Introduction to Partial Differential Equations with MATLAB

ISBN-10: 1584883731
ISBN-13: 9781584883739
Edition: 2004
Authors: Laurie Kelly
List price: $113.95
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Description: 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  More...

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Book details

List price: $113.95
Copyright year: 2004
Publisher: CRC Press LLC
Publication date: 9/29/2004
Binding: Hardcover
Pages: 688
Size: 6.50" wide x 9.50" long x 1.50" tall
Weight: 2.398
Language: English

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. The author uses MATLAB software to generate tables, figures, and exercises. Topics include Sturm-Liouville problems, adjoint and self-adjoint problems, the application of Green's functions to solving nonhomogeneous PDEs, and practical numerical methods used by engineers, including finite difference, finite element, and spectral methods. The book includes many important applications of PDEs.

Prelude to 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
Separation of Variables for Linear, Homogeneous PDEs
Eigenvalue Problems
Prelude to 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
Using Separation of Variables to Solve the Big Three PDEs
Prelude to Chapter 3
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
Prelude to Chapter 4
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
Prelude to Chapter 5
Characteristics
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
Prelude to Chapter 6
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
Prelude to Chapter 7
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
Prelude to Chapter 8
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 L[superscript 2] Norm and Convergence in the Mean
Generalized Fourier Series; Parseval's Equality and Completeness
Prelude to Chapter 9
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
Prelude to Chapter 10
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
Prelude to Chapter 11
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
Other Important Theorems
Existence and Uniqueness Theorems
A Menagerie of PDEs
MATLAB Code for Figures and Exercises
Answers to Selected Exercises
Index

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