Mathematical Methods for Physicists: A concise introduction - Site Map

SOME METHODS OF SOLUTION
by iteration or successive approximations, and begin with the approximation
u…x† u 0 …x† f …x†:
This approximation is equivalent to saying that the constant or the integral is
small. We then put this crude choice into the integral equation (11.2) under the
integral sign to obtain a second approximation:
u 1 …x† ˆf …x†‡
Z b
and the process is then repeated and we obtain
u 2 …x† ˆf …x†‡
Z b
a
K…x; t† f …t†dt ‡ 2 Z b
a
K…x; t† f …t†dt
a
Z b
a
K…x; t†K…t; t 0 † f …t 0 †dt 0 dt:
We can continue iterating this process, and the resulting series is known as the
Neumann series, or Neumann solution:
u…x† ˆf …x†‡
Z b
a
K…x; t† f …t†dt ‡ 2 Z b
This series can be written formally as
where
' 0 …x† ˆu 0 …x† ˆf …x†;
' 1 …x† ˆ
' 2 …x† ˆ
.
.
' n …x† ˆ
Z b
a
Z b Z b
a
K…x; t 1 † f …t 1 †dt 1 ;
a
Z b Z b
a
a
u n …x† ˆXn
iˆ1
a
Z b
a
i ' i …x†;
K…x; t 1 †K…t 1 ; t 2 † f …t 2 †dt 1 dt 2 ;
Z b
a
K…x; t†K…t; t 0 † f …t 0 †dt 0 dt ‡:
K…x; t 1 †K…t 1 ; t 2 †K…t n1 ; t n † f …t n †dt 1 dt 2 dt n :
>;
…11:14†
9
>=
…11:15†
The series (11.14) will converge for suciently small , when the kernel K…x; t† is
bounded. This can be checked with the Cauchy ratio test (Problem 11.4).
Example 11.2
Use the Neumann method to solve the integral equation
u…x† ˆf …x†‡ 1 2
Z 1
1
K…x; t†u…t†dt;
…11:16†
417