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Nonlinear Stochastic Operator Equations
1st Edition - December 28, 1986
Author: George Adomian
eBook ISBN:
9 7 8 - 1 - 4 8 3 2 - 5 9 0 9 - 3
Nonlinear Stochastic Operator Equations deals with realistic solutions of the nonlinear stochastic equations arising from the modeling of frontier problems in many fields of… Read more
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Nonlinear Stochastic Operator Equations deals with realistic solutions of the nonlinear stochastic equations arising from the modeling of frontier problems in many fields of science. This book also discusses a wide class of equations to provide modeling of problems concerning physics, engineering, operations research, systems analysis, biology, medicine. This text discusses operator equations and the decomposition method. This book also explains the limitations, restrictions and assumptions made in differential equations involving stochastic process coefficients (the stochastic operator case), which yield results very different from the needs of the actual physical problem. Real-world application of mathematics to actual physical problems, requires making a reasonable model that is both realistic and solvable. The decomposition approach or model is an approximation method to solve a wide range of problems. This book explains an inherent feature of real systems—known as nonlinear behavior—that occurs frequently in nuclear reactors, in physiological systems, or in cellular growth. This text also discusses stochastic operator equations with linear boundary conditions. This book is intended for students with a mathematics background, particularly senior undergraduate and graduate students of advanced mathematics, of the physical or engineering sciences.
Foreword
Preface
Acknowledgments
Chapter 1 Introduction
References
Chapter 2 Operator Equations and the Decomposition Method
2.1. Modeling, Approximation, and Reality
2.2. The Operator Equations
2.3. The Decomposition Method
2.4. Evaluation of the Inverse Operator L-1 and the y10 Term of the Decomposition for Initial or Boundary Conditions
References
Suggested Further Reading
Chapter 3 Expansion of Nonlinear Terms: The An Polynomials
3.1. Introduction
3.2. Calculation of the An Polynomials for Simple Nonlinear Operators
3.3. The An Polynomials for Differential Nonlinear Operators
3.4. Convenient Computational Forms for the An Polynomials
3.5. Linear Limit
3.6. Calculation of the An Polynomials for Composite Nonlinearities
References
Suggested Further Reading
Chapter 4 Solution of Differential Equations
4.1. General Method and Examples
4.2. Calculating a Simple Green's Function
4.3. Green's Function by Decomposition
4.4. Approximating Difficult Green's Functions
4.5. Polynomial Nonlinearities
4.6. Negative Power Nonlinearities
4.7. Decimal Power Nonlinearities
4.8. Product Nonlinearities
4.9. Anharmonic Oscillator Systems
4.10. Limiting Case: The Harmonic Oscillator
4.11. Extensions to Stochastic Oscillators
4.12. Asymptotic Solutions
References
Suggested Further Reading
Chapter 5 Coupled Nonlinear Stochastic Differential Equations
5.1. Deterministic Coupled Differential Equations
5.2. Stochastic Coupled Equations
5.3. Generalization to n Coupled Stochastic Differential Equations
Suggested Further Reading
Chapter 6 Delay Equations
6.1. Definitions
6.2. Solution of Delay Operator Equations
Reference
Suggested Further Reading
Chapter 7 Discretization versus Decomposition
7.1. Discretization
7.2. A Differential-Difference Equation
7.3. Difference Equations and the Decomposition Method
7.4. Some Remarks on Supercomputers
References
Suggested Further Reading
Chapter 8 Random Eigenvalue Equations
References
Suggested Further Reading
Chapter 9 Partial Differential Equations
9.1. Solving m-Dimensional Equations
9.2. Four-Dimensional Linear Partial Differential Equation
9.3. Nonlinear Partial Differential Equation
9.4. Some General Remarks
9.5. The Heat Equation
9.6. Inhomogeneous Heat Equation
9.7. Asymptotic Decomposition for Partial Differential Equations
Reference
Suggested Further Reading
Chapter 10 Algebraic Equations
Part I Polynomials
10.1. Quadratic Equations by Decomposition
10.2. Cubic Equations
10.3. Higher-Degree Polynomial Equations
10.4. Equation with Negative Power Nonlinearities
10.5. Equations with Noninteger Powers
10.6. Equations with Decimal Powers
10.7. Random Algebraic Equations
10.8. General Remarks
Part II Transcendental Equations
10.9. Trigonometric Equations
10.10. Exponential Cases
10.11. Logarithmic Equation: Purely Nonlinear Equations
10.12. Products of Nonlinear Functions
10.13. Hyperbolic Sine Nonlinearity
10.14. Composite Nonlinearities
Part III Inversion of Matrices
10.15. Discussion: Inversion by Decomposition
10.16. Convergence
10.17. Decomposition into Diagonal Matrices
10.18. Systems of Matrix Equations
10.19. Inversion of Random Matrices
10.20. Inversion of Very Large Matrices
References
Suggested Further Reading
Chapter 11 Convergence
11.1. The Convergence Question for the Nonlinear Case
11.2. Estimating the Radius of Convergence
11.3. On the Calculus
11.4. Some Remarks on Convergence
References
Chapter 12 Boundary Conditions
12.1. Linear Boundary Conditions
12.2. Treatment of Inhomogeneous Boundary Conditions
12.3. General Boundary Operators and Matrix Equations
12.4. Random Boundary Operators
12.5. Random Inhomogeneous Boundary Conditions
12.6. Linear Differential Equations with Linear Boundary Conditions
12.7. Deterministic Operator Equations with Random Input and Random Boundary Conditions
12.8. Stochastic Operator Equations with Linear Boundary Conditions
12.9. Linear Stochastic Equations with Nonlinear Boundary Conditions
12.10. Linear Differential Equations with Nonlinear Boundary Conditions
12.11. Coupled Nonlinear Stochastic Differential Equations
12.12. Coupled Linear Deterministic Equations with Coupled Boundary Conditions
12.13. Coupled Equations and Coupled Boundary Conditions
12.14. Boundary Conditions for Partial Differential Equations
12.15. Summary
Reference
Suggested Further Reading
Chapter 13 Intergral and Integro-Differential Operators
Chapter 14 On Systems of Nonlinear Partial Differential Equations
References
Chapter 15 Postlude
References
Index
Errata for "Stochastic Systems"
- No. of pages: 304
- Language: English
- Published: December 28, 1986
- Imprint: Academic Press
- eBook ISBN: 9781483259093