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