Page 472 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 31: Fredholm Equations with Poincere Goursat Kernels

In fact, conditions (26) are necessary because if equation (1) for = admits a certain solution Notes
0
(x), then from the equation itself, it follows that
x
x
x
x
x
f ( ) 0h ( )dx ( ) 0h ( )dx 0 0h ( )dx K ( , ) ( )dy
x
y
y
x
dx
x
x
( ) ( )dx ( )dy K ( , ) ( ) .
y
x
y
0h 0 0h
But, since and (x) are eigenvalue and corresponding eigenfunction of the associated Kernel,
0 0h
we have
K ( , ) ( )dx ( );
x
y
x
y
0 0h 0h
hence
x
f ( ) 0h ( )dx 0
x
Furthermore, conditions (26) are also sufficient, since from them it can be easily deduced that the
non-homogeneous system (7), which we shall write briefly as
b b ..., b ,
1 1 2 2 n n
reduces to only n r independent equations. Consequently we can now solve it readily (carrying
r unknowns on the right hand side), since the characteristic of matrix of the coefficients is exactly
p = n r.
We can reduce the system for the following reason: Let us multiply the previous equations by
*
*
*
B , B , ...B , respectively and add. Bearing in mind equations (25), we have
h1 h2 hr
n
*
B hk k [(1 a 11 )B h * 1 a B * 2 ... a B * ] 1
21 h
n
1 hn
k 1
[ a B * 1 (1 a 22 )B h * 2 ... a B * ] 2
12 h
2 hn
n
..................................................................
[ a B * a B * ... (1 a )B * ] 0,
1n h 1 2n h 2 nn hn n
while on the other side, by virtue of (26), we also have
n n
*
*
x
x
x
x
f
B b B Y ( ) f ( )dx ( ) ( )dx 0.
hk k hk k oh
k 1 k 1
Among other things, form (27) of the solution demonstrates the following obvious fact: the
general solution of equation (1) when D( ) = 0 can be considered as the sum of any particular
solution (x) and of the general solution (18) of the homogeneous equation.
Thus we have proved for PG Kernels the following basic Fredholm theorem, which will be
extended to general Kernels in the next section:
Fredholm’s integral equation of the second kind
y
y
x
x
( ) K ( , ) ( )dy f ( )
x
has, in general, one and only one solution of the class L given by the formula
2
x
( ) f ( ) H ( , ; ) ( )dy ,
y
x
x
y
f
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