Page 468 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 31: Fredholm Equations with Poincere Goursat Kernels

Such a Kernel has been mentioned in the case of Volterra integral equations and are called Notes
degenerate Kernels or Poincere Goursat Kernels i.e. P-G Kernels. In equation (2) the functions
g (x) and h (u), for i = 1, 2, ...n, are L -class. We will see that in this case the Fredholm integral
i i 2
equation can be reduced to an algebraic system of a linear equation in n unknown. Here g (x) and
i
h (x) i = 1, 2, ...n are independent in the basic interval (0, 1).
i
Substituting (2) in one we have
n
1
u
h
x
y ( ) g i ( ) ( ) ( )du f ( )
x
u
x
y
i
0
i 1
n
1
x
or y ( ) ( ) g i ( ) h i ( ) ( )du f ( )
u
x
u
y
x
0
i 1
If we put
1
h ( ) ( )du (x = 1, 2, ...n) ...(3)
u
u
g
k k
0
then equation (1) becomes
n
x
x
g
x
y ( ) f ( ) i i ( ) ...(4)
i 1
From equation (4) it is already seen that the difference y(x) f(x) must necessarily coincide with
a suitable linear combination of the functions g (x). Now multiply equation (4) by h (x) i = 1, 2,...n
i k
and integrate between 0 and 1 we have
n
1 1 1
x
x
x
x
f
h k ( ) ( )dx h k ( ) ( )dx i g i ( ) ( )dx
y
x
x
h
k
0 0 0
i 1
Defining
h
g i ( ) ( )dx a ik
x
x
k
...(5)
h i ( ) ( )dx b i
x
x
f
We have
n
a b ...(6)
k ik i k
i 1
We thus see that the unknowns , ... must satisfy the following system of linear equations
1 n
(1 a 11 ) 1 a 12 2 a 13 3 ...... a 1n n b 1
a 21 1 (1 a 22 ) 2 a 23 3 .... a 2n n b 2
 ...(7)
a n 1 1 a n 2 2 ...... (1 a nn ) n b n
To each set of solution ° , ° ... ° of this system there corresponds a solution of equation (1)
1 2 n
given by (4). Now the solution of equation exist if the determinant formed by the coefficients
i
in equation (7) defined by
1 a 11 a 12 a 13 ...... a 1n
a (1 a ) ............... a
21 22 2n
D ( ) a 31 ........................................
...(8)
...................................................
a n 1 a n 2 ..............(1 a nn )
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