Mathematical Methods for Physicists: A concise introduction - Site Map

SOME METHODS OF SOLUTION
Note that g…x† is a constant as far as the t integration is concerned, hence it may
be taken outside the integral sign and we have
Now
u…x† ˆf …x†‡ Xn
Z b
a
iˆ1
g i …x†
Z b
a
h i …t†u…t†dt:
h i …t†u…t†dt ˆ C i …ˆ const:†:
Substituting this into Eq. (11.7) and solving for u…t†, we obtain
u…t† ˆf …x†‡C Xn
iˆ1
g i …x†:
…11:7†
…11:8†
…11:9†
The value of C i may now be obtained by substituting Eq. (11.9) into Eq. (11.8).
The solution is only valid for certain values of , and we call these the eigenvalues
of the integral equation. The homogeneous equation has non-trivial solutions
only if is one of these eigenvalues; these solutions are called eigenfunctions of
the kernel (operator) K.
Example 11.1
As an example of this method, we consider the following equation:
u…x† ˆx ‡
Z 1
0
…xt 2 ‡ x 2 t†u…t†dt:
…11:10†
This is a Fredholm equation of the second kind, with f …x† ˆx and
K…x; t† ˆxt 2 ‡ x 2 t. If we de®ne
ˆ
then Eq. (11.10) becomes
Z 1
0
t 2 u…t†dt;
ˆ
Z 1
u…x† ˆx ‡ …x ‡ x 2 †:
0
tu…t†dt;
To determine A and B, we put Eq. (11.12) back into Eq. (11.11) and obtain
Solving this for and we ®nd
…11:11†
…11:12†
ˆ 1
4 ‡ 1 4 ‡ 1 5 ; ˆ 1
3 ‡ 1 3 ‡ 1 4
: …11:13†
60 ‡
ˆ
240 120 2 ; ˆ 80
240 120 2 ;
and the ®nal solution is
u…t†
ˆ…240 60†x ‡ 80x2
240 120 2 :
415