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Methods of Nonlinear Analysis
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Methods of Nonlinear Analysis
Authors: Richard Bellman Publisher: Elsevier Science Publication date: 1973 Publication language: Angielski Number of pages: 281 Publication formats: EAN: 9780080955711 ISBN: 9780080955711 Category: Mathematical foundations Kolekcja specjalna: Rok matematyki 2019 Publisher's index: S0076-5392(08)X6029-1 Bibliographic note: -
TOC
- Front Cover 2
- Methods of Nonlinear Analysis 5
- Copyright Page 6
- Preface 9
- Contents 13
- Contents of Volume I 19
-
Chapter 9. Upper and Lower Bounds via Duality
21
- 9.1. Introduction 21
- 9.2. Guiding Idea 22
- 9.3. A Simple Identity 22
- 9.4. Quadratic Functional: Scalar Case 23
- 9.5. min u J = max v H 25
- 9.6. The Functional l t o [u'2 + g(u)] dt 25
- 9.7. Geometric Aspects 27
- 9.8. Multidimensional Case 28
- Front Cover 2
- Methods of Nonlinear Analysis 5
- Copyright Page 6
- Preface 9
- Contents 13
- Contents of Volume I 19
-
Chapter 9. Upper and Lower Bounds via Duality
21
- 9.1. Introduction 21
- 9.2. Guiding Idea 22
- 9.3. A Simple Identity 22
- 9.4. Quadratic Functional: Scalar Case 23
- 9.5. min u J = max v H 25
- 9.6. The Functional l t o [u'2 + g(u)] dt 25
- 9.7. Geometric Aspects 27
- 9.8. Multidimensional Case 28
- 9.9. The Rayleigh–Ritz Method 29
- 9.10. Alternative Approach 30
- 9.11. J(u) = lto [u'2 + φ(t)u2] dt; General φ(t) 32
- 9.12. Geometric Aspects 33
- Miscellaneous Exercises 34
- Bibliography and Comments 39
-
Chapter 10. Caplygin's Method and Differential Inequalities
41
- 10.1. Introduction 41
- 10.2. The Caplygin Method 42
- 10.3. The Equation u' < a(t)u + f (t) 42
- 10.5. Elementary Approach 44
- 10.4. The Linear Differential Inequality L(u) < f ( t ) 44
- 10.6. An Integral Identity 45
- 10.8. Factorization of the Operator 46
- 10.7. Strengthening of Previous Result 46
- 10.9. Alternate Proof of Monotonicity 47
- 10.10. A Further Condition 48
- 10.11. Two-point Boundary Conditions 49
- 10.12. Variational Approach 49
- 10.13. A Related Parabolic Partial Differential Equation 51
- 10.14. Nonnegativity of u(t, s) 51
- 10.15. Limiting Behavior 53
- 10.16. Limiting Behavior: Energy Inequalities 54
- 10.18. Lyapunov Functions 56
- 10.17. Monotonicity of Maximum 56
- 10.19. Factorization of the nth-order Linear Operator 59
- 10.21. An Example 62
- 10.20. A Result for the nth-order Linear Differential Equation 62
- 10.22. Linear Systems 63
- 10.24. Partial Differential Equation—II 64
- Miscellaneous Exercises 64
- 10.23. Partial Differential Equation—I 64
- Bibliography and Comments 66
-
Chapter 11. Quasilinearization
70
- 11.1. Introduction 70
- 11.2. The Riccati Equation 71
- 11.3. Explicit Representation 72
- 11.4. Successive Approximations and Monotone Convergence 72
- 1 1.7. Newton–Raphson–Kantorovich Approximation 74
- 11.5. Maximum Interval of Convergence 74
- 11.6. Dini's Theorem and Uniform Convergence 74
- 11.8. Quadratic Convergence 75
- 11.9. Upper Bounds 76
- 11.10. u' = g(u, t ) 77
- 11.11. Random Equation 77
- 11.12. Upper and Lower Bounds 78
- 1 I. 13. Asymptotic Behavior 79
