Covers All 8 Chapters
SOLUTIONS MANUAL
,Contents
Preface ix
1 Introductory Concepts 1
1.2 Classification of Linear Integral Equations . . . . . . . . . . 1
1.3 Solution of an Integral Equation . . . . . . . . . . . . . . . 2
1.4 Converting Volterra Equation to an ODE . . . . . . . . . . 4
1.5 Converting IVP to Volterra Equation . . . . . . . . . . . . . 7
1.6 Converting BVP to Fredholm Equation . . . . . . . . . . . 11
1.7 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Fredholm Integral 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 Homogeneous Fredholm Equation . . . . . . . . . . . . . . . 35
2.9 Fredholm Integral Equation of the First Kind . . . . . . . . 39
3 Volterra Integral Equations 41
3.2 Adomian Decomposition Method . . . . . . . . . . . . . . . 41
3.3 The Variational Iteration Method . . . . . . . . . . . . . . . 54
3.4 The Series Solution Method . . . . . . . . . . . . . . . . . . 57
3.5 Converting 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
vii
,viii Contents
4 Fredholm Integro-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 Converting to Fredholm Integral Equations . . . . . . . . . 96
5 Volterra Integro-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 Converting to Volterra Equations . . . . . . . . . . . . . . . 107
5.7 Converting to Initial Value Problems . . . . . . . . . . . . . 110
5.8 The Volterra Integro-Differential Equations of the First
Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6 Singular Integral Equations 117
6.2 Abel’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.3 Generalized Abel’s Problem . . . . . . . . . . . . . . . . . . 122
6.4 The Weakly Singular Volterra Equations . . . . . . . . . . . 122
6.5 The Weakly Singular Fredholm Equations . . . . . . . . . . 130
7 Nonlinear Fredholm Integral Equations 133
7.2 Nonlinear Fredholm Integral 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 Integral Equations of the First
Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.4 Weakly-Singular Nonlinear Fredholm Integral Equations . . 153
8 Nonlinear Volterra Integral Equations 157
8.2 Nonlinear Volterra Integral 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 Integral 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-Singular Volterra Equation . . . . . . . . 173
, Chapter 1
Introductory Concepts
1.2 Classification of Linear Integral Equations
Exercises 1.2
1. Fredholm, linear, nonhomogeneous
2. Volterra, linear, nonhomogeneous
3. Volterra, nonlinear, nonhomogeneous
4. Fredholm, linear, homogeneous
5. Fredholm, linear, nonhomogeneous
6. Fredholm, nonlinear, nonhomogeneous
7. Fredholm, nonlinear, nonhomogeneous
8. Fredholm, linear, nonhomogeneous
9. Volterra, nonlinear, nonhomogeneous
10. Volterra, linear, nonhomogeneous
11. Volterra integro-differential equation, nonlinear
12. Fredholm integro-differential equation, linear
13. Volterra integro-differential equation, nonlinear
14. Fredholm integro-differential equation, linear
15. Volterra integro-differential equation, linear
Z x
16. u(x) = 1 + 4u(t)dt
0
Z x
17. u(x) = 1 + 3t2 u(t)dt
0
Z x
18. u(x) = 4 + u2 (t)dt
0
1
SOLUTIONS MANUAL
,Contents
Preface ix
1 Introductory Concepts 1
1.2 Classification of Linear Integral Equations . . . . . . . . . . 1
1.3 Solution of an Integral Equation . . . . . . . . . . . . . . . 2
1.4 Converting Volterra Equation to an ODE . . . . . . . . . . 4
1.5 Converting IVP to Volterra Equation . . . . . . . . . . . . . 7
1.6 Converting BVP to Fredholm Equation . . . . . . . . . . . 11
1.7 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Fredholm Integral 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 Homogeneous Fredholm Equation . . . . . . . . . . . . . . . 35
2.9 Fredholm Integral Equation of the First Kind . . . . . . . . 39
3 Volterra Integral Equations 41
3.2 Adomian Decomposition Method . . . . . . . . . . . . . . . 41
3.3 The Variational Iteration Method . . . . . . . . . . . . . . . 54
3.4 The Series Solution Method . . . . . . . . . . . . . . . . . . 57
3.5 Converting 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
vii
,viii Contents
4 Fredholm Integro-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 Converting to Fredholm Integral Equations . . . . . . . . . 96
5 Volterra Integro-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 Converting to Volterra Equations . . . . . . . . . . . . . . . 107
5.7 Converting to Initial Value Problems . . . . . . . . . . . . . 110
5.8 The Volterra Integro-Differential Equations of the First
Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6 Singular Integral Equations 117
6.2 Abel’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.3 Generalized Abel’s Problem . . . . . . . . . . . . . . . . . . 122
6.4 The Weakly Singular Volterra Equations . . . . . . . . . . . 122
6.5 The Weakly Singular Fredholm Equations . . . . . . . . . . 130
7 Nonlinear Fredholm Integral Equations 133
7.2 Nonlinear Fredholm Integral 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 Integral Equations of the First
Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.4 Weakly-Singular Nonlinear Fredholm Integral Equations . . 153
8 Nonlinear Volterra Integral Equations 157
8.2 Nonlinear Volterra Integral 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 Integral 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-Singular Volterra Equation . . . . . . . . 173
, Chapter 1
Introductory Concepts
1.2 Classification of Linear Integral Equations
Exercises 1.2
1. Fredholm, linear, nonhomogeneous
2. Volterra, linear, nonhomogeneous
3. Volterra, nonlinear, nonhomogeneous
4. Fredholm, linear, homogeneous
5. Fredholm, linear, nonhomogeneous
6. Fredholm, nonlinear, nonhomogeneous
7. Fredholm, nonlinear, nonhomogeneous
8. Fredholm, linear, nonhomogeneous
9. Volterra, nonlinear, nonhomogeneous
10. Volterra, linear, nonhomogeneous
11. Volterra integro-differential equation, nonlinear
12. Fredholm integro-differential equation, linear
13. Volterra integro-differential equation, nonlinear
14. Fredholm integro-differential equation, linear
15. Volterra integro-differential equation, linear
Z x
16. u(x) = 1 + 4u(t)dt
0
Z x
17. u(x) = 1 + 3t2 u(t)dt
0
Z x
18. u(x) = 4 + u2 (t)dt
0
1
