Nonlinear Stochastic Operator Equations

Nonlinear Stochastic Operator Equations

1st Edition - December 28, 1986

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  • Author: George Adomian
  • eBook ISBN: 9781483259093

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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.

Table of Contents

  • Foreword



    Chapter 1 Introduction


    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


    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


    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


    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


    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


    Suggested Further Reading

    Chapter 8 Random Eigenvalue Equations


    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


    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


    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


    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


    Suggested Further Reading

    Chapter 13 Intergral and Integro-Differential Operators

    Chapter 14 On Systems of Nonlinear Partial Differential Equations


    Chapter 15 Postlude



    Errata for "Stochastic Systems"

Product details

  • No. of pages: 304
  • Language: English
  • Copyright: © Academic Press 1986
  • Published: December 28, 1986
  • Imprint: Academic Press
  • eBook ISBN: 9781483259093

About the Author

George Adomian

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