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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
3.2.2 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
4.2.2 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
5.2.2 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
6.2.4 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
7.4.2 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
8.3 The Mixed Volterra-Fredholm Integral Equations
8.3.1 The Series Solution 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.
10.2.2 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.
11.2.2 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
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.
13.4 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.
15.3.4 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.
17.5 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.
18.3 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
Index

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