Partial Differential Equations of Applied Mathematics (1998 292s OCR pdf)_D.H. Sattinger.pdf

文档介绍：14
12 n=12
10
8
6
4
n=4
2
0
−2
−4
−3 −2 −1 0 1 2 3
Contents
1 The Heat Equation 1
The Fundamental Solution .................... 1
Fourier series & separation of variables ............. 12
Brownian Motion ......................... 19
Exercises ............................. 26
2 Laplace's Equation
Conservative vector fields and potentials .
Separation of Variables ..........
Analytic function theory ...
The Helmholtz equation . .
EXERCISES .......
3 The Wave Equation
The wave equation in . 3 dimensions . .
Characteristic curves in the plane ......
Characteristic surfaces in higher dimensions . .
The Klein Gordon Equation ........
EXERCISES .........
4 Fourier transforms and distributions
The Fourier Transform ........
Distributions ........
Fundamental Solutions ...
Exercises ...........
5 The Calculus of Variations
The Euler-Lagrange Equations .
CONTENTS
The Dirichlet Principle. .
Hilbert Space. . .
The Wave Equation. .
Eigenvalue Problems . . .
EXERCISES. . . . . .
6 Finite Element Methods
7 The Maximum Principle
ELLIPTICAND PARABOLIC INEQUALITIES .
MONOTONEMETHODS . . . . . . . . . . . .
CRASHCOURSE IN ELLIPTICEQUATIONS . . .
The method of moving planes . . . . . . . .
EXERCISES. . . . . . . . . ,
8 Nonlinear Evolution Equations 143
INTERPOLATIONINEQUALITIES . . . . . . . . . . . . . . . . . 143
Sobolev Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 146
9 Shocks and Gas Dynamics
Conservation Laws . . . . . ,
D7Alembert'sParadox .
Transonic flow . . . . . .
Hyperbolic Conservation Laws .
Encrgy and Entropy . . . . . .
10 Bifurcation in Dissipative Systems
Prototypical Examples . . . . . . .
Functional Analysis . . . . . . .
The L