Stochastic Systems

Name: Stochastic Systems | Monograph
Brand: Academic Press
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Monograph

Adomian(Autor*in)

Adomian(Herausgeber*in)

Academic Press

Erschienen am 14. Mai 2014

351 Seiten

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978-0-08-095675-6 (ISBN)

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Inhalt

Front Cover
Stochastic Systems
Copyright Page
Contents
Foreword
Preface
Chapter 1. Green's Functions and Systems Theory
1.1. Introduction: Some Remarks on the Mathematical Modeling of Dynamical Systems
1.2. Linearity and Superposition
1.3. The Concept of a Green's Function
1.4. Simple Input-Output Systems and Green's Functions
1.5. Operator Forms
1.6. Green's Function for the Inhomogeneous Sturm-Liouville Operator
1.7. Properties of the Green's Function
1.8. Evaluation of the Wronskian
1.9. Solution Using Abel's Formula
1.10. Use of Green's Function to Solve the Inhomogeneous Equation
1.11. Adjoint Operators
1.12. Green's Functions for Adjoint Operators
1.13. Symbolic Functions
1.14. Sturm-Liouville Differential Equation
1.15. Boundary Conditions Specified on a Finite Interval [a, b]
1.16. Series Expansions for G(x, ?)
1.17. Multiple-Input-Multiple-Output Systems
1.18. Bilinear Form of the Green's Function
1.19. Bilinear Form of the Green's Function for the Sturm-Liouville Differential Equation
1.20. Cases Where the Green's Function Does Not Exist
1.21. Multidimensional Green's Functions
1.22. Green's Functions for Initial Conditions
1.23. Approximate Calculation of Green's Functions
References
Chapter 2. A Basic Review of the Theory of Stochastic Processes
2.1. The Nature of a Stochastic Process
2.2. Stochastic Processes-Basic Definitions
2.3. Characterization and Classification of Stochastic Processes
2.4. Consistency Conditions on the Distribution
2.5. Some Simple Stochastic Processes
2.6. Time Dependences of Distributions
2.7. Statistical Measures of Stochastic Processes
2.8. Random Fields
2.9. The Calculus of Stochastic Processes
2.10. Expansions of Random Functions
2.11. Ergodic Theorems
2.12. Generalized Random Processes
References
Chapter 3. Stochastic Operators and Stochastic Systems
3.1. Stochastic Systems-Basic Concepts
3.2. Stochastic Green's Functions
3.3. Statistical Operators
3.4. Stochastic Green's Theorem
3.5. Determination of the Kernel from the Physical Process
References
Chapter 4. Linear Stochastic Differential Equations
4.1. Stochastic Differential Operators
4.2. The Differential Equation Formulation
4.3. Derivation of Stochastic Green's Theorem
4.4. Hierarchy or Averaging Method
4.5. Perturbation Theory
4.6. Connection between Perturbation Theory and the Hierarchy Method
4.7. The Decomposition Method
4.8. Differential Operator with One Random Coefficient
4.9. A Convenient Resolvent Kernel Formulation
4.10. Inverse Operator Form of the Decomposition Method Solution
4.11. Some Further Remarks on the Operator Identity (4.10.1)
4.12. General Form of the Stochastic Green's Function
4.13. Random Initial Conditions
4.14. Simplifying Green's Function Calculations for Higher Order Equations
References
Chapter 5. Nonlinear Stochastic Differential Equations
5.1. Generalization of the Decomposition Method to the Nonlinear Case
5.2. Symmetrized Solutions
5.3. Inverse Operator Method in the Nonlinear Case
5.4. Operator-Theoretic Solution for General Nonlinear Stochastic Systems
5.5 Product Nonlinearities
References
Chapter 6. Successive Improvements
6.1. Corrected Approximate Solution for the Linear and Nonlinear Cases
6.2. Error in the Approximate Mean Solution
References
Chapter 7. Computational Aspects and a General Method
7.1. A Comparison of the Inverse Method and the Symmetrized Method
7.2. Numerical Comparisons of Decomposition Solutions with Perturbation and Hierarchy Solutions
7.3. Quadrature Method
7.4. Simulation
7.5. A General Method
References
Chapter 8. The Deterministic Limit of Stochastic Theory
8.1. Numerical Solution of Differential Equations
8.2. A Comparison of the Decomposition Method and Picard's Approximations
References
Chapter 9. Comparisons and Relations
9.1. The Itô Equation
9.2. Canonical Expansions and the Itô Equation
9.3. General Expressions for the Stochastic Green's Function and the Perturbation Limit
References
Chapter 10. Stochastic Partial Differential Equations
10.1. The Linear Deterministic Case
10.2. Linear Stochastic Partial Differential Equations
10.3. Nonlinear Partial Differential Equations
References
Chapter 11. Analytical Foundations of Stochastic Operator Equations
11.1. Introduction
11.2. Mathematical Description of Stochastic Processes
11.3. Statistical Measures
11.4. Statistical Independence of Stochastic Processes
11.5. Stationarity Concepts
11.6. Ergodicity
11.7. Analytic Expressions for the Probability Density Functions
11.8. Stochastic Differential Equations
11.9. Abstract Structure
11.10. Existence and Uniqueness of Statistical Measures and Existence of the Inverse Operator
References
Chapter 12. Applications
12.1. The Stochastic Riccati Equation
12.2. On the Control of Stochastic Systems
12.3. The n-Compartment Stochastic Model in Biology and Medicine
12.4. Signal Processing and Filtering
12.5. The Random Harmonic Oscillator
12.6. Second-Order Differential Equations with One Random Coefficient
12.7. General Linear Operator with One Random Coefficient
12.8. An RC Circuit with Random Capacitance
12.9. On Stochastic Transport Equations
12.10. Random Wave Propagation
References
Supplementary References
Index

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