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Symmetry Analysis of Differential Equations
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Preface xi Acknowledgments xiii 1 An Introduction 1 1.1 What is a Symmetry? 1 1.2 Lie Groups 4 1.3 Invariance of Differential Equations 6 1.4 Some Ordinary Differential Equations 8 Exercises 12 2 Ordinary Differential Equations 15 2.1 Infinitesimal Transformations 19 2.2 Lie's Invariance Condition 23 Exercises 27 2.3 Standard Integration Techniques 28 2.3.1 Linear Equations 28 2.3.2 Bernoulli Equation 30 2.3.3 Homogeneous Equations 31 2.3.4 Exact Equations 32 2.3.5 Riccati Equations 35 Exercises 37 2.4 Infinitesimal Operator and Higher Order Equations 38 2.4.1 The Infinitesimal Operator 38 2.4.2 The Extended Operator 39 2.4.3 Extension to Higher Orders 40 2.4.4 First-Order Infinitesimals (revisited) 40 2.4.5 Second-Order Infinitesimals 41 2.4.6 The Invariance of Second-Order Equations 42 2.4.7 Equations of arbitrary order 43 2.5 Second-Order Equations 43 Exercises 55 2.6 Higher Order Equations 56 Exercises 61 2.7 ODE Systems 61 2.7.1 First Order Systems 61 2.7.2 Higher Order Systems 67 Exercises 71 3 Partial Differential Equations 73 3.1 First-Order Equations 73 3.1.1 What Do We Do with the Symmetries of PDEs? 77 3.1.2 Direct Reductions 80 3.1.3 The Invariant Surface Condition 83 Exercises 84 3.2 Second-Order PDEs 84 3.2.1 Heat Equation 84 3.2.2 Laplace's Equation 91 3.2.3 Burgers' Equation and a Relative 94 3.2.4 Heat Equation with a Source 100 Exercises 107 3.3 Higher Order PDEs 109 Exercises 115 3.4 Systems of PDEs 115 3.4.1 First-Order Systems 115 3.4.2 Second-Order Systems 120 Exercises 124 3.5 Higher Dimensional PDEs 126 Exercises 132 4 Nonclassical Symmetries and Compatibility 133 4.1 Nonclassical Symmetries 133 4.1.1 Invariance of the Invariant Surface Condition 135 4.1.2 The Nonclassical Method 137 4.2 Nonclassical Symmetry Analysis and Compatibility 146 4.3 Beyond Symmetries Analysis-General Compatibility 147 4.3.1 Compatibility with First-Order PDEs-Charpit's Method 149 4.3.2 Compatibility of Systems 157 4.3.3 Compatibility of the Nonlinear Heat Equation 159 Exercises 160 4.4 Concluding Remarks 161 Solutions 163 References 171 Index 175

DANIEL J. ARRIGO, PhD, is Professor in the Department ofMathematics at the University of Central Arkansas. The author ofover 30 journal articles, his research interests include theconstruction of exact solutions of PDEs; symmetry analysis ofnonlinear PDEs; and solutions to physically important equations,such as nonlinear heat equations and governing equations modelingof granular materials and nonlinear elasticity. In 2008, Dr. Arrigoreceived the Oklahoma-Arkansas Section of the MathematicalAssociation of America s Award for Distinguished Teaching ofCollege or University Mathematics.