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REFEREN CES . 156 9 Transforma.tion of a Boundary Value Problem to an Initial Value Problem . 157 9.0 Introduction . 157 9.1 Blasius Equation in Boundary Layer Flow . 157 9.2 Longitudinal Impact of Nonlinear Viscoplastic Rods . 163 9.3 Summary . 168 REFERENCES . . . . . . . . . . . . . . . . . . 168 . 10 From Nonlinear to Linear Differential Equa.tions Using Transformation Groups. . . . . . . . . . . . . . 169 . 10.1 From Nonlinear to Linear Differential Equations . 170 10.2 Application to Ordinary Differential Equations -Bernoulli's Equation . . . . . . . . . . . 173 10.3 Application to Partial Differential Equations -A Nonlinear Chemical Exchange Process . 178 10.4 Limitations of the Inspectional Group Method . 187 10.5 Summary . 188 REFERENCES . . . . 188 11 Miscellaneous Topics . 190 11.1 Reduction of Differential Equations to Algebraic Equations 190 11.2 Reduction of Order of an Ordinary Differential Equation . 191 11.3 Transformat.ion From Ordinary to Partial Differential Equations-Search for First Integrals . . . . . . " 193 . 11.4 Reduction of Number of Variables by Multiparameter Groups of Transformations . . . . . . . . .. . . . 194 11.5 Self-Similar Solutions of the First and Second Kind . . 202 11.6 Normalized Representation and Dimensional Consideration 204 REFERENCES .206 Problems . 208 .220 Index .. Chapter 1 INTRODUCTION AND GENERAL OUTLINE Physical problems in engineering science are often described by dif­ ferential models either linear or nonlinear. There is also an abundance of transformations of various types that appear in the literature of engineer­ ing and mathematics that are generally aimed at obtaining some sort of simplification of a differential model.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction and General Outline

Abstract
Physical problems in engineering science are often described by differential models either linear or nonlinear. There is also an abundance of transformations of various types that appear in the literature of engineering and mathematics that are generally aimed at obtaining some sort of simplification of a differential model.
R. Seshadri, T. Y. Na

Chapter 2. Concepts of Continuous Transformation Groups

Abstract
The foundation of the group — theoretic method is contained in the general theories of continuous transformation groups that were introduced and treated extensively by Lie 1, Lie and Engel 2 and Lie and Scheffers 3 in the latter part of the last century. Subsequently, the books by Cohen 4, Campbell 5, Eisenhart 6, Ovsjannikov 7, Bluman and Cole 8, have contributed greatly to the development and clarification of many of Lie’s theories, particularly its applications to the invariant solutions of differential equations. In the literature of engineering and applied sciences, the works of Birkhoff 9, Morgan 10, Hansen 11, Na, Abbott and Hansen 12, Na and Hansen 13 and Ames 14,15 give quite extensively the general theories involved in the similarity solutions of partial differential equations as applied to engineering problems. It is assumed, however, that the average engineer may not be thoroughly acquainted with the concepts of that branch of modern algebra designated as group theory. For this reason as well as for clarifications of the terms and concepts involved, a brief review of some of the key aspects of the theory of transformation groups will be given in this chapter. Emphasis will be placed on presenting the ideas of Lie groups in a simple and clear manner suitable for an engineer and scientist, instead of the rigorous and mathematically elegant approach used in the books by Eisenhart 6, Ovsjannikov 7 and Blumau and Cole 8.
R. Seshadri, T. Y. Na

Chapter 3. A Survey of Methods for Determining Similarity Transformations

Abstract
In this chapter, different methods for determining similarity transformations of partial differential equations will be discussed. A similarity transformation reduces the number of independent variables in the partial differential equations. The transformed system of equations and auxiliary conditions is known as a “similarity representation”.
R. Seshadri, T. Y. Na

Chapter 4. Application of Similarity Analysis to Problems in Science and Engineering

