# Nonlinear Stochastic Operator Equations

1st Edition - December 28, 1986

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• eBook ISBN: 9781483259093

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## Description

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

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

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

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

Chapter 6 Delay Equations

6.1. Definitions

6.2. Solution of Delay Operator Equations

Reference

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

Chapter 8 Random Eigenvalue Equations

References

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

Chapter 10 Algebraic Equations

Part I Polynomials

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

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

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"

## Product details

• No. of pages: 304
• Language: English