Page 292 - Handbook Of Integral Equations

n
b
16. y(x)+ A k exp(λ k |x – t|) y(t) dt = f(x), –∞ < a < b < ∞.
a
k=1
1 . Let us remove the modulus in the kth summand of the integrand:
◦

b x b
I k (x)= exp(λ k |x – t|)y(t) dt = exp[λ k (x – t)]y(t) dt + exp[λ k (t – x)]y(t) dt. (1)
a a x
Differentiating (1) with respect to x twice yields

x b

I = λ k exp[λ k (x – t)]y(t) dt – λ k exp[λ k (t – x)]y(t) dt,
k
a x
x b (2)
I =2λ k y(x)+ λ 2 exp[λ k (x – t)]y(t) dt + λ 2 exp[λ k (t – x)]y(t) dt,

k k k
a x
where the primes denote the derivatives with respect to x. By comparing formulas (1) and (2),
we find the relation between I and I k :

k
2
I =2λ k y(x)+ λ I k , I k = I k (x). (3)

k k
2 . With the aid of (1), the integral equation can be rewritten in the form
◦
n

y(x)+ A k I k = f(x). (4)
k=1
Differentiating (4) with respect to x twice and taking into account (3), we find that
n n
2
y (x)+ σ n y(x)+ A k λ I k = f (x), σ n =2 A k λ k . (5)

xx k xx
k=1 k=1
Eliminating the integral I n from (4) and (5) yields
n–1
2 2 2
2

y (x)+(σ n – λ )y(x)+ A k (λ – λ )I k = f (x) – λ f(x). (6)

xx n k n xx n
k=1
Differentiating (6) with respect to x twice and eliminating I n–1 from the resulting equation
with the aid of (6), we obtain a similar equation whose left-hand side is a second-order linear
n–2

differential operator (acting on y) with constant coefficients plus the sum B k I k .If we
k=1
successively eliminate I n–2 , I n–3 , ... , with the aid of double differentiation, then we finally
arrive at a linear nonhomogeneous ordinary differential equation of order 2n with constant
coefficients.
3 . The boundary conditions for y(x) can be found by setting x = a in the integral equation
◦
and all its derivatives. (Alternatively, these conditions can be found by setting x = a and x = b
in the integral equation and all its derivatives obtained by means of double differentiation.)
4.2-2. Kernels Containing Power-Law and Exponential Functions

b
γt
17. y(x) – λ (x – t)e y(t) dt = f(x).
a
γt
This is a special case of equation 4.9.8 with A =0, B = 1, and h(t)= e .

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 271