Abstract
Different methods for carrying out similarity analysis of partial differential equations were discussed in Chapter 3 with particular reference to the linear heat equation. The methods were classified into (1) direct methods and (2) group-theoretic methods. In the direct methods, the concept of group invariance is not explicitly invoked. They are straightforward and simple to apply. Since the direct methods are based on assumed transformations, the resulting solutions are restrictive. The group-theoretic methods on the other hand are based upon the invocation of invariance under groups of transformations of the partial differential equations and the auxiliary conditions. Group-theoretic methods such as Birkhoff-Morgan method and the Heliums-Churchill procedure start out by assuming a specific form of the group. Therefore, the resulting similarity solutions are restrictive. The simplicity of these methods is on account of the fact that only algebraic equations (resulting from invocation of invariance) need to be solved. On the other hand, deductive group procedures while being systematic and more rigorous and tedious.
R. Seshadri, T. Y. Na

Chapter 5. Similarity Analysis of Boundary Value Problems with Finite Boundaries

Abstract
It is commonly believed that similarity analysis of boundary value problems in science and engineering is domain-and-boundary condition limited, in that semi-infinite or infinite domains are required. A review of literature on similarity would, indeed, reveal that similarity solutions are mostly available for boundary value problems that lack a characteristic length in one or more coordinate directions. In his book, Hansen 1 points out that problems with finite boundaries associated with finite, non-zero values do not usually possess similarity solutions. Therefore, he suggests that a lack of characteristic length in a coordinate direction could be used as a hint to proceed with similarity analysis and seek possible solutions.
R. Seshadri, T. Y. Na

Chapter 6. On Obtaining Non-Similar Solutions from Similar Solutions

Abstract
We have seen in the earlier chapters, that a similarity representation is obtainable for a boundary value problem provided the governing differential equations and the associated boundary conditions are invariant under a group of transformations. However, if any of the equations and boundary conditions is not invariant under a group, then the problem becomes nonsimilar.
R. Seshadri, T. Y. Na

Chapter 7. Moving Boundary Problems Governed by Parabolic Equations

Abstract
In order to carry out similarity analysis of moving boundary problems governed by parabolic partial differential equations, it is necessary to ascertain whether the speed of propagation of the moving boundary is infinite or finite. This can usually be accomplished by carefully investigating the physical formulation of the boundary value problem. For the purpose of similarity analysis, such problems can be classified into (1) problems that involve a change of phase, and (2) problems without a change of phase. It will be seen that in boundary value problems with a change of phase, the moving boundary would advance with a finite speed of propagation. However, the moving boundary could propagate with either infinite or finite speed in problems where no phase change is involved.
R. Seshadri, T. Y. Na

Chapter 8. Similarity Analysis of Wave Propagation Problems

Abstract
A wave is any recognizable feature of disturbance that is transferred from one part of the medium to another with a recognizable velocity of propagation. There are two main classes of wave motion that can arise in physical situations: (1) propagation of waves along the characteristics of the governing hyperbolic partial differential equation, and (2) non-characteristic wave propagation for which the wave does not move along the characteristics. In this chapter, we will discuss different aspects of similarity analysis and the role of group invariance, as they relate to wave-propagation problems.
R. Seshadri, T. Y. Na

Chapter 9. Transformation of a Boundary Value Problem to an Initial Value Problem

Abstract
The method for transforming nonlinear boundary value problems to initial value problems was first introduced by Toepfer1 in 1912 in his attempt to solve Blasius’ equation in boundary layer theory by a series expansion method. About half a century later Klamkin2, based on the same reasoning, extended the method to a wider class of problems. Major extensions were made possible only when the transformation process was interpreted and re-examined by Na3 in terms of the continuous groups of transformations.
R. Seshadri, T. Y. Na

Chapter 10. From Nonlinear to Linear Differential Equations using Transformation Groups

Abstract
The mathematical descriptions of large number of physical problems arising in science and engineering manifest themselves as nonlinear differential equations. Since there is an abundance of methods for dealing with linear differential equations, a popular practice has been to introduce some form of approximation that would linearize the nonlinear equation. These approximations usually impose certain restrictions on the solutions. In this chapter, we will discuss procedures for deriving mappings based on group- theoretic motivations that transform a nonlinear differential equation into a linear differential equation.
R. Seshadri, T. Y. Na

Chapter 11. Miscellaneous Topics

Abstract
The reduction of differential equations to algebraic equations developed by Moran and Gaggioli1, represents yet another application of the group theory. The method was developed based on the notion of Birkhoff2, that whenever a system of equations transform invariantly under a group of transformations, solutions that are sought are also invariant under the same group.
R. Seshadri, T. Y. Na

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