Page 298 - Handbook Of Integral Equations

b
3
28. y(x)+ A sinh (λ|x – t|)y(t) dt = f(x).
a
3
Using the formula sinh β = 1 sinh 3β – 3 sinh β, we arrive at an equation of the form 4.3.29
4 4
with n =2:
b

1 3
y(x)+ A sinh(3λ|x – t|) – A sinh(λ|x – t|) y(t) dt = f(x).
4 4
a
n
b
29. y(x)+ A k sinh(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)= sinh(λ k |x–t|)y(t) dt = sinh[λ k (x–t)]y(t) dt+ sinh[λ k (t–x)]y(t) dt. (1)
a a x
Differentiating (1) with respect to x twice yields
x b

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

I =2λ k y(x)+ λ 2 sinh[λ k (x – t)]y(t) dt + λ 2 sinh[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 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.)
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 277