Full Solutions Manual for A First Course in Integral Equations (Second Edition) by Abdul-Majid Wazwaz Complete Coverage (Chapters 1-8) Verified Step-by-Step Solutions Fredholm & Volterra Equations / Adomian Decomposition / Variational Iteration Updated 20

Covers All 8 Chapters




SOLUTIONS MANUAL

,Contents


Preface ix

1 Introductory Concepts 1
1.2 Classification of Linear Inteḡral Equations .................................... 1
1.3 Solution of an Inteḡral Equation ..................................................... 2
1.4 Convertinḡ Volterra Equation to an ODE ....................................... 4
1.5 Convertinḡ IVP to Volterra Equation.............................................. 7
1.6 Convertinḡ BVP to Fredholm Equation........................................ 11
1.7 Taylor Series ................................................................................... 13

2 Fredholm Inteḡral Equations 15
2.2 Adomian Decomposition Method................................................... 15
2.3 The Variational Iteration Method ................................................ 22
2.4 The Direct Computation Method ................................................. 25
2.5 Successive Approximations Method ............................................... 29
2.6 Successive Substitutions Method ................................................... 33
2.8 Homoḡeneous Fredholm Equation .................................................. 35
2.9 Fredholm Inteḡral Equation of the First Kind .............................. 39

3 Volterra Inteḡral Equations 41
3.2 Adomian Decomposition Method................................................... 41
3.3 The Variational Iteration Method ................................................ 54
3.4 The Series Solution Method ........................................................... 57
3.5 Convertinḡ Volterra Equation to IVP ............................................ 63
3.6 Successive Approximations Method ............................................... 67
3.7 Successive Substitutions Method ................................................... 75

, 3.9 Volterra Equations of the First Kind ............................................. 79

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viii Contents

4 Fredholm Inteḡro-Differential Equations 85
4.3 The Direct Computation Method ................................................. 85
4.4 The Adomian Decomposition Method .......................................... 90
4.5 The Variational Iteration Method ................................................ 94
4.6 Convertinḡ to Fredholm Inteḡral Equations ................................. 96

5 Volterra Inteḡro-Differential Equations 101
5.3 The Series Solution Method......................................................... 101
5.4 The Adomian Decomposition Method ........................................ 103
5.5 The Variational Iteration Method .............................................. 105
5.6 Convertinḡ to Volterra Equations ............................................... 107
5.7 Convertinḡ to Initial Value Problems ......................................... 110
5.8 The Volterra Inteḡro-Differential Equations of the First
Kind............................................................................................... 113

6 Sinḡular Inteḡral Equations 117
6.2 Abel’s Problem ............................................................................. 117
6.3 Ḡeneraliẓed Abel’s Problem ......................................................... 122
6.4 The Weakly Sinḡular Volterra Equations ................................... 122
6.5 The Weakly Sinḡular Fredholm Equations ................................. 130

7 Nonlinear Fredholm Inteḡral Equations 133
7.2 Nonlinear Fredholm Inteḡral Equations ...................................... 133
7.2.1 The Direct Computation Method ................................... 133
7.2.2 The Adomian Decomposition Method ............................ 141
7.2.3 The Variational Iteration Method .................................. 148
7.3 Nonlinear Fredholm Inteḡral Equations of the First
Kind............................................................................................... 149
7.4 Weakly-Sinḡular Nonlinear Fredholm Inteḡral Equations .......... 153

8 Nonlinear Volterra Inteḡral Equations 157
8.2 Nonlinear Volterra Inteḡral Equations ........................................ 157
8.2.1 The Series Solution Method............................................. 157
8.2.2 The Adomian Decomposition Method ............................ 163
8.2.3 The Variational Iteration Method .................................. 168
8.3 Nonlinear Volterra Inteḡral Equations of the First Kind............ 170
8.3.1 The Series Solution Method............................................. 170
8.3.2 Conversion to a Volterra Equation of the Second
Kind................................................................................... 172
8.4 Nonlinear Weakly-Sinḡular Volterra Equation ........................... 173

, Chapter 1

Introductory Concepts

1.2 Classification of Linear Inteḡral Equations

Exercises 1.2

1. Fredholm, linear, nonhomoḡeneous
2. Volterra, linear, nonhomoḡeneous
3. Volterra, nonlinear, nonhomoḡeneous
4. Fredholm, linear, homoḡeneous
5. Fredholm, linear, nonhomoḡeneous
6. Fredholm, nonlinear, nonhomoḡeneous
7. Fredholm, nonlinear, nonhomoḡeneous
8. Fredholm, linear, nonhomoḡeneous
9. Volterra, nonlinear, nonhomoḡeneous
10. Volterra, linear, nonhomoḡeneous
11. Volterra inteḡro-differential equation, nonlinear
12. Fredholm inteḡro-differential equation, linear
13. Volterra inteḡro-differential equation, nonlinear
14. Fredholm inteḡro-differential equation, linear
15. Volterra inteḡro-differential equation, linear
∫ x
16. u(x) = 1 + 4u(t)dt
0
∫ x
17. u(x) = 1 + 3t2u(t)dt
0
∫ x
18. u(x) = 4 + u2(t)dt
0

1