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A First Course in Integral Equations (Solution Manual) By Abdul Majid Wazwaz

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A First Course in Integral Equations (Solution Manual) written by Abdul Majid Wazwaz . Engineering, physics and mathematics students, both advanced undergraduate and beginning graduate, need an integral equations textbook that simply and easily introduces the material. They also need a textbook that embarks upon their already acquired knowledge of regular integral calculus and ordinary differential equations. Because of these needs, this textbook was created. From many years of teaching, I have found that the available treatments of the subject are abstract. Moreover, most of them are based on comprehensive theories such as topological methods of functional analysis, Lebesgue integrals and Green functions. Such methods of introduction are not easily accessible to those who have not yet had a background in advanced mathematical concepts. This book is especially designed for those who wish to understand integral equations without having the extensive mathematical background. In this fashion, this text leaves out abstract methods, comprehensive methods and advanced mathematical topics.
From my experience in teaching and in guiding related senior seminar projects for advanced undergraduate students, I have found that the material can indeed be taught in an accessible manner. Students have showed both a lot of motivation and capability to grasp the subject once the abstract theories and difficult theorems were omitted. In my approach to teaching integral equations, I focus on easily applicable techniques and I do not emphasize such abstract methods as existence, uniqueness, convergence and Green functions. I have translated my means of introducing and fully teaching this subject into this text so that the intended user can take full advantage of the easily presented and explained material.

A First Course in Integral Equations (Solution Manual) written by Abdul Majid Wazwaz cover the following topics.

  • 1. Introductory Concepts
    1.1 Definitions
    1.2 Classification of Linear Integral Equations
    1.2.1 Fredholm Linear Integral Equations
    1.2.2 Volterra Linear Integral Equations
    1.2.3 Integro-Differential Equations
    1.2.4 Singular Integral Equations
    1.2.5 Volterra-Fredholm Integral Equations
    1.2.6 Volterra-Fredholm Integro-Differential Equations
    1.3 Solution of an Integral Equation
    1.4 Converting Volterra Equation to an ODE
    1.4.1 Differentiating Any Integral: Leibniz Rule
    1.5 Converting IVP to Volterra Equation
    1.6 Converting BVP to Fredholm Equation
    1.7 Taylor Series
    1.8 Infinite Geometric Series

  • 2. Fredholm Integral Equations
    2.1 Introduction
    2.2 The Adomian Decomposition Method
    2.2.1 The Modified Decomposition Method
    2.2.2 The Noise Terms Phenomenon
    2.3 The Variational Iteration Method
    2.4 The Direct Computation Method
    2.5 The Succe

  • 3.
    3.2 The Adomian Decomposition Method
    3.2.1 The Modified Decomposition Method
    3.2.2 The Noise Terms Phenomenon
    3.3 The Variational Iteration Method
    3.4 The Series Solution Method
    3.5 Converting Volterra Equation to IVP
    3.6 Successive Approximations Method
    3.7 The Method of Successive Substitutions
    3.8 Comparison between Alternative Methods
    3.9 Volterra Integral Equations of the First Kind
    3.9.1 The Series Solution Method
    3.9.2 Conversion of First Kind to Second Kind

  • 4. Fredholm Integro-Differential Equations
    4.1 Introduction
    4.2 Fredholm Integro-Differential Equations
    4.3 The Direct Computation Method
    4.4 The Adomian Decomposition Method
    4.4.1 The Modified Decomposition Method
    4.4.2 The Noise Terms Phenomenon
    4.5 The Variational Iteration Method
    4.6 Converting to Fredholm Integral Equations

  • 5. Volterra Integro-Differential Equations
    5.1 Introduction
    5.2 Volterra Integro-Differential Equations
    5.3 The Series Solution Method
    5.4 The Adomian Decomposition Method
    5.5 The Variational Iteration Method
    5.6 Converting to Volterra Integral Equation
    5.7 Converting to Initial Value Problems
    5.8 Volterra Integro-Differential Equations of the First Kind

  • 6. Singular Integral Equations
    6.1 Introduction
    6.2 Abel’s Problem
    6.3 The Generalized Abel’s Integral Equation
    6.4 The Weakly-Singular Volterra Integral Equations
    6.4.1 The Adomian Decomposition Method
    6.5 The Weakly-Singular Fredholm Integral Equations
    6.5.1 The Modified Decomposition Method

  • 7. Nonlinear Fredholm Integral Equations
    7.1 Introduction
    7.2 Nonlinear Fredholm Integral Equations of the Second Kind
    7.2.1 The Direct Computation Method
    7.2.2 The Adomian Decomposition Method
    7.2.3 The Variational Iteration Method
    7.3 Nonlinear Fredholm Integral Equations of the First Kind
    7.3.1 The Method of Regularization
    7.4 Nonlinear Weakly-Singular Fredholm Integral Equations
    7.4.1 The Modified Decomposition Method

  • 8. Nonlinear Volterra Integral Equations
    8.1 Introduction
    8.2 Nonlinear Volterra Integral Equations of the Second Kind
    8.2.1 The Series Solution Method
    8.2.2 The Adomian Decomposition Method
    8.2.3 The Variational Iteration Method
    8.3 Nonlinear Volterra Integral Equations of the First Kind
    8.3.1 The Series Solution Method
    8.3.2 Conversion to a Volterra Equation of the Second Kind
    8.4 Nonlinear Weakly-Singular Volterra Integral Equations
    8.4.1 The Modified Decomposition Method

  • 9. Applications of Integral Equations
    9.1 Introduction
    9.2 Volterra Integral Form of the Lane-Emden Equation
    9.2.1 Lane-Emden Equation of the First Kind
    9.2.2 Lane-Emden Equation of the Second Kind
    9.3 The Schlömilch’s Integral Equation
    9.3.1 The Linear Schlömilch’s Integral Equation
    9.3.2 The Method of Regularization
    9.3.3 The Nonlinear Schlömilch’s Integral Equation
    9.4 Bratu-Type Problems
    9.4.1 First Bratu-Type Problem
    9.4.2 Second Bratu-Type Problem
    9.4.3 Third Bratu-Type Problem
    9.5 Systems of Integral Equations
    9.5.1 Systems of Fredholm Integral Equations
    9.5.2 Systems of Volterra Integral Equations
    9.6 Numerical Treatment of Fredholm Integral Equations
    9.7 Numerical Treatment of Volterra Integral Equations

  • Appendix
    A Table of Indefinite Integrals
    B Integrals Involving Irrational Algebraic Functions
    C Series
    D The Error and the Gamma Functions
    Answers to Exercises

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