Page 470 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 31: Fredholm Equations with Poincere Goursat Kernels

At this point it is to be noticed that the only singular points of H(x, u, ) in the -plane are the Notes
roots of the equation
D( ) = 0 ...(15)

which will be called the eigenvalues of our Kernel K(x, u)

31.3 Eigenvalues and Eigenvectors

If D( ) = 0 the non-homogeneous equation (1) has no solution in general, because an algebraic
linear system with vanishing determinant can only be solved for certain values of the quantities
on the right hand side of equation (7).
Furthermore, from each non-trivial solution 0 , 0 ,..., 0 of the homogeneous algebraic system
1 2 n
we obtain a non-trivial solution of the homogeneous equation (10), which we call an
eigenfunction and vice versa.
To be precise, from the theory of algebraic systems of linear equations. We infer that, if
coincides with a certain eigenvalue for which the determinant D( ) has the characteristic
0 0
P(1 p n 1), and we put n p = r, then there are r solutions of the homogeneous system (7).
Furthermore, these solutions can be represented by formulae of the type
n
B C B C ... B C (k 1, 2, ..., ) ...(16)
k 1k 1 2k 2 rk r
where C , C ,..., C denote r arbitrary constants and
1 2 r
B 11 , B 12 , ... B 1n
........................
...(17)
B r 1 , B r 2 . ... B rn
are r arbitrarily fixed but linearly independent solutions of the system in question.
This shows that to each eigenvalue of index r = n p there corresponds a solution of the
0
homogeneous equation (10) of the form
x
( ) C ( )C ( ) ... C ( ) ...(18)
x
x
x
0 1 01 2 02 r 0r
where C , C ,...C are r arbitrary constants and
1 2 r
x
x
x
( ), ( ), ..., ( )
01 02 0r
are r linearly independent functions, which can be expressed in terms of the B as follows:
hk
n
r
x
x
( ) B g ( ) (h 1, 2, ..., ) ...(19)
0h hk k
k 1
Moreover, we can assume that these functions are normalized, i.e., that their norms are all equal
to unity,
2
x
r
0h ( )dx 1 (h 1, 2, ..., ) ...(20)
Al these eigenfunctions are annihilated by the Fredholm operator
F s [ oh ( )] 0. ...(21)
y
Using elementary transformations on the determinant (7), we can see that the index r = n p of
an eigenvalue is never larger than its multiplicity m as a root of the equation D( ) = 0. Moreover,
in the important case a = a we have
hk kh
r = m
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