Stochastic Systems
Monograph
Erschienen am 14. Mai 2014
351 Seiten
978-0-08-095675-6 (ISBN)
Beschreibung
Stochastic Systems
Weitere Details
Weitere Ausgaben
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
