Partial Differential Equations : An Introduction with Mathematica and Maple,9789812388155

Partial Differential Equations : An Introduction with Mathematica and Maple

by ;
Edition: 2nd
ISBN13: 9789812388155
ISBN10: 981238815X
Format: Hardcover
Pub. Date: 2004-06-01
Publisher(s): World Scientific Pub Co Inc
List Price: $105.92

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Table of Contents

1. First-order Partial Differential Equations 1(38)
1.1. Introduction
1(3)
1.2. Linear First-order Equations
4(7)
1.3. The Cauchy Problem for First-order Quasi-linear Equations
11(12)
1.4. General Solutions of Quasi-linear Equations
23(5)
1.5. Fully-nonlinear First-order Equations
28(11)
2. Second-order Partial Differential Equations 39(28)
2.1. Linear Equations
39(7)
2.2. Classification and Canonical Forms of Equations in Two Independent Variables
46(13)
2.3. Classification of Almost-linear Equations in Rn
59(8)
3. One Dimensional Wave Equation 67(30)
3.1. The Wave Equation on the Whole Line. D'Alembert Formula
67(11)
3.2. The Wave Equation on the Half-line. Reflection Method
78(6)
3.3. Mixed Problem for the Wave Equation
84(3)
3.4. Inhomogeneous Wave Equation
87(5)
3.5. Conservation of the Energy
92(5)
4. One Dimensional Diffusion Equation 97(26)
4.1. Maximum-minimum Principle for the Diffusion Equation
97(6)
4.2. The Diffusion Equation on the Whole Line
103(12)
4.3. Diffusion on the Half-line
115(3)
4.4. Inhomogeneous Diffusion Equation on the Whole Line
118(5)
5. Weak Solutions, Shock Waves and Conservation Laws 123(46)
5.1. Weak Derivatives and Weak Solutions
123(7)
5.2. Conservation Laws
130(10)
5.3. Burgers' Equation
140(13)
5.4. Weak Solutions. Riemann Problem
153(9)
5.5. Discontinuous Solutions of Conservation Laws. Rankine-Hugoniot Condition
162(7)
6. The Laplace Equation 169(30)
6.1. Harmonic Functions. Maximum-minimum Principle
169(4)
6.2. Green's Identities
173(9)
6.3. Green's Functions
182(3)
6.4. Green's Functions for a Half-space and Sphere
185(8)
6.5. Harnack's Inequalities and Theorems
193(6)
7. Fourier Series and Fourier Method for PDEs 199(56)
7.1. Fourier Series
199(18)
7.2. Orthonormal Systems. General Fourier Series
217(12)
7.3. Fourier Method for the Diffusion Equation
229(9)
7.4. Fourier Method for the Wave Equation
238(5)
7.5. Fourier Method for the Laplace Equation
243(12)
8. Diffusion and Wave Equations in Higher Dimensions 255(32)
8.1. The Diffusion Equation in Three Dimensional Space
255(7)
8.2. Fourier Method for the Diffusion Equation in Higher Dimensions
262(7)
8.3. Kirchoff's Formula for the Wave Equation. Huygens' Principle
269(7)
8.4. Fourier Method for the Wave Equation on the Plane. Nodal Sets
276(11)
References 287(4)
Answers and Hints to Exercises 291(10)
Index 301

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