Page 109 - Handbook Of Integral Equations

x

56. [AI ν (λx)+ BI µ (βt)]y(t) dt = f(x).
a
This is a special case of equation 1.9.6 with g(x)= AI ν (λx) and h(t)= BI µ (βt).

x
ν
57. (x – t) I ν [λ(x – t)]y(t) dt = f(x).
a
Solution:
2 n x
d 2 n–ν–1
y(x)= A – λ (x – t) I n–ν–1 [λ(x – t)] f(t) dt,
dx 2 a
2 Γ(ν +1) Γ(n – ν)

n–1
A = ,
λ Γ(2ν +1) Γ(2n – 2ν – 1)
1
where – < ν < n–1 and n =1, 2, ...
2 2
If the right-hand side of the equation is differentiable sufficiently many times and the
conditions f(a)= f (a)= ··· = f x (n–1) (a) = 0 are satisfied, then the solution of the integral

x
equation can be written in the form
x d 2 n
y(x)= A (x – t) n–ν–1 I n–ν–1 [λ(x – t)]F(t) dt, F(t)= 2 – λ 2 f(t).
a dt
•
Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).
x
58. (x – t) ν+1 I ν [λ(x – t)]y(t) dt = f(x).
a
Solution:
x

y(x)= g(t) dt,
a
where

d 2 n–ν–2
2 n t
g(t)= A – λ (t – τ) I n–ν–2 [λ(t – τ)] f(τ) dτ,
dt 2 a
2 Γ(ν +1) Γ(n – ν – 1)

n–2
A = ,
λ Γ(2ν +2) Γ(2n – 2ν – 3)
where –1< ν < n – 1 and n =1, 2, ...
2
If the right-hand side of the equation is differentiable sufficiently many times and the
conditions f(a)= f (a)= ··· = f x (n–1) (a) = 0 are satisfied, then the function g(t)defining the

x
solution can be written in the form
t n
d 2
g(t)= A (t – τ) n–ν–2 I n–ν–2 [λ(t – τ)]F(τ) dτ, F(τ)= 2 – λ 2 f(τ).
a dτ
•
Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).
x
√
59. I 0 λ x – t y(t) dt = f(x).
a
Solution:
d 2 x √
y(x)= J 0 λ x – t f(t) dt.
dx 2
a

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 87