Group Invariance in Engineering Boundary Value Problems

ISBN-13: 9781461295648 / Angielski / Miękka / 2011 / 224 str.

R. Seshadri; T. y. Na

Opis

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."

Spis treści

1 Introduction and General Outline.- References.- 2 Concepts of Continuous Transformation Groups.- 2.0 Introduction.- 2.1 Group Approach.- 2.2 Transformation Groups.- 2.3 The Concept of an Infinitesimal Transformation.- 2.4 Relation Between Global and Infinitesimal Groups Transformations.- 2.5 The Concept of Invariance.- 2.6 Invariance of Differential Equations Under Groups of Transformations.- 2.7 The Extended Group of Transformations.- 2.8 The Characteristic Function.- 2.9 Transformations Involving Two Independent Variables.- 2.10 Transformation Involving Two Dependent and Two Independent Variables.- 2.11 Dimensional and Affine Groups of Transformations.- 2.12 Summary.- References.- 3 A Survey of Methods for Determining Similarity.- 3.0 Introduction.- 3.1 Direct Methods.- 3.2 Group-Theoretic Methods.- 3.3 Summary.- References.- 4 Application of Similarity Analysis to Problems in Sciences and Engineering.- 4.0 Introduction.- 4.1 Laminar Two Dimensional Jet:Separation of Variable Method.- 4.2 Impact of Rods With Nonlinear Material Properties: Separation of Variables Method.- 4.3 Diffusion of Vorticity From a Line Vortex Immersed in a Quiescent Fluid: Dimensional Method.- 4.4 Laminar Boundary Layer Equation: Dimensional Method.- 4.5 Free Convection From a Vertical Needle: Heliums Churchill Method.- 4.6 Deflection of a Semi-Infinite Wedge Shaped Plate: Birkhoff-Morgan Method.- 4.7 Heated Jet: Birkhoff-Morgan Method.- 4.8 Unsteady One-Dimensional Gas Dynamics Equations: Characteristic Function Method.- 4.9 Summary.- References.- 5 Similarity Analysis of Boundary Value Problems With Finite Boundaries.- 5.0 Introduction.- 5.1 Boundary Value Problems With Moving Boundaries.- 5.2 Invariant Boundary and Surface Description.- 5.3 Invariance Under Groups Other Than Dimensional Groups.- 5.4 Summary.- References.- 6 On Obtaining Non-Similar Solutions From Similar Solutions.- 6.0 Introduction.- 6.1 Superposition of Similarity Solutions.- 6.2 The Use of Fundamental Solutions.- 6.3 Pseudo-Similarity Transformations.- 6.4 Similarity Solutions as Asymptotic Limits of the Non-Similar Problem Description.- 6.5 Summary.- References.- 7 Moving Boundary Problems Governed by Parabolic Equations.- 7.0 Introduction.- 7.1 Problems With Phase Change.- 7.2 Problems Without Phase Change.- 7.3 Summary.- References.- 8 Similarity Analysis of Wave Propagation Problems.- 8.0 General.- 8.1 Propagation Along Characteristics.- 8.2 Non-Characteristic Propagation:Shock Waves.- 8.3 Non-Characteristic Propagation:Uniform Propagation Regime.- 8.4 From Translation to Dimensional Group Invariance.- 8.5 Summary.- References.- 9 Transformation of a Boundary Value Problem to an Initial Value Problem.- 9.0 Introduction.- 9.1 Blasius Equation in Boundary Layer Flow.- 9.2 Longitudinal Impact of Nonlinear Viscoplastic Rods.- 9.3 Summary.- References.- 10 From Nonlinear to Linear Differential Equations Using Transformation Groups.- 10.1 From Nonlinear to Linear Differential Equations.- 10.2 Application to Ordinary Differential Equations Bernoulli’s Equation.- 10.3 Application to Partial Differential Equations A Nonlinear Chemical Exchange Process.- 10.4 Limitations of the Inspectional Group Method.- 10.5 Summary.- References.- 11 Miscellaneous Topics.- 11.1 Reduction of Differential Equations to Algebraic Equations.- 11.2 Reduction of Order of an Ordinary Differential Equation.- 11.3 Transformation From Ordinary to Partial Differential Equations-Search for First Integrals.- 11.4 Reduction of Number of Variables by Multiparameter Groups of Transformations.- 11.5 Self-Similar Solutions of the First and Second Kind.- 11.6 Normalized Representation and Dimensional Consideration.- References.- Problems.