Nonlinear Partial Differential Equations in Engineering

✍ Scribed by William F. Ames (editor)

• Publisher: Elsevier Science
• Year: 1972
• Tongue: English
• Leaves: 319
• Series: Mathematics in Science and Engineering; 18-II
• Category: Library

✦ Synopsis

In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; andmethods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory.As a result, the book represents a blend of new methods in general computational analysis,and specific, but also generic, techniques for study of systems theory ant its particularbranches, such as optimal filtering and information compression.

- Best operator approximation,- Non-Lagrange interpolation,- Generic Karhunen-Loeve transform- Generalised low-rank matrix approximation- Optimal data compression- Optimal nonlinear filtering

✦ Table of Contents

Nonlinear Partial Differential Equations in Engineering
Copyright Page
Contents
Preface
Contents of Volume I
Chapter 1. Analytic Techniques and Solutions
1.0 Introduction
1.1 Nonlinear Superposition Principles
1.2 Generation of Nonlinear Equations with Built-in Solutions
1.3 Employing the Wrong Equation to Find the Right Solution
1.4 Application of the Quasi-Linear Theory
1.5 Earnshaw's Procedure
1.6 Traveling-Wave Solutions
1.7 Arbitrary Functions
1.8 Equation Splitting
1.9 Inversion of Dependent and Independent Variables
1.10 Contact Transformations
1.11 Parametrization and the Legendre Transformation
1.12 Bäcklund Transformations
1.13 An Example Bäcklund Transformation
1.14 First Integrals
1.1 5 Development of First Integrals
1.16 Lagrange Series Solutions
1.17 Breakdown Theory of Jeffrey–Lax
1.18 Application of the Jeffrey–Lax Method
1.19 Dynamics of Moving Threadline
1.20 Ballooning Vibration of a Moving Threadline
References
Chapter 2. Applications of Modern Algebra
2.0 Introduction
2.1 The Similarity Method of Morgan
2.2 Application of the Morgan Method
2.3 Determination of Groups by Finite Transformations
2.4 Incorporation of the Auxiliary Conditions
2.5 Determination of Absolute Invariants
2.6 Example of Deductive Similarity Method
2.7 Similarity Formalism with Multiparameter Groups
2.8 Infinitesimal Transformations
2.9 Classical Determination of Infinitesimal Transformations
2.10 Nonclassical Determination of Infinitesimal Transformations
2.11 The Nonclassical Method and Simultaneous Equations
2.12 Some Similarity Literature
2.13 Transformation of Boundary-Value Problems into Initial-Value Problems—Single Equations
2.14 Transformation of Boundary-Value Problems into Initial-Value Problems- Simultaneous Equations
References
Chapter 3. Approximate Methods
3.0 Introduction
3.1 Weighted Residual Methods (WRM)
3.2 Novel Applications of WRM in Fluid Mechanics
3.3 WRM in Transport Phenomena—Some Recent Literature
3.4 WRM in Dynamics and Solid Mechanics
3.5 Comments on WRM Theory
3.6 Maximum Principles—Ordinary Differential Equations
3.7 Maximum Principles—Partial Differential Equations
3.8 Quasi Linearization
3.9 Regular Perturbation and Irregular Domains
3.10 Classical Regular Perturbation
3.11 The Perturbation Method of Keller et al
3.12 Singular Perturbation
3.13 Lighthill's Method of Strained Coordinates
3.14 Miscellaneous Asymptotic Procedures
References
Chapter 4. Numerical Methods
4.0 Introduction
A. Finite Elements
4.1 Introduction to Finite Elements
4.2 Formulation of Finite Element Characteristics
4.3 Theoretical Comments on Displacement Functions
4.4 Additional Elements in Two and Three Dimensions
4.5 Finite Elements and Field Problems
4.6 Finite Elements and Nonlinear Problems
B. Numerical Solutions in Fluid Mechanics
4.7 Preliminary Remarks
4.8 Finite Elements and Unsteady Flow
4.9 Stream Function—Vorticity Techniques
4.10 Primitive Variable Methods
4.11 Vector Potential Methods
C. Some New Directions
4.12 Introductory Remarks
4.13 Predictor–Corrector Methods
4.14 Functional Methods
4.15 Reformulation in New Independent Variables
References
Author Index
Subject Index