Page 111 - Handbook Of Integral Equations

x
√

68. (x – t) ν/2 I ν λ x – t y(t) dt = f(x).
a
Solution:
2 d

n–2 n x
n–ν–2 √
y(x)= x – t 2 J n–ν–2 λ x – t f(t) dt,
λ dx n a
where –1< ν < n – 1, n =1, 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
2
(n)

n–2 x
n–ν–2 √
y(x)= x – t 2 J n–ν–2 λ x – t f t (t) dt.
λ
a
•
Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).
x √
2 2 –1/4 2 2

69. x – t I –1/2 λ x – t y(t) dt = f(x).
0
Solution: √
x 2 2
2λ d cos λ x – t
y(x)= t √ f(t) dt.
π dx 0 x – t 2
2
∞ √

2 2 –1/4 2 2
70. t – x I –1/2 λ t – x y(t) dt = f(x).
x
Solution:
√
∞ 2 2
2λ d cos λ t – x
y(x)= – t √ f(t) dt.
π dx 2 2
x t – x
x
√
2 2 ν/2

71. x – t I ν λ x – t 2 y(t) dt = f(x), –1< ν <0.
2
0
Solution:
d x 2 2 –(ν+1)/2 √

2
y(x)= λ t x – t J –ν–1 λ x – t 2 f(t) dt.
dx
0
•
Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).
∞ √

2
2 ν/2
2
72. (t – x ) I ν λ t – x 2 y(t) dt = f(x), –1< ν <0.
x
Solution:
d ∞ 2 2 –(ν+1)/2 √
2
y(x)= –λ t (t – x ) J –ν–1 λ t – x 2 f(t) dt.
dx x
•
Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).
x

s
k
73. [At I ν (λx)+ Bx I µ (λt)]y(t) dt = f(x).
a
k
s
This is a special case of equation 1.9.15 with g 1 (x)= AI ν (λx), h 1 (t)= t , g 2 (x)= Bx , and
h 2 (t)= I µ (λt).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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