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 Voltari Equation to an ODE ......................................... 4
1.5 Converting IVP to Voltari Equation ................................................ 7
1.6 Converting BVP to Fred Holm Equation...................................... 11
1.7 Taylor Series ................................................................................... 13
2 Fred Holm Integral Equations 15
2.2 Domain Decomposition Method .................................................... 15
2.3 The Variation 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 Fred Holm Equation................................................. 35
2.9 Fred Holm Integral Equation of the First Kind ............................ 39
3 Voltari Integral Equations 41
3.2 Domain Decomposition Method .................................................... 41
3.3 The Variation Iteration Method................................................... 54
3.4 The Series Solution Method .......................................................... 57
3.5 Converting Voltari Equation to IVP .............................................. 63
3.6 Successive Approximations Method .............................................. 67
3.7 Successive Substitutions Method................................................... 75
3.9 Voltari Equations of the First Kind ............................................... 79
vii
,viii Contents
4 Fred Holm Integra-Differential Equations 85
4.3 The Direct Computation Method ................................................ 85
4.4 The Domain Decomposition Method............................................ 90
4.5 The Variation Iteration Method................................................... 94
4.6 Converting to Fred Holm Integral Equations ............................... 96
5 Voltari Integra-Differential Equations 101
5.3 The Series Solution Method ........................................................ 101
5.4 The Domain Decomposition Method.......................................... 103
5.5 The Variation Iteration Method................................................. 105
5.6 Converting to Voltari Equations ................................................. 107
5.7 Converting to Initial Value Problems......................................... 110
5.8 The Voltari Integra-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 Voltari Equations ..................................... 122
6.5 The Weakly Singular Fred Holm Equations ............................... 130
7 Nonlinear Fred Holm Integral Equations 133
7.2 Nonlinear Fred Holm Integral Equations ..................................... 133
7.2.1 The Direct Computation Method .................................. 133
7.2.2 The Domain Decomposition Method ............................. 141
7.2.3 The Variation Iteration Method..................................... 148
7.3 Nonlinear Fred Holm Integral Equations of the First
Kind............................................................................................... 149
7.4 Weakly-Singular Nonlinear Fred Holm Integral Equations......... 153
8 Nonlinear Voltari Integral Equations 157
8.2 Nonlinear Voltari Integral Equations ........................................... 157
8.2.1 The Series Solution Method............................................ 157
8.2.2 The Domain Decomposition Method ............................. 163
8.2.3 The Variation Iteration Method..................................... 168
8.3 Nonlinear Voltari Integral Equations of the First Kind .............. 170
8.3.1 The Series Solution Method............................................ 170
8.3.2 Conversion to a Voltari Equation of the Second
Kind .................................................................................. 172
8.4 Nonlinear Weakly-Singular Voltari Equation.............................. 173
, Chapter 1
Introductory Concepts
1.2 Classification of Linear Integral Equations
Exercises 1.2
1. Fred Holm, linear, nonhomogeneous
2. Voltari, linear, nonhomogeneous
3. Voltari, nonlinear, nonhomogeneous
4. Fred Holm, linear, homogeneous
5. Fred Holm, linear, nonhomogeneous
6. Fred Holm, nonlinear, nonhomogeneous
7. Fred Holm, nonlinear, nonhomogeneous
8. Fred Holm, linear, nonhomogeneous
9. Voltari, nonlinear, nonhomogeneous
10. Voltari, linear, nonhomogeneous
11. Voltari integral-differential equation, nonlinear
12. Fred Holm integral-differential equation, linear
13. Voltari integral-differential equation, nonlinear
14. Fred Holm integral-differential equation, linear
15. Voltari integral-differential equation, linear
∫X
16. u(x) = 1 + 4u (t) dot
0
∫ X
17. u(x) = 1 + 3t2u (t) dot
0
∫ X
18. u(x) = 4 + u2 (t) dot
