Page 338 - Handbook Of Integral Equations

n
b
20. y(x) – λ g k (x)h k (t) y(t) dt = f(x), n =2, 3, ...
a
k=1
The characteristic values of the integral equation (counting the multiplicity, we have exactly
n of them) are the roots of the algebraic equation
∆(λ)=0,
where
–1
1 – λs 11 –λs 12 ··· –λs 1n s 11 – λ s 12 ··· s 1n
–1
–λs 21 1 – λs 22 ··· –λs 2n s 21 s 22 – λ ··· s 2n
∆(λ)= . . . . =(–λ) n . . . . ,
. . . . . . . . . . . . . . . .

–1
–λs n1 –λs n2 ··· 1 – λs nn s n1 s n2 ··· s nn – λ
and the integrals
b

s mk = h m (x)g k (x) dx; m, k =1, ... , n,
a
are assumed to be convergent.
Solution with regular λ:
n

y(x)= f(x)+ λ A k g k (x),
k=1
where the constants A k form the solution of the following system of algebraic equations:
n b

A m – λ s mk A k = f m , f m = f(x)h m (x) dx, m =1, ... , n.
a
k=1
The A k can be calculated by Cramer’s rule:
A k = ∆ k (λ)/∆(λ),
where

1 – λs 11 ··· –λs 1k–1 f 1 –λs 1k+1 ··· –λs 1n

–λs 21 ··· –λs 2k–1 f 2 –λs 2k+1 ··· –λs 2n
∆ k (λ)= .
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

–λs n1 ··· –λs nk–1 f n –λs nk+1 ··· 1 – λs nn
For solutions of the equation in the case in which λ is a characteristic value, see Subsec-
tion 11.2-2.
•
Reference: S. G. Mikhlin (1960).

4.9-2. Equations With Difference Kernel: K(x, t)= K(x – t)
π
21. y(x)= λ K(x – t)y(t) dt, K(x)= K(–x).
–π
Characteristic values:
π
1 1
λ n = , a n = K(x) cos(nx) dx (n =0, 1, 2, ... ).
πa n π –π
The corresponding eigenfunctions are
(2)
(1)
y 0 (x)=1, y (x) = cos(nx), y (x) = sin(nx)(n =1, 2, ... ).
n n
For each value λ n with n ≠ 0, there are two corresponding linearly independent eigenfunctions
(1)
(2)
y (x) and y (x).
n
n
•
Reference: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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