Handbook of Differential Equations

PrefaceIntroductionHow to Use This BookI.A Definitions and Concepts 1 Definition of Terms 2 Alternative Theorems 3 Bifurcation Theory 4 A Caveat for Partial Differential Equations 5 Chaos in Dynamical Systems 6 Classification of Partial Differential Equations 7 Compatible Systems 8 Conservation Laws 9 Differential Resultants 10 Existence and Uniqueness Theorems 11 Fixed Point Existence Theorems 12 Hamilton-Jacobi Theory 13 Inverse Problems 14 Limit Cycles 15 Natural Boundary Conditions for a PDE 16 Normal Forms: Near-Identity Transformations 17 Self-Adjoint Eigenfunction Problems 18 Stability Theorems 19 Sturm-Liouville Theory 20 Variational Equations 21 Well-Posedness of Differential Equations 22 Wronskians and Fundamental SolutionsI.B Transformations 23 Canonical Forms 24 Canonical Transformations 25 Darboux Transformation 26 An Involutory Transformation 27 Liouville Transformation - 1 28 Liouville Transformation - 2 29 Reduction of Linear ODEs to a First Order System 30 Prüfer Transformation 31 Modified Prüfer Transformation 32 Transformations of Second Order Linear ODEs - 1 33 Transformations of Second Order Linear ODEs - 2 34 Transformation of an ODE to an Integral Equation 35 Miscellaneous ODE Transformations 36 Reduction of PDEs to a First Order System 37 Transforming Partial Differential Equations 38 Transformations of Partial Differential EquationsII Exact Analytical Methods 39 Introduction to Exact Analytical Methods 40 Look Up Technique 41 Look Up ODE FormsII.A Exact Methods for ODEs 42 An N-th Order Equation 43 Use of the Adjoint Equation 44 Autonomous Equations 45 Bernoulli Equation 46 Clairaut's Equation 47 Computer-Aided Solution 48 Constant Coefficient Linear Equations 49 Contact Transformation 50 Delay Equations 51 Dependent Variable Missing 52 Differentiation Method 53 Differential Equations with Discontinuities* 54 Eigenfunction Expansions* 55 Equidimensional-in-x Equations 56 Equidimensional-in-y Equations 57 Euler Equations 58 Exact First Order Equations 59 Exact Second Order Equations 60 Exact N-th Order Equations 61 Factoring Equations* 62 Factoring Operators* 63 Factorization Method 64 Fokker-Planck Equation 65 Fractional Differential Equations* 66 Free Boundary Problems* 67 Generating Functions* 68 Green's Functions* 69 Homogeneous Equations 70 Method of Images* 71 Integrable Combinations 72 Integral Representations: Laplace's Method* 73 Integral Transforms: Finite Intervals* 74 Integral Transforms: Infinite Intervals* 75 Integrating Factors* 76 Interchanging Dependent and Independent Variables 77 Lagrange's Equation 78 Lie Groups: ODEs 79 Operational Calculus* 80 Pfaffian Differential Equations 81 Reduction of Order 82 Riccati Equation 83 Matrix Riccati Equations 84 Scale Invariant Equations 85 Separable Equations 86 Series Solution* 87 Equations Solvable for x 88 Equations Solvable for y 89 Superposition* 90 Method of Undetermined Coefficients* 91 Variation of Parameters 92 Vector Ordinary Differential EquationsII.B Exact Methods for PDEs 93 Bäcklund Transformations 94 Method of Characteristics 95 Characteristic Strip Equations 96 Conformai Mappings 97 Method of Descent 98 Diagonalization of a Linear System of PDEs 99 Duhamel's Principle 100 Exact Equations 101 Hodograph Transformation 102 Inverse Scattering 103 Jacobi's Method 104 Legendre Transformation 105 Lie Groups: PDEs 106 Poisson Formula 107 Riemann's Method 108 Separation of Variables 109 Similarity Methods 110 Exact Solutions to the Wave Equation 111 Wiener-Hopf TechniqueIII Approximate Analytical Methods 112 Introduction to Approximate Analysis 113 Chaplygin's Method 114 Collocation 115 Dominant Balance 116 Equation Splitting 117 Floquet Theory 118 Graphical Analysis: The Phase Plane 119 Graphical Analysis: The Tangent Field 120 Harmonic Balance 121 Homogenization 122 Integral Methods 123 Interval Analysis 124 Least Squares Method 125 Lyapunov Functions 126 Equivalent Linearization and Nonlinearization 127 Maximum Principles 128 McGarvey Iteration Technique 129 Moment Equations: Closure 130 Moment Equations: Itô Calculus 131 Monge's Method 132 Newton's Method 133 Padé Approximants 134 Perturbation Method: Method of Averaging 135 Perturbation Method: Boundary Layer Method 136 Perturbation Method: Functional Iteration 137 Perturbation Method: Multiple Scales 138 Perturbation Method: Regular Perturbation 139 Perturbation Method: Strained Coordinates 140 Picard Iteration 141 Reversion Method 142 Singular Solutions 143 Soliton Type Solutions 144 Stochastic Limit Theorems 145 Taylor Series Solutions 146 Variational Method: Eigenvalue Approximation 147 Variational Method: Rayleigh-Ritz 148 WKB MethodIV.A Numerical Methods: Concepts 149 Introduction to Numerical Methods 150 Definition of Terms for Numerical Methods 151 Available Software 152 Finite Difference Methodology 153 Finite Difference Formulas 154 Excerpts from GAMS 155 Grid Generation 156 Richardson Extrapolation 157 Stability: ODE Approximations 158 Stability: Courant Criterion 159 Stability: Von Neumann TestIV.B Numerical Methods for ODEs 160 Analytic Continuation* 161 Boundary Value Problems: Box Method 162 Boundary Value Problems: Shooting Method* 163 Continuation Method* 164 Continued Fractions 165 Cosine Method* 166 Differential Algebraic Equations 167 Eigenvalue/Eigenfunction Problems 168 Euler's Forward Method 169 Finite Element Method* 170 Hybrid Computer Methods* 171 Invariant Imbedding* 172 Multigrid Methods* 173 Parallel Computer Methods 174 Predictor-Corrector Methods 175 Runge-Kutta Methods 176 Stiff Equations* 177 Integrating Stochastic Equations 178 Weighted Residual Methods*IV.C Numerical Methods for PDEs 179 Boundary Element Method 180 Differential Quadrature 181 Domain Decomposition 182 Elliptic Equations: Finite Differences 183 Elliptic Equations: Monte Carlo Method 184 Elliptic Equations: Relaxation 185 Hyperbolic Equations: Method of Characteristics 186 Hyperbolic Equations: Finite Differences 187 Lattice Gas Dynamics 188 Method of Lines 189 Parabolic Equations: Explicit Method 190 Parabolic Equations: Implicit Method 191 Parabolic Equations: Monte Carlo Method 192 Pseudo-Spectral MethodMathematical NomenclatureDifferential Equation IndexIndex