Group Invariance in Engineering Boundary Value Problems

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.