Front Cover Dynamic Programming and Partial Differential Equations Copyright Page Contents Preface CHAPTER 1. INTRODUCTION CHAPTER 2. QUADRATIC VARIATIONAL PROBLEMS 1. Introduction 2. Variational Approach 3. Positive Definiteness, Existence, and Uniqueness of Solution 4. Computational Aspects 5. Vector–Matrix Case 6. Rayleigh–Ritz Method 7. Bubnov–Galerkin Method Bibliography and Comment CHAPTER 3. DYNAMIC PROGRAMMING 1. Introduction 2. Difference Equations 3. Functional Equation 4. Principle of Optimality 5. Nonstationary Case 6. Quadratic Functions 7. Minimum Convolution 8. Acceleration of Calculation 9. Differential Equations 10. Quadratic Case 11. Minimum Convolutions 12. Tridiagonal Matrices Bibliography and Comment CHAPTER 4. THE POTENTIAL EQUATION 1. Introduction 2. The Euler–Langrange Equation 3. Inhomogeneous and Nonlinear Cases 4. Green’s Function 5. One-Dimensional Case 6. Two-Dimensional Case 7. Discretization 8. Rectangular Region 9. Rigorous Aspects 10. Associated Minimization Problem 11. Approximation from Above 12. Discussion 13. Semidiscretization 14. Irregular Grid 15. Solution of the Difference Equations 16. Iterative Solutions 17. Limitations of the Iterative Approach Miscellaneous Exercises Bibliography and Comment CHAPTER 5. DYNAMIC PROGRAMMING AND ELLIPTIC EQUATIONS 1. The Potential Equation 2. Discretization 3. Matrix–Vector Formulation 4. Dynamic Programming 5. Recurrence Equations 6. The Calculations 7. Nonsingularity 8. Stability 9. Discussion 10. Efficiency 11. Example 12. Deferred Passage to the Limit 13. General Linear Equations 14. Irregular Regions 15. Higher Order Equations 16. Distributed Control Bibliography and Comment CHAPTER 6. INVARIANT IMBEDDING 1. Invariant Imbedding 2. The Riccati Transformation 3. Single Sweep Methods 4. Discretization 5. Recurrence Relations 6. Relation to Dynamic Programming 7. Nonsingularity and Stability 8. Relation to Gaussian Elimination 9. Relation to the Riccati Equation 10. Invariant Imbedding 11. Continuous Invariant Imbedding 12. Generalized Riccati Transformations 13. The Biharmonic Equation 14. Random Walk 15. Invariant Imbedding and Random Walk 16. Another Imbedding Bibliography and Comment CHAPTER 7. IRREGULAR REGIONS 1. Introduction 2. Irregular Regions 3. Case I: Order uR > Order uR–1 4. Example 5. Case II: Order uR < Order UR–1 6. Example 7. Nonsingularity and Stability 8. Removal of Restrictions 9. Examples 10. General Linear Equations 11. Other Boundary Conditions 12. Three Dimensional Equations 13. The Biharmonic Equation 14. Invariant Imbedding and Difference Equations 15. A Second Approach 16. Matrix–Vector Equations 17. General Regions Bibliography and Comment CHAPTER 8. SPECIAL COMPUTATIONAL METHODS 1. Direct versus Iterative Methods 2. The Characteristic Values of Q 3. Kronccker Product 4. Kronecker Sums 5. An Example 6. Another Direct Method 7. Diagonal Decomposition 8. Point Iterative Methods 9. The Successive Overrelaxation Method 10. Block Iterative Methods 11. Alternating-Direction Implicit Methods 12. Discussion Bibliography and Comment CHAPTER 9. UNCONVENTIONAL DIFFERENCE METHODS 1. Introduction 2. Invariant Imbedding 3. The Equation ut = uux 4. Approximating Finite Difference Equation 5. Convergence 6. Improvement of Accuracy 7. Differential Quadrature Bibliography and Comment CHAPTER 10. PARABOLIC EQUATIONS 1. The Heat Equation 2. Properly Posed Problems 3. Consistency and Stability 4. Explicit Methods 5. Implicit Methods 6. Crank–Nicholson Method 7. Alternating-Direction Implicit Methods 8. The Laplace Transform 9. Gaussian Quadrature 10. Inversion of the Laplace Transform 11. Computational Aspects Bibliography and Comment CHAPTER 11. NONLI NEAR EQUATIONS AND QUASlLl NEARlZATlON 1. Introduction 2. Successive Approximations 3. Quasilinearization 4. An Example 5. The Equation uxx + uyy = u2 6. A Differential Inequality 7. Monotonicity 8. Maximum Domain of Convergence 9. Quadratic Convergence 10. Computational Aspects 11. Example 12. Identification Problems 13. The Least-Squares Criterion 14. Newton–Raphson–Kantorovich Method 15. The Sensitivity Equations 16. Quasilinearization 17. Example Miscellaneous Exercises Bibliography and Comment APPENDIX. COMPUTER PROGRAMS Program 1. Dynamic Programming Program 2. Riccati Transformation Program 3. Invariant Imbedding Program 4. Quasilinearization Author Index Subject Index Mathematics in Science and Engineering
