Nonlinear Partial Differential Equations
A Symposium on Methods of Solution
- 1st Edition - September 20, 2014
- Imprint: Academic Press
- Editor: W. F. Ames
- Language: English
- Paperback ISBN:9 7 8 - 1 - 4 8 3 2 - 0 8 0 7 - 7
- eBook ISBN:9 7 8 - 1 - 4 8 3 2 - 2 1 5 0 - 2
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… Read more
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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.
List of Contributors
Foreword
Preface
Generalized Similarity Analysis of Partial Differential Equations
Introduction
Types of Similarity Analyses
Free Parameter Analysis
Separation of Variables Method
The Group Theory Approach
References
Vector Eigenfunction Expansions for the Growth of Taylor Vortices in the Flow between Rotating Cylinders
1. Introduction
2. The Governing Equations
3. The Linear Problem
4. The Growth of Taylor Vortices
5. Eigenfunction Expansions
6. Discussion
References
New Methods for the Solution of Partial Differential Equations
1. Introduction
2. Partial Differential Equations in Dynamic Programming
3. Quasilinearization
4. Novel Difference Techniques
5. Novel Difference Techniques
6. Infinite Systems of Ordinary Differential Equations
7. Laplace Transform Techniques
8. Quadrature Techniques
9. Perturbation Techniques
References
Ad hoc Exact Techniques for Nonlinear Partial Differential Equations
1. Introduction
2. Separation of Variables
3. Further Specific Forms
4. Assumed Relations between Dependent Variables
5. Equations Equivalent to Linear Forms
6. Equation Splitting
7. Equation Splitting and the Navier-Stokes Equations
References
The Lubrication Approximation Applied to Non-Newtonian Flow Problems: a Perturbation Approach
1. Introduction
2. The Lubrication Approximation
3. Equations of State for Non-Newtonian Fluids
4. Perturbation and Iterative Solution Scheme
5. Extension to Include Unsteadiness, Compressibility, and Heat Effects
6. Discussion
References
The Computation of Compressible Boundary-Layer Flow
Text
References
Integral Equations for Nonlinear Problems in Partial Differential Equations
Introduction
1. Boundary Value Problems for Elliptic Equations
2. Upper and Lower Function for Volterra Equations with Monotonic Integrands
3. A Nonlinear Initial Value Problem
References
Electrical Problems Modeled by Nonlinear Partial Differential Equations
Text
References
Difference Methods and Soft Solutions
1. Soft Solutions
2. Weak Solutions
3. Exact Difference Methods
4. Second Order Equations
References
Numerical Solution of the Nonlinear Equations for Two-Phase Flow through Porous Media
Introduction
The Differential Equations
Solution by Finite Difference Equations
Evaluation of the Nonlinear Coefficients S'
Limiting Form of Equations at Zero Capillary Pressure
Use of "Upstream" Values of Coefficients KN and KW
Existence of Discontinuity
Possible Improvements
References
An Extrapolated Crank-Nicolson Difference Scheme for Quasilinear Parabolic Equations
Text
References
Heat Transfer to the Endwall of a Shocktube. A Variational Analysis
Introduction
A Least-Error Problem for Transport Experiments
The Shocktube Experiment
The Thermal Conduction Model
Energy Equation Transformations
Variational Formulation
A Computational Procedure
Some Numerical Results
Discussion
References
A Synergetic Approach to Problems of Nonlinear Dispersive Wave Propagation and Interaction
I. Introduction
II. The Synergetic Approach
III. The Nonlinear One-Dimensional Lattice
IV. Solitons, the Korteweg-de Vries Equation, and Some Computational Results
V. Synergetics—Future Directions
References
Uniformization of Asymptotic Expansions
I. Introduction
II. The Uniformization Method
III. Results and Open Problems
References
High Order Accurate Difference Methods in Hydrodynamics
1. Introduction
2. Trends in Lagrange Calculations
3. Eulerian Calculations in Three Independent Variables
4. Two Step Lax-Wendroff Schemes
5. Instabilities of the Nonlinear Type
6. Navier-Stokes Equations
7. Conclusions
References
Nonlinear Problems in the Dynamics of Thin Shells
Text
References
Index
- Edition: 1
- Published: September 20, 2014
- Imprint: Academic Press
- Language: English
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