Page 463 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes solving the integral equation is to be chosen. Here the parameter also plays an important part.
So if is small as well as the Kernel K(x, t) is continuous along with its partial derivatives, we can
use the method of successive approximation.
Let us consider first the equation (2) of Fredholm integral equation of the second kind. To a zero
approximation

x
y ( )  f ( ).
x
If we substitute this value of y(x) in the integral (2) we get
b
t
x
f ( )  f ( ) K ( , ) ( )dt ...(3)
x
f
x
t
a
x
x
x
or y ( ) f ( ) 1 ( )
b
t
x
f
x
t
where 1 ( ) K ( , ) ( )dt ...(4)
a
So to a first approximation y(x) is given by (2). To get an improvement over the above
approximation we put this new value of y(x) given by (3) into (2) to improve the solution as
follows:
b b
x
u
t
t
f
u
y ( ) f ( ) K ( , ) f ( ) K ( , ) ( )du dt
t
x
x
a a
b b b
t
t
u
f
x
t
t
x
x
f ( ) K ( , ) ( )dt 2 K ( , )dt K ( , ) ( )du
f
u
a a a
or y ( ) f ( ) 1 ( ) 2 2 ( ) ...(5)
x
x
x
x
where
b b
t
x
u
u
( ) K ( , )dt K ( , ) ( )du
x
f
t
2
a a
b b
K
x
u
t
du f ( ) K ( , ) ( , )dt
t
u
a a
b
u
u
f
x
x
or 2 ( ) du K 2 ( , ) ( ) ...(6a)
a
b
x
x
u
t
u
t
K
where K 2 ( , ) K ( , ) ( , )dt ...(6b)
a
We can improve the accuracy by taking more powers of in y(x) i.e. we may write
x
y ( ) f ( ) 2 3 ... n ... ...(7)
x
1 2 3 n
where , are given by (4) and (6a) and other s are given by
1 2
b
u
( ) du K ( , ) ( ) for n = 1, 2, .... ...(8)
x
x
f
u
n n
a
th
and the n Kernel K (x, u) given by
n
b
x
K n ( , ) K r ( , u 1 )K n r (u 1 , )du 1 [n = 2, 3, 4,...; r = 1, 2, ...n 1 ...(9)
x
u
u
a
while K (x, u) = K(x, u)
1
n
x
x
x
Thus y ( ) f ( ) i i ( ) .... for any n ...(10)
i 1
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