Nonlinear Partial Differential Equations

Nonlinear Partial Differential Equations

A Symposium on Methods of Solution

1st Edition - January 1, 1967

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  • Editor: W. F. Ames
  • eBook ISBN: 9781483221502

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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.

Table of Contents

  • List of Contributors



    Generalized Similarity Analysis of Partial Differential Equations


    Types of Similarity Analyses

    Free Parameter Analysis

    Separation of Variables Method

    The Group Theory Approach


    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


    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


    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


    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


    The Computation of Compressible Boundary-Layer Flow



    Integral Equations for Nonlinear Problems in Partial Differential Equations


    1. Boundary Value Problems for Elliptic Equations

    2. Upper and Lower Function for Volterra Equations with Monotonic Integrands

    3. A Nonlinear Initial Value Problem


    Electrical Problems Modeled by Nonlinear Partial Differential Equations



    Difference Methods and Soft Solutions

    1. Soft Solutions

    2. Weak Solutions

    3. Exact Difference Methods

    4. Second Order Equations


    Numerical Solution of the Nonlinear Equations for Two-Phase Flow through Porous Media


    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


    An Extrapolated Crank-Nicolson Difference Scheme for Quasilinear Parabolic Equations



    Heat Transfer to the Endwall of a Shocktube. A Variational Analysis


    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



    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


    Uniformization of Asymptotic Expansions

    I. Introduction

    II. The Uniformization Method

    III. Results and Open Problems


    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


    Nonlinear Problems in the Dynamics of Thin Shells




Product details

  • No. of pages: 332
  • Language: English
  • Copyright: © Academic Press 1967
  • Published: January 1, 1967
  • Imprint: Academic Press
  • eBook ISBN: 9781483221502

About the Editor

W. F. Ames

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