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Linear and Nonlinear Integral Equations: Methods and Applications by Abdul-Majid Wazwaz



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Linear and Nonlinear Integral Equations: Methods and Applications written by Abdul-Majid Wazwaz , Saint Xavier University, Chicago. Linear and Nonlinear Integral Equations: Methods and Applications is a self-contained book divided into two parts. Part I offers a comprehensive and systematic treatment of linear integral equations of the first and second kinds. The text brings together newly developed methods to reinforce and complement the existing procedures for solving linear integral equations. The Volterra integral and integro-differential equations, the Fredholm integral and integro-differential equations, the Volterra-Fredholm integral equations, singular and weakly singular integral equations, and systems of these equations, are handled in this part by using many different computational schemes. Selected worked-through examples and exercises will guide readers through the text. Part II provides an extensive exposition on the nonlinear integral equations and their varied applications, presenting in an accessible manner a systematic treatment of ill-posed Fredholm problems, bifurcation points, and singular points. Selected applications are also investigated by using the powerful Padé approximants. This book is intended for scholars and researchers in the fields of physics, applied mathematics and engineering. It can also be used as a text for advanced undergraduate and graduate students in applied mathematics, science and engineering, and related fields. Dr. Abdul-Majid Wazwaz is a Professor of Mathematics at Saint Xavier University in Chicago, Illinois, USA.

Linear and Nonlinear Integral Equations: Methods and Applications written by Abdul-Majid Wazwaz cover the following topics.

  • Part I Linear Integral Equations

  • 1. Preliminaries
    1.1 Taylor Series
    1.2 Ordinary Differential Equations
    1.2.1 First Order Linear Differential Equations
    1.2.2 Second Order Linear Differential Equations
    1.2.3 The Series Solution Method
    1.3 Leibnitz Rule for Differentiation of Integrals
    1.4 Reducing Multiple Integrals to Single Integrals
    1.5 Laplace Transform
    1.5.1 Properties of Laplace Transforms
    1.6 Infinite Geometric Series
    References

  • 2. Introductory Concepts of Integral Equations
    2.1 Classification of Integral Equations
    2.1.1 Fredholm Integral Equations
    2.1.2 Volterra Integral Equations
    2.1.3 Volterra-Fredholm Integral Equations
    2.1.4 Singular Integral Equations
    2.2 Classification of Integro-Differential Equations
    2.2.1 Fredholm Integro-Differential Equations
    2.2.2 Volterra Integro-Differential Equations
    2.2.3 Volterra-Fredholm Integro-Differential Equations
    2.3 Linearity and Homogeneity
    2.3.1 Linearity Concept
    2.3.2 Homogeneity Concept
    2.4 Origins of Integral Equations
    2.5 Converting IVP to Volterra Integral Equation
    2.5.1 Converting Volterra Integral Equation to IVP
    2.6 Converting BVP to Fredholm Integral Equation
    2.6.1 Converting Fredholm Integral Equation to BVP
    2.7 Solution of an Integral Equation
    References

  • 3. Volterra Integral Equations
    3.1 Introduction
    3.2 Volterra Integral Equations of the Second Kind
    3.2.1 The Adomian Decomposition Method
    3.2.2 The Modified Decomposition Method
    3.2.3 The Noise Terms Phenomenon
    3.2.4 The Variational Iteration Method
    3.2.5 The Successive Approximations Method
    3.2.6 The Laplace Transform Method
    3.2.7 The Series Solution Method
    3.3 Volterra Integral Equations of the First Kind
    3.3.1 The Series Solution Method
    3.3.2 The Laplace Transform Method
    3.3.3 Conversion to a Volterra Equation of the Second Kind
    References 0.118

  • 4. Fredholm Integral Equations
    4.1 Introduction
    4.2 Fredholm Integral Equations of the Second Kind
    4.2.1 The Adomian Decomposition Method
    4.2.2 The Modified Decomposition Method
    4.2.3 The Noise Terms Phenomenon
    4.2.4 The Variational Iteration Method
    4.2.5 The Direct Computation Method
    4.2.6 The Successive Approximations Method
    4.2.7 The Series Solution Method
    4.3 Homogeneous Fredholm Integral Equation
    4.3.1 The Direct Computation Method
    4.4 Fredholm Integral Equations of the First Kind
    4.4.1 The Method of Regularization
    4.4.2 The Homotopy Perturbation Method
    References

  • 5. Volterra Integro-Differential Equations
    5.1 Introduction
    5.2 Volterra Integro-Differential Equations of the Second Kind
    5.2.1 The Adomian Decomposition Method
    5.2.2 The Variational Iteration Method
    5.2.3 The Laplace Transform Method
    5.2.4 The Series Solution Method
    5.2.5 Converting Volterra Integro-Differential Equations to Initial Value Problems
    5.2.6 ConvertingVolterra Integro-Differential Equation to Volterra Integral Equation
    5.3 Volterra Integro-Differential Equations of the First Kind
    5.3.1 Laplace Transform Method
    5.3.2 The Variational Iteration Method
    References

