytical portion. The re-designed content will be extremely useful for students of mathematics, physics and engineering who would like to focus on the practical aspects of using the theory of PDEs for modeling and later while taking various courses in numerical analysis, computer science, PDE-based programming, and optimization.
Included in this new edition is a substantial amount of material on reviewing computational methods for solving ODEs (symbolically and numerically), visualizing solutions of PDEs, using MATLAB's symbolic programming toolbox, and applying various numerical schemes for computing with regard to numerical solutions in practical applications, along with suggestions for topics of course projects.
Students will use sample MATLAB and Python codes available online for their practical experiments and for completing computational lab assignments and course projects.
Chapter 1. Introduction
What are Partial Differential Equations?
PDEs We Can Already Solve
Initial and Boundary Conditions
Linear PDEs – Definitions
Linear PDEs – The Principle of Superposition
The Method of Characteristics I
The Method of Characteristics II
Separation of Variables for Linear, Homogeneous PDEs
Eigenvalue Problems
Chapter 2. The Big Three PDEs
Second-Order, Linear, Homogeneous PDEs with Constant Coefficients
The Heat Equation and Diffusion
The Wave Equation and the Vibrating String
Initial and Boundary Conditions for the Heat and Wave Equations
Laplace's Equation – The Potential Equation
D'Alembert's Solution for the Infinite String Problem
General Second-Order Linear PDEs and Characteristics
Using Separation of Variables to Solve the Big Three PDEs
Chapter 3. Using MATLAB for Solving Differential Equations and Visualizing Solutions
Visualizing Solutions of ODEs
Symbolic Math Toolbox for Solving ODEs
Solving BVPs Numerically Using bvp4(5)c
Solving PDEs Numerically Using pdepe
Exercises for Chapter 3
Lab Assignment #1: Review Chapters 1-3
Chapter 4. Fourier Series
Introduction
Properties of Sine and Cosine
The Fourier Series
The Fourier Series, Continued
Fourier Sine and Cosine Series
Chapter 5. Solving the Big Three PDEs on Finite Domains
Solving the Homogeneous Heat Equation for a Finite Rod
Solving the Homogeneous Wave Equation for a Finite String
Solving the Homogeneous Laplace’s Equation on a Rectangular Domain
Nonhomogeneous Problems
Chapter 6. Review of Numerical Methods for Solving ODEs
Approaches to Solving First-Order IVPs
Numerical Solutions Using Euler's Method
Numerical Solutions Using Runge–Kutta Methods
Solving Higher-Order ODEs Numerically
Implicit Approximations for BVPs
Exercises for Chapter 6
Chapter 7. Solving PDEs Using Finite Difference Approximations
Numerical Solutions for the Heat Equation
Explicit Scheme for the Wave Equation
Numerical Schemes for Laplace's Equation
Numerical Solution of First-Order PDEs
Exercises for Chapter 7
Lab Assignment #2: Review Chapters 6-7
Lab Assignment #3: Review Chapters 4-7
Chapter 8. Integral Transforms
The Laplace Transform for PDEs
Fourier Sine and Cosine Transforms
The Fourier Transform
The Infinite and Semi-Infinite Heat Equations
Other Integral Transforms and Integral Equations
Chapter 9. Using MATLAB's Symbolic Math Toolbox with Integral Transforms
Integral Transforms via Symbolic Programming
Solving ODEs Using the Laplace Transform in MATLAB
Symbolic Solution of PDEs Using the Laplace Transform
Symbolic Solution of PDEs Using the Fourier Transform
Exercises for Chapter 9
Lab Assignment #4: Review Chapters 8-9
Chapter 10. PDEs in Higher Dimensions
PDEs in Higher Dimensions: Examples and Derivations
The Heat and Wave Equations on a Rectangle; Multiple Fourier Series
Laplace's Equation in Polar Coordinates: Poisson's Integral Formula
Interlude 1: Bessel Functions
Interlude 2: The Legendre Polynomials
The Wave and Heat Equations in Polar Coordinates
Problems in Spherical Coordinates
The Infinite Wave Equation and Multiple Fourier Transforms
MATLAB Exercises for Chapter 10
Lab Assignment #5: Review Chapters 7 & 10
Chapter 11. Overview of Spectral, Finite Element, and Finite Volume Methods
Spectral Methods
Finite Element Methods
Finite Volume Methods
Exercises for Chapter 11
