Partial differential equations with numerical methods

Stig Larsson, Vidar Thomée (auth.)9783540017721, 9783540887058, 3540017720, 3540887059

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.

Table of contents :
Front Matter….Pages I-XI
Introduction….Pages 1-14
A Two-Point Boundary Value Problem….Pages 15-24
Elliptic Equations….Pages 25-41
Finite Difference Methods for Elliptic Equations….Pages 43-49
Finite Element Methods for Elliptic Equations….Pages 51-76
The Elliptic Eigenvalue Problem….Pages 77-94
Initial-Value Problems for Ordinary Differential Equations….Pages 95-108
Parabolic Equations….Pages 109-127
Finite Difference Methods for Parabolic Problems….Pages 129-148
The Finite Element Method for a Parabolic Problem….Pages 149-161
Hyperbolic Equations….Pages 163-183
Finite Difference Methods for Hyperbolic Equations….Pages 185-199
The Finite Element Method for Hyperbolic Equations….Pages 201-216
Some Other Classes of Numerical Methods….Pages 217-224
Back Matter….Pages 225-261