  • 6. Fredholm Integro-Differential Equations
    6.1 Introduction
    6.2 Fredholm Integro-Differential Equations of the Second Kind
    6.2.1 The Direct Computation Method
    6.2.2 The Variational Iteration Method
    6.2.3 The Adomian Decomposition Method
    6.2.4 The Series Solution Method
    References

  • 7. Abel’s Integral Equation and Singular Integral Equations.
    7.1 Introduction
    7.2 Abel’s Integral Equation
    7.2.1 The Laplace Transform Method
    7.3 The Generalized Abel’s Integral Equation
    7.3.1 The Laplace Transform Method.
    7.3.2 The Main Generalized Abel Equation
    7.4 The Weakly Singular Volterra Equations
    7.4.1 The Adomian Decomposition Method
    7.4.2 The Successive Approximations Method
    7.4.3 The Laplace Transform Method
    References

  • 8. Volterra-Fredholm Integral Equations
    8.1 Introduction
    8.2 The Volterra-Fredholm Integral Equations
    8.2.1 The Series Solution Method
    8.2.2 The Adomian Decomposition Method
    8.3 The Mixed Volterra-Fredholm Integral Equations
    8.3.1 The Series Solution Method
    8.3.2 The Adomian Decomposition Method
    8.4 The Mixed Volterra-Fredholm Integral Equations in Two Variables
    8.4.1 The Modified Decomposition Method
    References.

  • 9. Volterra-Fredholm Integro-Differential Equations
    9.1 Introduction
    9.2 The Volterra-Fredholm Integro-Differential Equation.
    9.2.1 The Series Solution Method.
    9.2.2 The Variational Iteration Method.
    9.3 The Mixed Volterra-Fredholm Integro-Differential Equations
    9.3.1 The Direct Computation Method
    9.3.2 The Series Solution Method
    9.4 The Mixed Volterra-Fredholm Integro-Differential Equations in Two Variables
    9.4.1 The Modified Decomposition Method
    References.

  • 10. Systems of Volterra Integral Equations
    10.1 Introduction.
    10.2 Systems of Volterra Integral Equations of the Second Kind.
    10.2.1 The Adomian Decomposition Method.
    10.2.2 The Laplace Transform Method
    10.3 Systems of Volterra Integral Equations of the First Kind
    10.3.1 The Laplace Transform Method
    10.3.2 Conversion to a Volterra System of the Second Kind
    10.4 Systems of Volterra Integro-Differential Equations
    10.4.1 The Variational Iteration Method
    10.4.2 The Laplace Transform Method
    References.

  • 11. Systems of Fredholm Integral Equations.
    11.1 Introduction.
    11.2 Systems of Fredholm Integral Equations.
    11.2.1 The Adomian Decomposition Method.
    11.2.2 The Direct Computation Method
    11.3 Systems of Fredholm Integro-Differential Equations.
    11.3.1 The Direct Computation Method
    11.3.2 The Variational Iteration Method
    References.

  • 12. Systems of Singular Integral Equations
    12.1 Introduction.
    12.2 Systems of Generalized Abel Integral Equations
    12.2.1 Systems of Generalized Abel Integral Equations in Two Unknowns
    12.2.2 Systems of Generalized Abel Integral Equations in Three Unknowns
    12.3 Systems of the Weakly Singular Volterra Integral Equations
    12.3.1 The Laplace Transform Method
    12.3.2 The Adomian Decomposition Method
    References 0.383

  • Part II Nonlinear Integral Equations

  • 13. Nonlinear Volterra Integral Equations
    13.1 Introduction
    13.2 Existence of the Solution for Nonlinear Volterra Integral Equations.
    13.3 Nonlinear Volterra Integral Equations of the Second Kind
    13.3.1 The Successive Approximations Method.
    13.3.2 The Series Solution Method.
    13.3.3 The Adomian Decomposition Method.
    13.4 Nonlinear Volterra Integral Equations of the First Kind
    13.4.1 The Laplace Transform Method
    13.4.2 Conversion to a Volterra Equation of the Second Kind.
    13.5 Systems of Nonlinear Volterra Integral Equations.
    13.5.1 Systems of Nonlinear Volterra Integral Equations of the Second Kind.
    13.5.2 Systems of Nonlinear Volterra Integral Equations of the First Kind.
    References.

