Page 473 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes where H(x, y; ) is the resolvent Kernel. H(x, y; ) is an analytic function , and if | | < || || it
1
is given by the Neumann series
y
x
y
x
y
x
y
x
H ( , ; ) K ( , ) K 2 ( , ) 2 K 3 ( , ) ...,
where K , K ,... are the iterated Kernels. The only exceptions are the singular points of H(x, y; )
2 3
which coincide with the zeros (called eigenvalues) of an analytic function D( ) of . In the case
of a PG Kernel, D( ) is a polynomial.
If = is a root of multiplicity m 1 of the equation D( ) = 0, then the homogeneous equation
0
y
y
x
( ) K ( , ) ( )dy 0
x
has r linearly independent non-trivial solutions, called eigenfunctions, where r, the index of the
eigenvalue, satisfies the condition 1 r m. The same is true of the associated homogeneous
equation.
y
y
( ) K ( , ) ( )dy 0.
x
x
However, if = the non-homogeneous equation has solutions (exactly r solutions) if and
0
only if the given function f(x) is orthogonal to all the eigenfunctions of the associated homogeneous
equation.
A very important alternative theorem can immediately be deduced as a corollary:
Alternative Theorem: If the homogeneous Fredholm integral equation has only the trivial
solution, then the corresponding non-homogeneous equation always has one and only one
solution. On the contrary, if the homogeneous equation has some non-trivial solutions, then the
non-homogeneous integral equation has either no solution or an infinity of solutions, depending
on the given function f(x).
But even this corollary has been proved only for PG Kernels.
Self Assessment

1. The Kernel of Fredholm integral equation
2
t
x
t
y
x
y ( ) f ( ) K ( , ) ( )dt
x
0
is given by
1
x
t
k ( , ) 2 sin(vx ) sin[(v 1) ]
t
V
v 1
Find the iterated Kernel.
t
x
K ( , )
2
sin u
Hint : Use the relation lim u .
0
31.4 Summary
 Fredholm integral equation of the second kind is studied with the help of Poincere Goursat
Kernels.

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