| The present book includes a comprehensive treatment of dimensional analysis. rhere is a full discussion of aspects of Lie groups of point transformations (point symmetries), contact symmetries, and higher-order symmetries that are essential for filing solutions of differential equations. No knowledge of group theory is assumed. Emphasis is placed on explicit algorithms to discover symmetries and integrating factors admitted by a given differential equation and to construct solutions and first integrals resulting from such symmetries and integrating factors. This book should be particularly suitable for applied mathematicians, engineers, and scientists interested in how to find systematically explicit solutions of differential equations. Almost all examples are taken from physical and engineering problems including those concerned with heat conduction, wave propagation, and fluid flow. |
Preface Introduction 1 Dimensional Analysis， Modeling，and Invariance 1.1 Introduction 1.2 Dimensional Analysis: Buckingham Pi-Theorem 1.2.1 Assumptions Behind Dimensional Analysis 1.2.2 Conclusions from Dimensional Analysis 1.2.3 Proof of the Buckingham Pi-Theorem 1.2.4 Examples 1.3 Application of Dimensional Analysis to PDEs 1.3.1 Examples 1.4 Generalization of Dimensional Analysis: Invariance of PDEs Under Scalings of Variables 1.5 Discussion 2 Lie Groups of Transformations and Infinitesimal Transformations 2.1 Introduction 2.2 Lie Groups of Transformations 2.2.1 Groups 2.2.2 Examples of Groups 2.2.3 Group of Transformations 2.2.4 One-Parameter Lie Group of Transformations 2.2.5 Examples of One-Parameter Lie Groups of Tranformations 2.3 Infinitesimal Transformations 2.3.1 First Fundamental Theorem of Lie 2.3.2 Examples Illustrating Liet's First Fundamenatl Theorem 2.3.3 Infinitesimal Generatiors 2.3.4 Invariant Functions 2.3.5 Canonical Coordinates 2.3.6 Examples of Sets of Canonical Coordinates 2.4 Point Transformations and Extended Transformation(Prolongations) 2.4.1 Extended Group of Point Transformations:One Dependent and One Independent Variable …… 3 Ordinary Differential Equations(ODEs) 4 Partial Differential Equations(PDEs) References Author Index Subject Index |
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