Partial Differential Equations With Numerical

This is the softcover reprint of the very popular hardcover edition. The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering. The main theme is the integration of the theory of linear PDEs and the numerical solution of such equations. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. As preparation, the two-point boundary value problem and the initial-value problem for ODEs are discussed in separate chapters. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. Some background on linear functional analysis and Sobolev spaces, and also on numerical linear algebra, is reviewed in two appendices.

Introduction	p. 1
Background	p. 1
Notation and Mathematical Preliminaries	p. 4
Physical Derivation of the Heat Equation	p. 7
Problems	p. 12
A Two-Point Boundary Value Problem	p. 15
The Maximum Principle	p. 15
Green's Function	p. 18
Variational Formulation	p. 20
Problems	p. 23
Elliptic Equations	p. 25
Preliminaries	p. 25
A Maximum Principle	p. 26
Dirichlet's Problem for a Disc. Poisson's Integral	p. 28
Fundamental Solutions. Green's Function	p. 30
Variational Formulation of the Dirichlet Problem	p. 32
A Neumann Problem	p. 35
Regularity	p. 37
Problems	p. 38
Finite Difference Methods for Elliptic Equations	p. 43
A Two-Point Boundary Value Problem	p. 43
Poisson's Equation	p. 46
Problems	p. 49
Finite Element Methods for Elliptic Equations	p. 51
A Two-Point Boundary Value Problem	p. 51
A Model Problem in the Plane	p. 57
Some Facts from Approximation Theory	p. 60
Error Estimates	p. 63
An A Posteriori Error Estimate	p. 66
Numerical Integration	p. 67
A Mixed Finite Element Method	p. 71
Problems	p. 73
The Elliptic Eigenvalue Problem	p. 77
Eigenfunction Expansions	p. 77
Numerical Solution of the Eigenvalue Problem	p. 88
Problems	p. 93
Initial-Value Problems for ODEs	p. 95
The Initial Value Problem for a Linear System	p. 95
Numerical Solution of ODEs	p. 101
Problems	p. 106
Parabolic Equations	p. 109
The Pure Initial Value Problem	p. 109
Solution by Eigenfunction Expansion	p. 114
Variational Formulation. Energy Estimates	p. 120
A Maximum Principle	p. 122
Problems	p. 124
Finite Difference Methods for Parabolic Problems	p. 129
The Pure Initial Value Problem	p. 129
The Mixed Initial-Boundary Value Problem	p. 138
Problems	p. 146
The Finite Element Method for a Parabolic Problem	p. 149
The Semidiscrete Galerkin Finite Element Method	p. 149
Some Completely Discrete Schemes	p. 156
Problems	p. 159
Hyperbolic Equations	p. 163
Characteristic Directions and Surfaces	p. 163
The Wave Equation	p. 166
First Order Scalar Equations	p. 169
Symmetric Hyperbolic Systems	p. 173
Problems	p. 181
Finite Difference Methods for Hyperbolic Equations	p. 185
First Order Scalar Equations	p. 185
Symmetric Hyperbolic Systems	p. 192
The Wendroff Box Scheme	p. 196
Problems	p. 198
The Finite Element Method for Hyperbolic Equations	p. 201
The Wave Equation	p. 201
First Order Hyperbolic Equations	p. 205
Problems	p. 216
Some Other Classes of Numerical Methods	p. 217
Collocation methods	p. 217
Spectral Methods	p. 218
Finite Volume Methods	p. 219
Boundary Element Methods	p. 221
Problems	p. 223
Some Tools from Mathematical Analysis	p. 225
Abstract Linear Spaces	p. 225
Function Spaces	p. 231
The Fourier Transform	p. 238
Problems	p. 240
Orientation on Numerical Linear Algebra	p. 245
Direct Methods	p. 245
Iterative Methods. Relaxation, Overrelaxation, and Acceleration	p. 246
Alternating Direction Methods	p. 248
Preconditioned Conjugate Gradient Methods	p. 249
Multigrid and Domain Decomposition Methods	p. 250
Bibliography	p. 253
Index	p. 257
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Partial Differential Equations With Numerical Methods

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