Page 595 - Handbook Of Integral Equations

11.13. The Method of Approximating a Kernel by a
Degenerate One

11.13-1. Approximation of the Kernel

For the approximate solution of the Fredholm integral equation of the second kind
b

y(x) – K(x, t)y(t) dt = f(x), a ≤ x ≤ b, (1)
a
where, for simplicity, the functions f(x) and K(x, t) are assumed to be continuous, it is useful to
replace the kernel K(x, t) by a close degenerate kernel

n

K (n) (x, t)= g k (x)h k (t). (2)
k=0
Let us indicate several ways to perform such a change. If the kernel K(x, t) is differentiable
with respect to x on [a, b] sufficiently many times, then, for a degenerate kernel K (n) (x, t), we can
take a finite segment of the Taylor series:

n m
(x – x 0 ) (m)
K (n) (x, t)= K x (x 0 , t), (3)
m!
m=0
where x 0 ∈ [a, b]. A similar trick can be applied for the case in which K(x, t) is differentiable with
respect to t on [a, b] sufficiently many times.
To construct a degenerate kernel, a finite segment of the double Fourier series can be used:
n n
p q
K (n) (x, t)= a pq (x – x 0 ) (t – t 0 ) , (4)
p=0 q=0

where
1 ∂ p+q
a pq = K(x, t) , a ≤ x 0 ≤ b, a ≤ t 0 ≤ b.
p! q! ∂x ∂t q x=x 0
p
t=t 0
A continuous kernel K(x, t) admits an approximation by a trigonometric polynomial of period 2l,
where l = b – a.
For instance, we can set

n
1 kπx
K (n) (x, t)= a 0 (t)+ a k (t) cos , (5)
2 l
k=1
where the a k (t)(k = 0,1,2, ... ) are the Fourier coefficients

b
2 pπx
a k (t)= K(x, t) cos dx. (6)
l a l
A similar decomposition can be obtained by interchanging the roles of the variables x and t.A
finite segment of the double Fourier series can also be applied by setting, for instance,

n
1 mπt
a k (t) ≈ a k0 + a km cos , k =0, 1, ... , n, (7)
2 l
m=1

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 578