  • 14. Nonlinear Volterra Integro-Differential Equations
    14.1 Introduction.
    14.2 Nonlinear Volterra Integro-Differential Equations of the Second Kind.
    14.2.1 The Combined Laplace Transform-Adomian Decomposition Method
    14.2.2 The Variational Iteration Method
    14.2.3 The Series Solution Method
    14.3 Nonlinear Volterra Integro-Differential Equations of the First Kind
    14.3.1 The Combined Laplace Transform-Adomian Decomposition Method
    14.3.2 Conversion to Nonlinear Volterra Equation of the Second Kind.
    14.4 Systems of Nonlinear Volterra Integro-Differential Equations.
    14.4.1 The Variational Iteration Method
    14.4.2 The Combined Laplace Transform-Adomian Decomposition Method
    References.

  • 15. Nonlinear Fredholm Integral Equations
    15.1 Introduction.
    15.2 Existence of the Solution for Nonlinear Fredholm Integral Equations.
    15.2.1 Bifurcation Points and Singular Points
    15.3 Nonlinear Fredholm Integral Equations of the Second Kind.
    15.3.1 The Direct Computation Method
    15.3.2 The Series Solution Method.
    15.3.3 The Adomian Decomposition Method.
    15.3.4 The Successive Approximations Method.
    15.4 Homogeneous Nonlinear Fredholm Integral Equations.
    15.4.1 The Direct Computation Method490
    15.5 Nonlinear Fredholm Integral Equations of the First Kind
    15.5.1 The Method of Regularization.
    15.5.2 The Homotopy Perturbation Method
    15.6 Systems of Nonlinear Fredholm Integral Equations
    15.6.1 The Direct Computation Method
    15.6.2 The Modified Adomian Decomposition Method
    References.

  • 16. Nonlinear Fredholm Integro-Differential Equations
    16.1 Introduction.
    16.2 Nonlinear Fredholm Integro-Differential Equations
    16.2.1 The Direct Computation Method
    16.2.2 The Variational Iteration Method
    16.2.3 The Series Solution Method.
    16.3 Homogeneous Nonlinear Fredholm Integro-Differential Equations.
    16.3.1 The Direct Computation Method
    16.4 Systems of Nonlinear Fredholm Integro-Differential Equations.
    16.4.1 The Direct Computation Method
    16.4.2 The Variational Iteration Method
    References.

  • 17. Nonlinear Singular Integral Equations.
    17.1 Introduction.
    17.2 Nonlinear Abel’s Integral Equation.
    17.2.1 The Laplace Transform Method
    17.3 The Generalized Nonlinear Abel Equation
    17.3.1 The Laplace Transform Method
    17.3.2 The Main Generalized Nonlinear Abel Equation
    17.4 The Nonlinear Weakly-Singular Volterra Equations.
    17.4.1 The Adomian Decomposition Method.
    17.5 Systems of Nonlinear Weakly-Singular Volterra Integral Equations.
    17.5.1 The Modified Adomian Decomposition Method
    References

  • 18. Applications of Integral Equations
    18.1 Introduction.
    18.2 Volterra’s Population Model
    18.2.1 The Variational Iteration Method
    18.2.2 The Series Solution Method.
    18.2.3 The Pad´e Approximants
    18.3 Integral Equations with Logarithmic Kernels
    18.3.1 Second Kind Fredholm Integral Equation with a Logarithmic Kernel
    18.3.2 First Kind Fredholm Integral Equation with a Logarithmic Kerne
    18.3.3 Another First Kind Fredholm Integral Equation with a Logarithmic Kernel.
    18.4 The Fresnel Integrals.
    18.5 The Thomas-Fermi Equation
    18.6 Heat Transfer and Heat Radiation.
    18.6.1 Heat Transfer: Lighthill Singular Integral Equation
    18.6.2 Heat Radiation in a Semi-Infinite Solid
    References

  • Appendix
    A Table of Indefinite Integrals.
    A.1 Basic Forms
    A.2 Trigonometric Forms.
    A.3 Inverse Trigonometric Forms
    A.4 Exponential and Logarithmic Forms.
    A.5 Hyperbolic Forms
    A.6 Other Forms.
    B Integrals Involving Irrational Algebraic Functions.
    B.1 Integrals Involving tn vx-t , n is an integer, n  0
    B.2 Integrals Involving tn2vx-t , n is an odd integer, n  1
    C Series Representations
    C.1 Exponential Functions Series
    C.2 Trigonometric Functions
    C.3 Inverse Trigonometric Functions
    C.4 Hyperbolic Functions
    C.5 Inverse Hyperbolic Functions
    C.6 Logarithmic Functions.
    D The Error and the Complementary Error Functions
    D.1 The Error Function
    D.2 The Complementary Error Function
    E Gamma Function
    F Infinite Series
    F.1 Numerical Series
    F.2 Trigonometric Series
    G The Fresnel Integrals
    G.1 The Fresnel Cosine Integral
    G.2 The Fresnel Sine Integral
    Answers
    Index

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