An Introduction to Partial Differential Equations with MATLAB 2nd Edition by Matthew Coleman ISBN 1439898464 978-1439898468

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Product details: 

ISBN 10: 1439898464 

ISBN 13: 978-1439898468

Author: Matthew P. Coleman

An Introduction to Partial Differential Equations with MATLAB®, Second Edition illustrates the usefulness of PDEs through numerous applications and helps students appreciate the beauty of the underlying mathematics. Updated throughout, this second edition of a bestseller shows students how PDEs can model diverse problems, including the flow of heat, the propagation of sound waves, the spread of algae along the ocean’s surface, the fluctuation in the price of a stock option, and the quantum mechanical behavior of a hydrogen atom.

Suitable for a two-semester introduction to PDEs and Fourier series for mathematics, physics, and engineering students, the text teaches the equations based on method of solution. It provides both physical and mathematical motivation as much as possible. The author treats problems in one spatial dimension before dealing with those in higher dimensions. He covers PDEs on bounded domains and then on unbounded domains, introducing students to Fourier series early on in the text.

Each chapter’s prelude explains what and why material is to be covered and considers the material in a historical setting. The text also contains many exercises, including standard ones and graphical problems using MATLAB. While the book can be used without MATLAB, instructors and students are encouraged to take advantage of MATLAB’s excellent graphics capabilities. The MATLAB code used to generate the tables and figures is available in an appendix and on the author’s website.

Table of contents: 

Prelude to Chapter 1
1 Introduction
1.1 What are Partial Differential Equations?
1.2 PDEs We Can Already Solve ..
1.3 Initial and Boundary Conditions
1.4 Linear PDEs
Definitions . .
1.5 Linear PDEs The Principle of Superposition
1.6 Separation of Variables for Linear, Homogeneous PDES 1.7 Eigenvalue Problems
Prelude to Chapter 2
2 The Big Three PDEs
2.1 Second-Order, Linear, Homogeneous PDEs with Constant Co-
efficients
2.2 The Heat Equation and Diffusion
2.3 The Wave Equation and the Vibrating String
2.4 Initial and Boundary Conditions for the Heat and Wave Equa- tions.
2.5 Laplace’s Equation The Potential Equation
2.6 Using Separation of Variables to Solve the Big Three PDEs

Prelude to Chapter 3
3 Fourier Series
3.1 Introduction
3.2 Properties of Sine and Cosine
3.3 The Fourier Series
3.4 The Fourier Series, Continued
3.5 The Fourier Series Proof of Pointwise Convergence
3.6 Fourier Sine and Cosine Series
3.7 Completeness

Prelude to Chapter 4
4 Solving the Big Three PDEs on Finite Domains

4.1 Solving the Homogeneous Heat Equation for a Finite Rod

4.2 Solving the Homogeneous Wave Equation for a Finite String

 4.3 Solving the Homogeneous Laplace’s Equation on a Rectangular
Domain…
4.4 Nonhomogeneous Problems

Prelude to Chapter 5
5 Characteristics
5.1 First-Order PDEs with Constant Coefficients
5.2 First-Order PDEs with Variable Coefficients
5.3 The Infinite String
5.4 Characteristics for Semi-Infinite and Finite String Problems
5.5 General Second-Order Linear PDEs and Characteristics

Prelude to Chapter 6
6 Integral Transforms
6.1 The Laplace Transform for PDEs
6.2 Fourier Sine and Cosine Transforms
6.3 The Fourier Transform
6.4 The Infinite and Semi-Infinite Heat Equations
6.5 Distributions, the Dirac Delta Function and Generalized Fourier Transforms
6.6 Proof of the Fourier Integral Formula

Prelude to Chapter 7
7 Special Functions and Orthogonal Polynomials
7.1 The Special Functions and Their Differential Equations
7.2 Ordinary Points and Power Series Solutions; Chebyshev, Hermite and Legendre Polynomials
7.3 The Method of Frobenius; Laguerre Polynomials

7.4 Interlude: The Gamma Function
7.5 Bessel Functions
7.6 Recap: A List of Properties of Bessel Functions and Orthogonal Polynomials

Prelude to Chapter 8
8 Sturm-Liouville Theory and Generalized Fourier Series
8.1 Sturm-Liouville Problems
8.2 Regular and Periodic Sturm-Liouville Problems
8.3 Singular Sturm-Liouville Problems; Self-Adjoint Problems
8.4 The Mean-Square or L2 Norm and Convergence in the Mean
8.5 Generalized Fourier Series; Parseval’s Equality and Completeness

Prelude to Chapter 9

9 PDEs in Higher Dimensions
9.1 PDEs in Higher Dimensions: Examples and Derivations
9.2 The Heat and Wave Equations on a Rectangle; Multiple Fourier Series
9.3 Laplace’s Equation in Polar Coordinates: Poisson’s Integral Formula
9.4 The Wave and Heat Equations in Polar Coordinates
9.5 Problems in Spherical Coordinates
9.6 The Infinite Wave Equation and Multiple Fourier Transforms
9.7 Postlude: Eigenvalues and Eigenfunctions of the Laplace Operator; Green’s Identities for the Laplacian

Prelude to Chapter 10
10 Nonhomogeneous Problems and Green’s Functions
10.1 Green’s Functions for ODEs
10.2 Green’s Function and the Dirac Delta Function
10.3 Green’s Functions for Elliptic PDEs (I): Poisson’s Equation in Two Dimensions
10.4 Green’s Functions for Elliptic PDEs (II): Poisson’s Equation in Three Dimensions; the Helmholtz Equation
10.5 Green’s Functions for Equations of Evolution

Prelude to Chapter 11
11 Numerical Methods
11.1 Finite Difference Approximations for ODEs

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Matthew Coleman,An Introduction,Partial Differential