EQUIVALENCE TRANSFORMATIONS FOR CLASSES OF DIFFERENTIAL EQUATIONS by Ian Lisle B.Sc. (Australian Environmental Studies) Griffith University M.Sc.St. (Mathematics) University of Queensland A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MATHEMATICS INSTITUTE OF APPLIED MATHEMATICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February, 1992 © Ian Lisle, 1992 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. ! further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allow/ed without my written permission. Department of Maihemaiics The University of British Columbia Vancouver, Canada Date 15^^ Apr,'!, mi DE-6 (2/88) Abstract We consider classes C of differential equations characterized by the presence of arbitrary ele• ments, that is, arbitrary functions or constants. Based on an idea of Ovsiannikov, we develop a systematic theory of equivalence transformations, that is, point changes of variables which map every equation in C to another equation in C. Examples of nontrivial groups of equivalence transformations are found for some linear wave and nonlinear diffusion convection systems, and used to clarify some previously known results. We show how equivalence transformations may be inherited as symmetries of equations in C, leading to a partial symmetry classification for the class C. New symmetry results for a potential system form of the nonlinear diffusion convection equation are derived by this procedure. Finally we show how to use equivalence group information to facilitate complete symmetry classification for a class of differential equations. The method relies on the geometric concept of a moving frame, that is, an arbitrary (possibly noncommuting) basis for differential operators on the space of independent and dependent variables. We show how to choose a frame which is invariant under the action of the equivalence group, and how to rewrite the determining equations for symmetries in terms of this frame. A symmetry classification algorithm due to Reid is modified to deal with the case of noncommuting operators. The result is an algorithm which combines features of Reid's classification algorithm and Cartan's equivalence method. The method is applied to the potential diffusion convection example, and yields a complete symmetry classification in a particularly elegant form. Table of Contents Abstract ii List of Tables vii List of Figures viii Acknowledgments ix 1 Introduction 1 1.1 Differential equations and their transformation 1 1.2 Equivalence of differential equations: Examples 4 1.3 Symmetries and differential equations 12 1.4 Equivalence transformations 15 1.5 Symmetry classification problem 17 2 Transformation Groups and Differential Equations 22 2.1 Transformation groups 22 2.1.1 Transformations, Lie groups 22 2.1.2 Infinitesimal operators 26 2.1.3 Invariant surface 32 2.2 Extension 33 2.2.1 Notation for derivatives 33 2.2.2 Extension of transformation 34 2.2.3 Extension of group operator 38 2.3 Differential equations eind symmetry 39 2.3.1 Differential equations 39 2.3.2 Symmetries of differential equations 41 2.3.3 Algorithmic construction of symmetries 45 3 The Equivalence Group 48 3.1 Class of differential equations 48 3.1.1 Decoupled systems of d.e.'s 48 3.1.2 Class of d.e.'s 49 3.2 Equivalence transformations 52 3.3 Infinitesimal augmented transformations 60 3.3.1 Infinitesimal augmented transformations 60 3.3.2 Algebra of equivalence operators 64 3.3.3 Algorithm for construction of equivalence group 70 3.3.4 Proposition on form of infinitesimals 72 3.3.5 Structure of the equivalence group 74 3.4 Examples of equivalence groups 80 3.4.1 Boltzmann's similarity solution for nonlinear diffusion 80 3.4.2 Nonlinear diffusion-convection equations 88 3.4.3 Wave equations 98 3.4.4 Hamilton's equations 103 4 Symmetry Group Classification 108 4.1 Symmetry classification problem 108 4.1.1 Example: scalar diffusion convection 109 4.2 Partial symmetry classification 110 4.2.1 Symmetry inherited from equivalence group 110 4.2.2 Optimal system of subalgebras 115 4.2.3 Partial symmetry classification for nonlinear diffusion convection 117 4.3 Modification of Reid algorithm 129 4.3.1 Moving frame and determining equations 131 4.3.2 Frame Reid method 139 4.4 Invariant frame 150 4.4.1 Augmented frame 150 4.4.2 Invariant frame 155 4.4.3 Differential invariants 156 4.4.4 Tresse basis 159 4.5 Symmetry classification 162 4.5.1 Invariant form of group classification 165 4.5.2 Potential diffusion convection system 175 5 Conclusion 196 5.1 Further work 196 5.1.1 Isovector method for frame determining system 197 5.2 Conclusions 201 A Algorithms for Frame Systems 203 A.l Reduction to frame involutive form 203 A. 1.1 Orthonomic form 203 A. 1.2 Reduced orthonomic form 205 A. 1.3 Involutive form 206 A.2 Group classification 207 B Structure Constants 210 C Similarity Solution for Nonlinear Diffusion 215 C.l Power law diffusivity 216 C.1.1 Phase reduction 217 C.1.2 Exact shooting 218 C.l.3 Series solution 219 C.2 Modified power law diffusivity 220 C.2.1 Phase reduction 221 C.2.2 Exact shooting 221 C.2.3 Series solution 222 C.3 Discussion 223 Bibliography 225 List of Tables 3.1 Commutator table of equivalence algebra of nonlinear diffusion potential system. 