Generalized Collocation Methods

, by Bellomo, Nicola; Lods, Bertrand; Revelli, Roberto; Ridolfi, Luca

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ISBN: 9780817645250 | 081764525X
Cover: Hardcover
Copyright: 10/19/2007

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This book examines various mathematical tools-based on generalized collocation methods-to solve nonlinear problems related to partial differential and integrodifferential equations. Covered are specific problems and models related to vehicular traffic flow, population dynamics, wave phenomena, heat convection and diffusion, transport phenomena, and pollution.

Preface	p. ix
Mathematical Models and Problems in Applied Sciences	p. 1
Introduction	p. 1
Some Models in Applied Sciences	p. 2
Dimensional Analysis	p. 16
Classification of Models and Mathematical Problems	p. 20
Guidelines for the Application of Collocation Methods	p. 28
Problems	p. 30
Lagrange and Sinc Collocation Interpolation Methods	p. 33
Introduction	p. 33
Collocation Methods in One Space Dimension	p. 34
Collocation and Interpolation in Two Space Dimensions	p. 39
Examples and Applications	p. 41
Critical Analysis	p. 50
Problems	p. 54
Nonlinear Initial Value Problems in Unbounded Domains	p. 57
Introduction	p. 57
On the Solution of Initial Value Problems	p. 58
On the Solution of the Third-Order KdV Model	p. 62
On the Solution of the Fifth-Order KdV Model	p. 66
Additional Applications and Discussion	p. 68
On the Solution of Ordinary Differential Equations	p. 71
Problems	p. 75
Nonlinear Initial-Boundary Value Problems in One Space Dimension	p. 77
Introduction	p. 77
Problems with Linear Boundary Conditions	p. 78
Traffic Flow Model with Dirichlet Boundary Conditions	p. 85
Nonlinear Diffusion Models with Neumann and Robin Boundary Conditions	p. 92
Problems with Known Analytic Solutions	p. 95
On the Selection of the Number of Nodes	p. 99
Problems	p. 101
Initial-Boundary Value Problems in Two Space Dimensions	p. 103
Introduction	p. 103
Mathematical Problems	p. 104
Solution Methods	p. 108
Initial-Boundary Value Problems on a Slab	p. 113
Application: The Heat Equation over a Square	p. 115
Reaction-Diffusion Equations	p. 118
Critical Analysis	p. 122
Problems	p. 125
Additional Mathematical Tools for Nonlinear Problems	p. 127
Introduction	p. 127
On the Statement of Some Ill-Posed Problems	p. 129
Problems with Nonlinear Boundary Conditions	p. 133
Problems with Unspecified Boundary Conditions	p. 138
On the Identification of Source Terms	p. 141
Collocation Methods and Integro-Differential Equations	p. 152
Collocations and Orthogonal Approximation	p. 157
Critical Analysis	p. 162
Problems	p. 164
Scientific Programs	p. 165
Introduction	p. 165
Lagrange Interpolation with Chebychev and Equally Spaced Collocations	p. 169
Lagrange and Sinc Interpolation in One Space Dimension	p. 170
Error Computation in One Space Dimension	p. 171
Gaussian Function in Two Space Dimensions	p. 172
Error Computation in Two Space Dimensions	p. 172
Solution of the Third-Order KdV Model	p. 173
Solution of the Fifth-Order KdV Model	p. 175
Traffic Flow Model (1 Variable) with Dirichlet Boundary Conditions	p. 176
Traffic Flow Model (2 Variables) with Dirichlet Boundary Conditions	p. 176
Nonlinear Diffusion Model with Neumann Boundary Conditions	p. 177
Nonlinear Diffusion Model with Robin Boundary Conditions	p. 178
Problems with Known Analytic Solutions	p. 179
Initial-Boundary Value Problem on a Slab	p. 180
Heat Equation over a Square	p. 181
Reaction-Diffusion Equations	p. 182
Vehicular Traffic Model with Nonlinear Boundary Conditions	p. 183
References	p. 185
Subject Index	p. 191
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