An Analytic Algorithm for Generalized Abel Integral Equation

Analytic algorithm for generalized Abel integral equation 229
y ( x)
= x
2
+
27
40
x
8
3
−
x
∫
0
y ( t )
( x − t )
2
with the exact Solution y ( x)
= x .
1
3
dt ,
Case 1(a) Homotopy perturbation method
27 8
2
3
Taking L 0 ( x)
= x + x , the various iterates are obtained from equation ( 12 ).
40
The first few iterates are as follows:
L ( x ) =
1
−
27
40
x
8
3
4 ⎛ ⎛ 2 ⎞ ⎞
x ⎜ Γ ⎜ ⎟ ⎟
3
2 ( )
⎝ ⎝ ⎠
L x =
⎠
12
x
L ( x)
= −
3
4
−
3
⎛ ⎛ 2 ⎞ ⎞
⎜ Γ⎜
⎟ ⎟
⎝ ⎝ 3 ⎠ ⎠
12
2 x
+
3
10
27
⎛ 3 ⎛
⎜ Γ ⎜
⎝ ⎝
⎛ 13
Γ ⎜
⎝ 3
2
3
⎞
⎟
⎠
⎞
⎟
⎠
⎞
⎟
⎠
10 2 11
3 ⎛ ⎞ ⎛ ⎞
x Γ ⎜ ⎟ Γ ⎜ ⎟
⎝ 3 ⎠ ⎝ 3 ⎠
,
⎛ 13 ⎞
40 Γ ⎜ ⎟
⎝ 3 ⎠
14 ⎛ 2
3 ⎛ ⎞ ⎞
243 x ⎜ Γ⎜
⎟ ⎟
⎝ ⎝ 3 ⎠
−
⎠
6160
The Fig.1 shows the absolute error between the exact solution y ( x )
approximate solution y a ( x ) obtained from (11) by truncating it at level n=13.
4. 10� 7
3. 10� 7
2. 10� 7
1. 10� 7
2
,
3
, ...
0.2 0.4 0.6 0.8 1.0
Figure 1. The absolute error for Example1, case 1(a) (n=13).
Case 1(b) Now choosing a different initial guess
L ( x)
= x,
the following iterates of the solution are obtained
0
9 5
2
3
L 1 ( x ) =
− x + x − x +
10
27
40
x
8
3
,
(20)
and the