70 3.2 Commutator table for equivalence operators of scalar diffusion convection equation 91 3.3 Commutation relations of equivalence algebra of scalar wave equation 100 4.1 Symmetry classification for scalar nonlinear diffusion convection equation Ill 4.2 Optimal system of subalgebras for nonlinear diffusion convection potential system. 121 4.3 Partial symmetry classification for diffusion convection potential system: Case K{u) = 0 (diffusion equations) 123 4.4 Partial symmetry classification for diffusion convection potential system: Case with nonlinear convection 124 4.5 Nonlocal symmetries inherited from equivalence group of diffusion convection potential system: Case K{u) = 0 (diffusion equations) 125 4.6 Nonlocal symmetries inherited from equivalence group of diffusion convection potential system: Case K{u) ^0 126 4.7 Commutator table of equivalence algebra of nonlinear diffusion equation 168 4.8 Commutation relations for equivalence algebra of diffusion convection potential system 176 vn List of Figures 3.1 Relationship between linearizable diffusion convection potential systems 98 4.1 Classification tree for symmetries of nonlinear diffusion equation 173 4.2 Preliminary classification tree for potential diffusion convection system 189 4.3 Complete symmetry classification tree for diffusion convection potential system. 195 C.l Relation between concentration, flux, and spatial coordinate for Boltzmann's similarity solution with power law diffusivity 217 C.2 Phase portrait for Boltzmann similarity solution with power law diffusivity . 218 vm Acknowledgments I wish to thank my supervisor Dr. G.W. Bluman for originally suggesting the study of equiva• lence transformations, and for his guidance, insight and unfailing sense of direction. Particular thanks are due for his careful and speedy reading of the drafts of my thesis. My wholehearted appreciation also to Dr. Greg Reid for his constant encouragement and support, for his patient explanations of his symmetry classification algorithm, and for making available the MAPLE implementation of this algorithm. Alan Boulton was a constant source of enthusiasm over the last two years, participating in many lively discussions. The insightful lec• tures of Dr. K.Y. Lam on differential geometry provided me with the tools which made §4.3-§4.5 possible. The problem described in Appendix C resulted from discussions with Dr. J.Y. Par- lange. My fellow graduate students have maintained me with their friendship over the years, and have also enriched my knowledge of mathematics, computing, French, and softball. My most heartfelt appreciation to Maria ChiaroUa for helping maintain my sanity when things did not go well. Honourable mention to the Stanley Cup playoffs, the few sunny days in winter, and Piper's Ale. Funding for much of my stay in Canada was provided by the Canadian Commonwealth Scholarship and Fellowship Administration, and their support is gratefully acknowledged. This thesis was typeset using M?gX. Chapter 1 Introduction 1.1 Differential equations and their transformation In dealing with differential equations, a common situation is that one wishes to analyze simul• taneously a whole class of equations of some given type. It is natural to consider 'the class of second order ordinary differential equations' or 'the nonlinear diffusion equation' ut = [D{u)u^]^ Here w, D are arbitrary (smooth) functions of their arguments, at least in some suitable domain of definition. Thus the entire class of equations under consideration is specified by allowing these arbitrary elements to range over all possible functional forms. In this dissertation, I will be concerned with the transformation properties of a given class C of differential equations. Attention will be restricted to invertible 'point' transformations, which act on a coordinate space of the independent and dependent variables. These are the usual 'changes of variables' in differential equations. For (1.1) for example, the most general such change of variables is x' = Fix,y) y' = Gix,y) (1.2) (subject to the Jacobian FxGy — G^Fy being nonzero). Any transformation applied to the variables in a differentiïJ equation (d.e.) yields another differential equation.