An Introduction to Partial Differential Equations with MATLAB

ISBN : 9781032639383

Author : Matthew P. Coleman

Publisher : Crc Press

Year : 2025

Language : English

Type : Book

Description : 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

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