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 Voltari Equation to an ODE ......................................... 4
1.5 Converting IVP to Voltari Equation ................................................ 7
1.6 Converting BVP to Fred Holm Equation...................................... 11
1.7 Taylor Series ................................................................................... 13
2 Fred Holm Integral Equations 15
2.2 Domain Decomposition Method .................................................... 15
2.3 The Variation 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 Fred Holm Equation................................................. 35
2.9 Fred Holm Integral Equation of the First Kind ............................ 39
3 Voltari Integral Equations 41
3.2 Domain Decomposition Method .................................................... 41
3.3 The Variation Iteration Method................................................... 54
3.4 The Series Solution Method .......................................................... 57
3.5 Converting Voltari Equation to IVP .............................................. 63
3.6 Successive Approximations Method .............................................. 67
3.7 Successive Substitutions Method................................................... 75
3.9 Voltari Equations of the First Kind ............................................... 79
vii
,viii Contents
4 Fred Holm Integra-Differential Equations 85
4.3 The Direct Computation Method ................................................ 85
4.4 The Domain Decomposition Method............................................ 90
4.5 The Variation Iteration Method................................................... 94
4.6 Converting to Fred Holm Integral Equations ............................... 96
5 Voltari Integra-Differential Equations 101
5.3 The Series Solution Method ........................................................ 101
5.4 The Domain Decomposition Method.......................................... 103
5.5 The Variation Iteration Method................................................. 105
5.6 Converting to Voltari Equations ................................................. 107
5.7 Converting to Initial Value Problems......................................... 110
5.8 The Voltari Integra-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 Voltari Equations ..................................... 122
6.5 The Weakly Singular Fred Holm Equations ............................... 130
7 Nonlinear Fred Holm Integral Equations 133
7.2 Nonlinear Fred Holm Integral Equations ..................................... 133
7.2.1 The Direct Computation Method .................................. 133
7.2.2 The Domain Decomposition Method ............................. 141
7.2.3 The Variation Iteration Method..................................... 148
7.3 Nonlinear Fred Holm Integral Equations of the First
Kind............................................................................................... 149
7.4 Weakly-Singular Nonlinear Fred Holm Integral Equations......... 153
8 Nonlinear Voltari Integral Equations 157
8.2 Nonlinear Voltari Integral Equations ........................................... 157
8.2.1 The Series Solution Method............................................ 157
8.2.2 The Domain Decomposition Method ............................. 163
8.2.3 The Variation Iteration Method..................................... 168
8.3 Nonlinear Voltari Integral Equations of the First Kind .............. 170
8.3.1 The Series Solution Method............................................ 170
8.3.2 Conversion to a Voltari Equation of the Second
Kind .................................................................................. 172
8.4 Nonlinear Weakly-Singular Voltari Equation.............................. 173
, Chapter 1
Introductory Concepts
1.2 Classification of Linear Integral Equations
Exercises 1.2
1. Fred Holm, linear, nonhomogeneous
2. Voltari, linear, nonhomogeneous
3. Voltari, nonlinear, nonhomogeneous
4. Fred Holm, linear, homogeneous
5. Fred Holm, linear, nonhomogeneous
6. Fred Holm, nonlinear, nonhomogeneous
7. Fred Holm, nonlinear, nonhomogeneous
8. Fred Holm, linear, nonhomogeneous
9. Voltari, nonlinear, nonhomogeneous
10. Voltari, linear, nonhomogeneous
11. Voltari integral-differential equation, nonlinear
12. Fred Holm integral-differential equation, linear
13. Voltari integral-differential equation, nonlinear
14. Fred Holm integral-differential equation, linear
15. Voltari integral-differential equation, linear
∫X
16. u(x) = 1 + 4u (t) dot
0
∫ X
17. u(x) = 1 + 3t2u (t) dot
0
∫ X
18. u(x) = 4 + u2 (t) dot
0
1
