Generalized Collocations Methods: Solutions to Nonlinear Problems

✍ Scribed by N. Bellomo, Bertrand Lods, Roberto Revelli

• Publisher: Springer
• Year: 2007
• Tongue: English
• Leaves: 207
• Series: Modeling and Simulation in Science, Engineering and Technology
• Edition: 1
• Category: Library

✦ Synopsis

Analysis of nonlinear models and problems is crucial in the application of mathematics to real-world problems. This book approaches this important topic by focusing on collocation methods for solving nonlinear evolution equations and applying them to a variety of mathematical problems. These include wave motion models, hydrodynamic models of vehicular traffic flow, convection-diffusion models, reaction-diffusion models, and population dynamics models. The book may be used as a textbook for graduate courses on collocation methods, nonlinear modeling, and nonlinear differential equations. Examples and exercises are included in every chapter.

✦ Table of Contents

Cover......Page 1
Generalized Collocation Methods Solutions to Nonlinear Problems......Page 4
ISBN-13: 9780817645250 e-ISBN: 9780817646103......Page 5
Contents......Page 6
Aims and Users......Page 10
Contents......Page 11
Critical Analysis......Page 13
1.1 Introduction......Page 14
1.2 Some Models in Applied Sciences......Page 15
1.3 Dimensional Analysis......Page 29
1.4 Classification of Models and Mathematical Problems......Page 33
1.5 Guidelines for the Application of Collocation Methods......Page 41
1.6 Problems......Page 43
2.1 Introduction......Page 46
2.2 Collocation Methods in One Space Dimension......Page 47
2.3 Collocation and Interpolation in Two Space Dimensions......Page 52
2.4 Examples and Applications......Page 54
2.5 Critical Analysis......Page 63
2.6 Problems......Page 67
3.1 Introduction......Page 70
3.2 On the Solution of Initial Value Problems......Page 71
3.3 On the Solution of the Third-Order KdV Model......Page 75
3.4 On the Solution of the Fifth-Order KdV Model......Page 79
3.5 Additional Applications and Discussion......Page 81
3.6 On the Solution of Ordinary Differential Equations......Page 84
3.7 Problems......Page 88
4.1 Introduction......Page 90
4.2 Problems with Linear Boundary Conditions......Page 91
4.3 Traffic Flow Model with Dirichlet Boundary Conditions......Page 98
4.4 Nonlinear Diffusion Models with Neumann and Robin Boundary Conditions......Page 105
4.5 Problems with Known Analytic Solutions......Page 108
4.6 On the Selection of the Number of Nodes......Page 112
4.7 Problems......Page 114
5.1 Introduction......Page 116
5.2 Mathematical Problems......Page 117
5.3 Solution Methods......Page 121
5.4 Initial-Boundary Value Problems on a Slab......Page 126
5.5 Application: The Heat Equation over a Square......Page 128
5.6 Reaction-Diffusion Equations......Page 131
5.7 Critical Analysis......Page 135
5.8 Problems......Page 138
6.1 Introduction......Page 140
6.2 On the Statement of Some Ill-Posed Problems......Page 142
6.3 Problems with Nonlinear Boundary Conditions......Page 146
6.4 Problems with Unspecified Boundary Conditions......Page 151
6.5 On the Identification of Source Terms......Page 154
6.6 Collocation Methods and Integro-Differential Equations......Page 165
6.7 Collocations and Orthogonal Approximation......Page 170
6.8 Critical Analysis......Page 175
6.9 Problems......Page 177
Appendix Scientific Programs......Page 178
References......Page 198
Subject Index......Page 204