Appendix A: Important Definitions and Theorems
Appendix B: Bessel's Equation and the Method of Frobenius
Appendix C: A Menagerie of PDEs
Appendix D: Review of Math with MATLAB
Appendix E: Answers to Selected Exercises
References
Index
Biography
Dr. Matthew P. Coleman is a Professor Emeritus of Mathematics at Fairfield University, CT, where he taught from 1989 until his retirement in 2019. He received his Ph.D. in Applied Mathematics from Penn State University in 1989 under the guidance of Dr. Goong Chen. While at Fairfield, Dr. Coleman taught almost every undergraduate course in the curriculum, along with a number of graduate courses. In addition, he was department chair for ten years, did a brief stint as associate dean (though he was happy when it was over!), and was a visitor at Texas A&M, NYU, and National Taiwan University. Dr. Coleman’s main research area is Control Theory and, more specifically, the vibration and damping of distributed systems. He has published numerous articles in this area, while collaborating with people from numerous universities, in mathematics, physics, and various branches of engineering. In addition, he has authored the first two editions of the textbook “An Introduction to Partial Differential Equations with MATLAB”. Dr. Vladislav Bukshtynov is an Assistant Professor at the Dept. of Mathematical Sciences of Florida Institute of Technology (Florida Tech) since 2015 after finishing his 3-year postdoctoral term at the Dept. of Energy Resources Engineering at Stanford University and having his Ph.D. degree in Computational Engineering & Science at McMaster University in 2012. As a Professor, he actively teaches and advises students from various fields: applied and computational math, operations research, and different engineering majors. His teaching experience includes Multivariable Calculus, Honors ODE/PDE courses for undergrad students, Applied Discrete Math, and Linear/Nonlinear Optimization for senior undergrads and graduates. As a researcher, Dr. Bukshtynov leads his research group with several dynamic scientific directions and ongoing collaborations for various cross-institutional and interdisciplinary projects. His current interests lie in but are not limited to the areas of applied and computational mathematics focusing on combining theoretical and numerical methods for various problems in computational/numerical optimization, control theory, and inverse problems. Besides being an expert in applying numerous optimization techniques, either analytically or numerically, Dr. Bukshtynov’s expertise includes PDE-based modeling for various engineering applications. For example, he pioneered system reduction techniques using a 4D VAR method and earned the 2012 Cecil Graham Doctoral Dissertation Award from Canadian Applied and Industrial Mathematics Society (CAIMS). At Stanford, he was part of a large collaborative research project to develop efficient computational and optimization algorithms for solving oil field management problems for petroleum reservoir models. At Florida Tech, Dr. Bukshtynov develops novel techniques suitable for reconstructing medical/computational images via solving inverse problems using multiscale simulation and optimization, optimal solution space multilevel parameterization, dynamical re-parameterization, and numerical methods for regularization. In addition, he has profound expertise in HPC programming. His particular strength is his ability to create hybrid computational frameworks by combining and tuning for optimal joint work scientific software of various types. As one of many examples, Dr. Bukshtynov is an author and active developer of EIT-OPT, an all-purpose open-structure multifaceted optimization framework for reconstructing biomedical images and early cancer detection via electrical impedance tomography. In 2023, Dr. Bukshtynov published a book “Computational Optimization: Success in Practice” with CRC Press to share his extensive experience in practical aspects of computational optimization with graduate students of math, computer science, engineering, and all who explore optimization techniques at different levels for educational or research purposes.
