Dynamic programming and partial differential equations, Volume 88 (Mathematics in Science and Engineering)

✍ Scribed by Angel (editor)

• Publisher: Academic Press
• Year: 1972
• Tongue: English
• Leaves: 219
• Category: Library

✦ Table of Contents

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