An Introduction to Partial Differential Equations with MATLAB

An Introduction to Partial Differential Equations with MATLAB

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An Introduction to Partial Differential Equations with MATLAB exposes the basic ideas critical to the study of PDEs-- characteristics, integral transforms, Green's functions, and, most importantly, Fourier series and related topics. The author approaches the subject from a motivational perspective, detailing equations only after a need for them has been established. He uses MATLAB (R) software to solve exercises and to generate tables and figures. This volume includes examples of many important PDEs and their applications. The first chapter introduces PDEs and makes analogies to familiar ODE concepts, then strengthens the connection by exploring the method of separation of variables. Chapter 2 examines the "Big Three" PDEs-- the heat, wave, and Laplace equations, and is followed by chapters explaining how these and other PDEs on finite intervals can be solved using the Fourier series for arbitrary initial and boundary conditions. Chapter 5 investigates characteristics for both first- and second-order linear PDEs, the latter revealing how the Big Three equations are important far beyond their original application to physical problems. The book extends the Fourier method to functions on unbounded domains, gives a brief introduction to distributions, then applies separation of variables to PDEs in higher dimensions, leading to the special funtions, including the orthogonal polynomials. Other topics include Sturm-Liouville problems, adjoint and self-adjoint problems, the application of Green's functions to solving nonhomogeneous PDEs, and an examination of practical numerical methods used by engineers, including the finite difference, finite element, and spectral methods.show more

Product details

  • Electronic book text | 688 pages
  • Taylor & Francis Ltd
  • Chapman & Hall/CRC
  • United States
  • 1000 equations; 5 Tables, black and white; 114 Illustrations, black and white
  • 0203504879
  • 9780203504871

Table of contents

Introduction What are Partial Differential Equations? PDEs We Can Already Solve Initial and Boundary Conditions Linear PDEs--Definitions Linear PDEs--The Principle of Superposition Separation of Variables for Linear, Homogeneous PDEs Eigenvalue Problems 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 Using Separation of Variables to Solve the Big Three PDEs Fourier Series Introduction Properties of Sine and Cosine The Fourier Series The Fourier Series, Continued The Fourier Series---Proof of Pointwise Convergence Fourier Sine and Cosine Series Completeness Solving the Big Three PDEs 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 Characteristicsfor Linear PDEs First-Order PDEs with Constant Coefficients First-Order PDEs with Variable Coefficients D'Alembert's Solution for the Wave Equation--The Infinite String Characteristics for Semi-Infinite and Finite String Problems General Second-Order Linear PDEs and Characteristics Integral Transforms The Laplace Transform for PDEs Fourier Sine and Cosine Transforms The Fourier Transform The Infinite and Semi-Infinite Heat Equations Distributions, the Dirac Delta Function and Generalized Fourier Transforms Proof of the Fourier Integral Formula Bessel Functions and Orthogonal Polynomials The Special Functions and Their Differential Equations Ordinary Points and Power Series Solutions; Chebyshev, Hermite and Legendre Polynomials The Method of Frobenius; Laguerre Polynomials Interlude: The Gamma Function Bessel Functions Recap: A List of Properties of Bessel Functions and Orthogonal Polynomials Sturm-Liouville Theory and Generalized Fourier Series Sturm-Liouville Problems Regular and Periodic Sturm-Liouville Problems Singular Sturm-Liouville Problems; Self-Adjoint Problems The Mean-Square or L2 Norm and Convergence in the Mean Generalized Fourier Series; Parseval's Equality and Completeness 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 The Wave and Heat Equations in Polar Coordinates Problems in Spherical Coordinates The Infinite Wave Equation and Multiple Fourier Transforms Postlude: Eigenvalues and Eigenfunctions of the Laplace Operator; Green's Identities for the Laplacian Nonhomogeneous Problems and Green's Functions Green's Functions for ODEs Green's Function and the Dirac Delta Function Green's Functions for Elliptic PDEs (I): Poisson's Equation in Two Dimensions Green's Functions for Elliptic PDEs (II): Poisson's Equation in Three Dimensions; the Helmholtz Equation Green's Function's for Equations of Evolution Numerical Methods Finite Difference Approximations for ODEs Finite Difference Approximations for PDEs Spectral Methods and the Finite Element Method References Uniform Convergence; Differentiation and Integration of Fourier Series Important Theorems: Limits, Derivatives, Integrals, Series, and Interchange of Operations Existence and Uniqueness Theorems A Menagerie of PDEs MATLAB Code for Figures and Exercises Answers to Selected Exercisesshow more