Quasilinearization and Invariant Imbedding
With Applications to Chemical Engineering and Adaptive Control
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Mathematics in Science and Engineering, Volume 41: Quasilinearization and Invariant Imbedding presents a study on the use of two concepts for obtaining numerical solutions of boundary-value problems—quasilinearization and invariant imbedding. This book emphasizes that the invariant imbedding approach reformulates the original boundary-value problem into an initial value problem by introducing new variables or parameters, while the quasilinearization technique represents an iterative approach combined with linear approximations. This volume focuses on analytical aspects that are concerned with actual convergence rates and computational requirements, considering various efficient algorithms that are suited for various types of boundary-value problems. This publication is a good reference for chemical and control engineers and scientists interested in obtaining numerical solutions of boundary-value problems in their particular fields.
Table of Contents
Preface
Chapter 1. Introductory Concepts
1. Introduction
2. Quasilinearization
3. Invariant Imbedding
4. Invariant Imbedding versus the Classical Approach
5. Numerical Solution of Ordinary Differential Equations
6. Numerical Solution Terminologies
References
Chapter 2. Quasilinearization
1. Introduction
2. Nonlinear Boundary-Value Problems
3. Linear Boundary-Value Problems
4. Finite-Difference Method for Linear Differential Equations
5. Discussion
6. Newton-Raphson Method
7. Discussion
8. Quasilinearization
9. Discussion
10. Existence and Convergence
11. Existence
12. Convergence
13. Maximum Operation and Differential Inequalities
14. Construction of a Monotone Sequence
15. Approximation in Policy Space and Dynamic Programming
16. Discussion
17. Systems of Differential Equations
References
Chapter 3. Ordinary Differential Equations
1. Introduction
2. A Second-Order Nonlinear Differential Equation
3. Recurrence Relation
4. Computational Procedure
5. Numerical Results
6. Stability Problem in Numerical Solution—The Fixed Bed Reactor
7. Finite-Difference Method
8. Systems of Algebraic Equations Involving Tridiagonal Matrices
9. Numerical Results
10. Stability Problem with High Peclet Number
11. Adiabatic Tubular Reactor with Axial Mixing
12. Numerical Results
13. Discussion
14. Unstable Initial-Value Problems
15. Discussion
16. Systems of Differential Equations
17. Computational Considerations
18. Simultaneous Solution of Different Iterations
References
Chapter 4. Parameter Estimation
1. Introduction
2. Parameter Estimation and the "Black Box" Problem
3. Parameter Estimation and the Experimental Determination of Physical Constants
4. A Multipoint Boundary-Value Problems
5. The Least Squares Approach
6. Computational Procedure for a Simpler Problems
7. Numerical Results
8. Nonlinear Boundary Condition
9. Random Search Technique
10. Numerical Results
11. Discussion
12. Parameter Up-Dating
13. Discussion
14. Estimation of Chemical Reaction Rate Constants
15. Differential Equations with Variable Coefficients
16. An Example
17. III-Conditioned Systems
18. Numerical Results
19. Discussion
20. An Empirical Approximation
21. Numerical Results
22. A Second Approximation
23. Numerical Results
24. Differential Approximation
25. A Second Formulation
26. Computational Aspects
27. Discussion
References
Chapter 5. Optimization
1. Introduction
2. Optimum Temperature Profiles in Tubular Reactors
3. Numerical Results
4. Discussion
5. Back and Forth Integration
6. Two Consecutive Gaseous Reactions
7. Optimum Pressure Profile in Tubular Reactor
8. Numerical Results
9. Optimum Temperature Profile with Pressure as Parameter
10. Numerical Results and Procedures
11. Calculus of Variations with Control Variable Inequality Constraint
12. Calculus of Variations with Pressure Drop in the Reactor
13. Pontryagin's Maximum Principle
14. Discussion
15. Optimum Feed Conditions
16. Partial Derivative Evaluation
17. Conclusions
References
Chapter 6. Invariant Imbedding
1. Introduction
2. The Invariant Imbedding Approach
3. An Example
4. The Missing Final Condition
5. Determination of x and y in Terms of r and s
6. Discussion
7. Alternate Formulations—I
8. Linear and Nonlinear Systems
9. The Riccati Equation
10. Alternate Formulations—II
11. The Reflection and Transmission Functions
12. Systems of Differential Equations
13. Large Linear Systems
14. Computational Considerations
15. Dynamic Programming
16. Discussion
References
Chapter 7. Quasilinearization and Invariant Imbedding
1. Introduction
2. The Predictor-Corrector Formula
3. Discussion
4. Linear Boundary-Value Problems
5. Numerical Results
6. Optimum Temperature Profiles in Tubular Reactors
7. Numerical Results
8. Discussion
9. Dynamic Programming and Quasilinearization—I
10. Discussion
11. Linear Differential Equations
12. Dynamic Programming and Quasilinearization—II
13. Further Reduction in Dimensionality
14. Discussion
References
Chapter 8. Invariant Imbedding, Nonlinear Filtering, and the Estimation of Variables and Parameters
1. Introduction
2. An Estimation Problem
3. Sequential and Nonsequential Estimates
4. The Invariant Imbedding Approach
5. The Optimal Estimates
6. Equation for the Weighting Function
7. A Numerical Example
8. Systems of Differential Equations
9. Estimation of State and Parameter—An Example
10. A More General Criterion
11. An Estimation Problem with Observational Noise and Disturbance Input
12. The Optimal Estimate—A Two-Point Boundary-Value Problem
13. Invariant Imbedding
14. A Numerical Example
15. Systems of Equations with Observational Noises and Disturbance Inputs
16. Discussion
References
Chapter 9. Parabolic Partial Differential Equations—Fixed Bed Reactors with Axial Mixing
1. Introduction
2. Isothermal Reactor with Axial Mixing
3. An Implicit Difference Approximation
4. Computational Procedure
5. Numerical Results—Isothermal Reactor
6. Adiabatic Reactor with Axial Mixing
7. Numerical Results—Adiabatic Reactor
8. Discussion
9. Influence of the Packing Particles
10. The Linearized Equations
11. The Difference Equations
12. Computational Procedure—Fixed Bed Reactor
13. Numerical Results—Fixed Bed Reactor
14. Conclusion
References
Appendix I. Variational Problems with Parameters
1. Introduction
2. Variational Equations with Parameters
3. Simpler End Conditions
4. Calculus of Variations with Control Variable Inequality Constraint
5. Pontryagin's Maximum Principle
References
Appendix II. The Functional Gradient Technique
1. Introduction
2. The Recurrence Relations
3. Numerical Example
4. Discussion
References
Author Index
Subject Index
Product details
- No. of pages: 350
- Language: English
- Copyright: © Academic Press 1968
- Published: January 1, 1968
- Imprint: Academic Press
- eBook ISBN: 9781483266756
About the Author
E. Stanley Lee
About the Editor
Richard Bellman
Affiliations and Expertise
Departments of Mathematics,
Electrical Engineering, and Medicine
University of Southern California
Los Angeles, California
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