Ames Nonlinear Partial Differential Equations

A Symposium on Methods of Solution
1. Auflage 2014
ISBN: 978-1-4832-2150-2
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 332 Seiten, Web PDF

Nonlinear Partial Differential Equations: A Symposium on Methods of Solution is a collection of papers presented at the seminar on methods of solution for nonlinear partial differential equations, held at the University of Delaware, Newark, Delaware on December 27-29, 1965. The sessions are divided into four Symposia: Analytic Methods, Approximate Methods, Numerical Methods, and Applications. Separating 19 lectures into chapters, this book starts with a presentation of the methods of similarity analysis, particularly considering the merits, advantages and disadvantages of the methods. The subsequent chapters describe the fundamental ideas behind the methods for the solution of partial differential equation derived from the theory of dynamic programming and from finite systems of ordinary differential equations. These topics are followed by reviews of the principles to the lubrication approximation and compressible boundary-layer flow computation. The discussion then shifts to several applications of nonlinear partial differential equations, including in electrical problems, two-phase flow, hydrodynamics, and heat transfer. The remaining chapters cover other solution methods for partial differential equations, such as the synergetic approach. This book will prove useful to applied mathematicians, physicists, and engineers.

Autoren/Hrsg.

Ames, W. F.

Inhaltsverzeichnis

1;Front Cover;1
2;Nonlinear Partial Differential Equations: A Symposium on Methods of Solution;4
3;Copyright Page;5
4;Table of Contents;12
5;List of Contributors;6
6;Foreword;8
7;Preface;10
8;Chapter 1. Generalized Similarity Analysis of Partial Differential Equations;20
8.1;Introduction;20
8.2;Types of Similarity Analyses;21
8.3;Free Parameter Analysis;22
8.4;Separation of Variables Method;27
8.5;The Group Theory Approach;29
8.6;References;36
9;Chapter 2. Vector Eigenfunction Expansions for the Growth of Taylor Vortices in the Flow between Rotating Cylinders;38
9.1;1. Introduction;38
9.2;2. The Governing Equations;40
9.3;3. The Linear Problem;44
9.4;4. The Growth of Taylor Vortices;50
9.5;5. Eigenfunction Expansions;53
9.6;6. Discussion;57
9.7;References;60
10;Chapter 3. New Methods for the Solution of Partial Differential Equations;62
10.1;1. Introduction;62
10.2;2. Partial Differential Equations in Dynamic Programming;63
10.3;3. Quasilinearization;65
10.4;4. Novel Difference Techniques;65
10.5;5. Novel Difference Techniques;66
10.6;6. Infinite Systems of Ordinary Differential Equations;67
10.7;7. Laplace Transform Techniques;67
10.8;8. Quadrature Techniques;68
10.9;9. Perturbation Techniques;71
10.10;References;71
11;Chapter 4. Ad hoc Exact Techniques for Nonlinear Partial Differential Equations;74
11.1;1. Introduction;74
11.2;2. Separation of Variables;75
11.3;3. Further Specific Forms;78
11.4;4. Assumed Relations between Dependent Variables;81
11.5;5. Equations Equivalent to Linear Forms;85
11.6;6. Equation Splitting;87
11.7;7. Equation Splitting and the Navier-Stokes Equations;88
11.8;References;91
12;Chapter 5. The Lubrication Approximation Applied to Non-Newtonian Flow Problems: A Perturbation Approach;92
12.1;1. Introduction;92
12.2;2. The Lubrication Approximation;94
12.3;3. Equations of State for Non-Newtonian Fluids;95
12.4;4. Perturbation and Iterative Solution Scheme;100
12.5;5. Extension to Include Unsteadiness, Compressibility, and Heat Effects;118
12.6;6. Discussion;121
12.7;References;125
13;Chapter 6. The Computation of Compressible Boundary-Layer Flow;128
13.1;Text;128
13.2;References;142
14;Chapter 7. Integral Equations for Nonlinear Problems in Partial Differential Equations;144
14.1;Introduction;144
14.2;1. Boundary Value Problems for Elliptic Equations;144
14.3;2. Upper and Lower Function for Volterra Equations with Monotonic Integrands;150
14.4;3. A Nonlinear Initial Value Problem;154
14.5;References;158
15;Chapter 8. Electrical Problems Modeled by Nonlinear Partial Differential Equations;160
15.1;Text;160
15.2;References;179
16;Chapter 9. Difference Methods and Soft Solutions;180
16.1;1. Soft Solutions;180
16.2;2. Weak Solutions;184
16.3;3. Exact Difference Methods;184
16.4;4. Second Order Equations;187
16.5;References;189
17;Chapter 10. Numerical Solution of the Nonlinear Equations for Two-Phase Flow through Porous Media;190
17.1;Introduction;190
17.2;The Differential Equations;190
17.3;Solution by Finite Difference Equations;195
17.4;Evaluation of the Nonlinear Coefficients S';197
17.5;Limiting Form of Equations at Zero Capillary Pressure;200
17.6;Use of "Upstream" Values of Coefficients KN and KW;204
17.7;Existence of Discontinuity;205
17.8;Possible Improvements;208
17.9;References;209
18;Chapter 11. An Extrapolated Crank-Nicolson Difference Scheme for Quasilinear Parabolic Equations;212
18.1;Text;212
18.2;References;220
19;Chapter 12. Heat Transfer to the Endwall of a Shocktube. A Variational Analysis;222
19.1;Introduction;222
19.2;A Least-Error Problem for Transport Experiments;223
19.3;The Shocktube Experiment;226
19.4;The Thermal Conduction Model;229
19.5;Energy Equation Transformations;230
19.6;Variational Formulation;231
19.7;A Computational Procedure;233
19.8;Some Numerical Results;235
19.9;Discussion;237
19.10;References;240
20;Chapter 13. A Synergetic Approach to Problems of Nonlinear Dispersive Wave Propagation and Interaction;242
20.1;I. Introduction;242
20.2;II. The Synergetic Approach;243
20.3;III. The Nonlinear One-Dimensional Lattice;244
20.4;IV. Solitons, the Korteweg-de Vries Equation, and Some Computational Results;252
20.5;V. Synergetics—Future Directions;264
20.6;References;275
21;Chapter 14. Uniformization of Asymptotic Expansions;278
21.1;I. Introduction;278
21.2;II. The Uniformization Method;284
21.3;III. Results and Open Problems;293
21.4;References;296
22;Chapter 15. High Order Accurate Difference Methods in Hydrodynamics;298
22.1;1. Introduction;298
22.2;2. Trends in Lagrange Calculations;298
22.3;3. Eulerian Calculations in Three Independent Variables;300
22.4;4. Two Step Lax-Wendroff Schemes;302
22.5;5. Instabilities of the Nonlinear Type;307
22.6;6. Navier-Stokes Equations;308
22.7;7. Conclusions;309
22.8;References;309
23;Chapter 16. Nonlinear Problems in the Dynamics of Thin Shells;310
23.1;Text;310
23.2;References;325
24;Index;328