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Taylor & Francis
An Introduction to Partial Differential Equations with MATLAB, Second Edition / Edition 2

An Introduction to Partial Differential Equations with MATLAB, Second Edition / Edition 2

by Matthew P. Coleman


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Product Details

ISBN-13: 9781439898468
Publisher: Taylor & Francis
Publication date: 07/05/2013
Series: Chapman & Hall/CRC Applied Mathematics & Nonlinear Science Series , #27
Edition description: New Edition
Pages: 683
Product dimensions: 6.40(w) x 9.30(h) x 1.40(d)

Table of Contents

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
Properties of Sine and Cosine
The Fourier Series
The Fourier Series, Continued
The Fourier Series—Proof of Pointwise Convergence
Fourier Sine and Cosine Series

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

First-Order PDEs with Constant Coefficients
First-Order PDEs with Variable Coefficients
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 Functions for Equations of Evolution

Numerical Methods
Finite Difference Approximations for ODEs
Finite Difference Approximations for PDEs
Spectral Methods and the Finite Element Method

Appendix A: Uniform Convergence; Differentiation and Integration of Fourier Series
Appendix B: Other Important Theorems
Appendix C: Existence and Uniqueness Theorems
Appendix D: A Menagerie of PDEs
Appendix E: MATLAB Code for Figures and Exercises
Appendix F: Answers to Selected Exercises



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