- 11.14. Multidimensional Riccati Equation 81
- 11.15. Two-point Boundary Value Problems 82
- 11.16. Maximum Interval of Convergence 84
- 11.1 7. Quadratic Convergence 85
- 11.18. Discussion 86
- 11.19. Computational Feasibility 87
- 11.21. Parabolic Equations 89
- 11.20. Elliptic Equations 89
- 1 1.22. Minimum and Maximum Principles 90
- Miscellaneous Exercises 91
- Bibliography and Comments 92
-
Chapter 12. Dynamic Programming
95
- 12.1. Introduction 95
- 12.2. Multistage Processes 96
- 12.3. Continuous Version 96
- 12.4. Multistage Decision Processes 97
- 12.6. Functional Equations 98
- 12.5. Stochastic and Adaptive Processes 98
- 12.7. Infinite Stage Process 100
- 12.8. Policy 101
- 12.9. Approximation in Policy Space 102
- 12.10. Discussion 103
- 12.11. Calculus of Variations as a Multistage Decision Process 103
- 12.12. A New Formalism 105
- 12.13. The Principle of Optimality 107
- 12.14. Quadratic Case 108
- 12.15. Multidimensional Case 109
- 12.17. Stability 111
- 12.1 6. Computational Feasibility 111
- 12.18. Computational Feasibility: General Case—I 113
- 12.20. The Curse of Dimensionality 115
- 12.21. Constraints 115
- 12.19. Computational Feasibility: General Case—II 115
- 12.22. Two-point Boundary Value Problems 116
- 12.23, Rigorous Aspects 116
- 12.24. The Equation ut = φ(x, ux) 118
- 12.24. The Equation ut = .p(x, u,) 118
- 12.25. The Equation ut = q(x, us=) 119
- 12.25. The Equation ut = φ(x, uxx) 119
- 12.26. Generalized Semigroup Theory and Nonlinear Equations 120
- 12.28. Adaptive Polygonal Approximation 121
- 12.27. Ill-conditioned Systems 121
- 12.29. Partial Differential Equations 124
- 12.30. Successive Approximations to Reduce Dimensionality 125
- Miscellaneous Exercises 126
- Bibliography and Comments 132
-
Chapter 13. Invariant Imbedding
136
- 13.1. Introduction 136
- 13.3. Functional Equation Approach 137
- 13.2. Maximum Altitude 137
- 13.4. Invariant Imbedding 139
- 13.5. Inhomogeneous Medium 139
- 13.6. Computational Aspects 140
- 13.7. Two-point Boundary Value Problems 141
- 13.8. A Simple Transport Process 142
- 13.9. Invariant Imbedding: Particle Counting 143
- 13.10. Perturbation Analysis 144
- 13.11. Linear Systems 147
- 13.1 2. Linear Transport Process 147
- 13.14. Invariant Imbedding 150
- 13.13. Reflection and Transmission Matrices 150
- 13.15. Equivalence of Analytic Formulations 152
- 13.16. Conservation Relation 153
- 13.17. Internal Fluxes 154
- 13.18. Asymptotic Behavior 155
- 13.19. Multidimensional Case 157
- 13.20. Asymptotic Phase 158
- 13.21. Wave Propagation 160
- 13.22. The Bremmer Series 162
- 13.23. Localization Principles 163
- 13.25. The Classical Solution to Wave Propagation through an Inhomogeneous Medium 164
- 13.24. Localization for Wave Motion 164
- 13.27. Invariant Imbedding 166
- 13.26. Riccati Equation 166
- 13.28. Functional Equations 167
- 13.29. Random Walk 168
- 13.30. Classical Formulation 169
- 13.31. Invariant Imbedding 170
- 13.32. Determination of u(a, a + 1 ) 170
- Miscellaneous Exercises 172
- Bibliography and Comments 173
-
Chapter 14. The Theory of Iteration
175
- 14.1. Introduction 175
- 14.2. Iteration 176
- 14.3. Abel–Schröder Functional Equation 177
