Page 433 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes x
u
x
K
u
where K 2 ( , ) K ( , u 1 ) (u 1 , )du 1
x
u
Similarly, we find in general
x
u
x
f
x
( ) K ( , ) ( )du (n = 1, 2, 3, ....) ...(8)
u
n n
0
Where the integrated Kernels are defined as
K (x, u) K(x, u), K (x, u), K (x, u)......
1 2 3
are defined by the recurrence formula
x
)
u
x
K n 1 ( , ) K ( , u K n (u 1 , )du 1 (n = 1, 2, 3, ....) ...(9)
x
u
1
0
Moreover, it is easily seen that we also have
x
)
u
u
x
x
K ( , ) K ( , u K (u , )du
n 1 1 1 n 1 1
0
x ...(9)
)
u
K ( , u K (u 1 , )du 1 r 0 s 0 n 1
x
1
0 0 r 0 s
where r = 1, s = n.
0 0
x x
,
u
x
u
x
)
Now K n 1 ( , ) K 1 ( , u 1 ) K 1 (u u K n 1 (u 2 , )du du 1
1
2
2
0 u 1
Interchanging the integrals we have
x x
,
)
K n 1 ( , ) K n 1 (u 2 , )du 2 K 1 ( , u K 1 (u u 2 )du 1
u
u
x
x
1
1
0 u 2
x
K n 1 (u 2 , ) K 2 ( , u 2 )du 2
u
x
0
x
)
K 2 ( , u K n 1 (u 2 , )du 2
x
u
2
0
In the same way we get
x
)
u
u
x
K n 1 ( , ) K 3 ( , u K n 2 (u 2 , )du 2
x
2
0
and so on. So we may write
x
)
x
u
u
x
K n 1 ( , ) K r ( , u K s (u 2 , )du 2 where (r = 1, 2,...n, s = n r + 1) ...(10)
2
0
Now from equation (7)
n
y n ( ) v v ( )
x
x
v 0
n x
v
x
f
u
K v ( , ) ( )du
u
0
v 0
n x
x
u
u
x
f
f ( ) v K v ( , ) ( )du
0
v 1
x n
x
or y n ( ) f ( ) K v ( , ) f ( )du
x
u
u
x
0
v 1
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