Page 559 - Handbook Of Integral Equations

11.6-5. Solution of the Nonhomogeneous Equation

Let us represent an integral equation
b
y(x) – λ K(x, t)y(t) dt = f(x), a ≤ x ≤ b, (17)
a
where the parameter λ is not a characteristic value, in the form

b
y(x) – f(x)= λ K(x, t)y(t) dt (18)
a
and apply the Hilbert–Schmidt theorem to the function y(x) – f(x):

∞

y(x) – f(x)= A k ϕ k (x),
k=1
b b b

A k = [y(x) – f(x)]ϕ k (x) dx = y(x)ϕ k (x) dx – f(x)ϕ k (x) dx = y k – f k .
a a a
Taking into account the expansion (8), we obtain

b ∞

y k
λ K(x, t)y(t) dt = λ ϕ k (x),
a λ k
k=1
and thus
y k λ k f k λf k
λ = y k – f k , y k = , A k = . (19)
λ k λ k – λ λ k – λ
Hence,
∞
f k
y(x)= f(x)+ λ ϕ k (x). (20)
λ k – λ
k=1
However, if λ is a characteristic value, i.e.,
λ = λ p = λ p+1 = ··· = λ q , (21)

then, for k ≠ p, p +1, ... , q, the terms (20) preserve their form. For k = p, p +1, ... , q, formula (19)
implies the relation f k = A k (λ – λ k )/λ, and by (21) we obtain f p = f p+1 = ··· = f q = 0. The last
relation means that
b

f(x)ϕ k (x) dx =0
a
for k =p, p+1, ... , q, i.e., the right-hand side of the equation must be orthogonal to the eigenfunctions
that correspond to the characteristic value λ.
In this case, the solutions of Eqs. (17) have the form

q
∞
f k
y(x)= f(x)+ λ ϕ k (x)+ C k ϕ k (x), (22)
λ k – λ
k=1 k=p
where the terms in the first of the sums (22) with indices k = p, p +1, ... , q must be omitted (for
these indices, f k and λ – λ k vanish in this sum simultaneously). The coefficients C k in the second
sum are arbitrary constants.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 542