An Analytic Algorithm for Generalized Abel Integral Equation

230 S. Kumar, O. P. Singh and S. Dixit
L ( x ) =
2
9
10
x
5
3
−
27
40
x
8
3
+
x
⎛ ⎛ 2
⎜ Γ ⎜
⎝ ⎝ 3
⎛ 10 ⎞
Γ ⎜ ⎟
⎝ 3 ⎠
7
3
⎞ ⎞
⎟ ⎟
⎠ ⎠
2
10 ⎛ 2
3 ⎛ ⎞ ⎞
3 x ⎜ Γ ⎜ ⎟ ⎟
⎝ ⎝ 3 ⎠
−
⎠
⎛ 10 ⎞
5 Γ ⎜ ⎟
⎝ 3 ⎠
The Fig. (2) is drawn at the same level of truncation, n (=13) as was the case of Figure
(1).
2.5 10� 6
2. 10� 6
1.5 10� 6
1. 10� 6
5. 10� 7
0.2 0.4 0.6 0.8 1.0
Figure 2. The absolute error for Example 1, case 1(b) (n=13).
From Figs.1 and 2, one observes the dependence of the convergence rate of the series
(13) on the initial choice ( x).
y o
Case 1(c) Modified Homotopy perturbation method
Writing ( ) ∑ ( ) .
0
∞
2
f x = k i x where k 0 ( x ) = x , k 1 ( x ) =
i =
27
40
8
3 x and k i ( x)
= 0 for
i ≥ 2,
we get L
2
( x ) = x .
0
Hence, the various iterates are as follows:
0
p : L ( x ) = x
p
p
1
2
:
:
0
L ( x)
=
1
L
2
27
40
x
( x ) =
2
,
8
3
x
∫
0
−
x
∫
0
L
L ( t)
0
( x − t)
1
( t )
( x − t )
1
3
Therefore, one can see that Ln ( x)
= 0,
for all n ≥ 1,
and hence,
1
3
dt
dt
=
=
0,
0 .
2
,...