- 14.4. Formal Analysis 177
- 14.5. Koenig's Representation 178
- 14.6. Majorization 179
- 14.8. Stochastic Effects 181
- 14.7. Refined Asymptotic Behavior 181
- 14.9. Stochastic Iteration 182
- 14.10. Stochastic Matrices 183
- 14.11. Random Initial Values 185
- 14.12. Imbedding 186
- 14.13. Differential Equations 187
- 14.14. Associated Partial Differential Equation 188
- 14.15. Multidimensional Case 189
- 14.16. A Counterexample 190
- 14.17. Statement of Result 192
- 14.18. Differential Equations 192
- 14.19. Commensurable Characteristic Roots 194
- 14.20. Two-point Boundary Values 195
- 14.21. Constant Right-hand Incident Flux 196
- 14.22. Associated Partial Differential Equation 197
- 14.23. Branching Processes 198
- 14.24. Closure of Operations 199
- 14.25. Number of Arithmetic Operations 200
- Miscellaneous Exercises 201
- Bibliography and Comment 208
-
Chapter 15. Infinite Systems of Ordinary Differential Equations and Truncation
211
- 15.1. Introduction 211
- 15.2. Ordinary Differential Equations and Difference Methods 212
- 15.4. Round-off Error 213
- 15.3. Stability Considerations 213
- 15.5. Linear Partial Differential Equations 215
- 15.7. Preservation of Properties 217
- 15.6. Partial Discretization 217
- 15.8. Orthonormal Expansions and Infinite Systems 219
- 15.9. Truncation 220
- 15.10. Associated Equation 221
- 15.12. The Fejer Sum 222
- 15.11. Discussion of Convergence of u(n) 222
- 15.13. Modified Partial Differential Equation 223
- 15.14. Modified Truncation 223
- 15.15. Infinite Systems of Ordinary Differential Equations 224
- 15.16. Proof of Theorem 226
- 15.17. The "Principe des Réduites" 228
- 15.18. Random Walk 231
- 15.20. Monotone Convergence 232
- 15.19. The "Principe des RCduites" 232
- 15.21. Existence of a Solution 233
- 15.22. Closure of the Process 234
- Miscellaneous Exercises 235
- Bibliography and Comment 239
-
Chapter 16. Integral and Differential Quadrature
241
- 16.1. Introduction 241
- 16.2. Laplace Transform 241
- 16.3. Simplifying Properties of the Laplace Transform 242
- 16.4. Examples 242
- 16.6. Abelian and Tauberian Results 244
- 16.5. Complex Inversion Formula 244
- 16.7. Interpolation Technique 245
- 16.8. Selective Calculation 245
- 16.9. Impossibility 246
- 16.10. Quadrature Techniques 247
- 16.11. Numerical Inversion of the Laplace Transform 249
- 16.12. Explicit Inverse of (xik) 250
- 16.13. Example of Ill-conditioning 251
- 16.14. Nonlinear Equations—I 252
- 16.15. Nonlinear Equations—II 253
- 16.16. Quasilinearization 253
- 16.1 7. Tychonov Regularization 255
- 16.18. Self-consistent Methods 256
- 16.19. An Imbedding Technique 257
- 16.20. Nonlinear Summability 258
- 16.21. Time-dependent Transport Process 258
- 16.22. Radiative Transfer 259
- 16.23. Radiative Transfer via Invariant Imbedding 260
- 16.24. Time-dependent Case 262
- 16.25. Error Analysis 263
- 16.26. Differential Quadrature 264
- 16.27. Application to Partial Differential Equations 265
- 16.28. Identification Problems 265
- 16.30. Differential Quadrature 266
- 16.29. Quasilinearization 266
- Miscellaneous Exercises 267
- Bibliography and Comment 271
- Author Index 275
- Subject Index 279