Page 104 - Handbook Of Integral Equations

x
√

23. (x – t) 1/4 J 1/2 λ x – t y(t) dt = f(x).
a
Solution:
2 x √
2 d cosh λ x – t
y(x)= √ f(t) dt.
πλ dx 2 x – t
a
x √

24. (x – t) 3/4 J 3/2 λ x – t y(t) dt = f(x).
a
Solution: √
2 3/2 d 3 x cosh λ x – t
y(x)= √ 3/2 3 √ f(t) dt.
πλ dx a x – t
x √

25. (x – t) n/2 J n λ x – t y(t) dt = f(x), n = 0,1,2, ...
a
Solution:
2 d

n n+2 x √
y(x)= I 0 λ x – t f(t) dt.
λ dx n+2
a
x
2n–3 √

26. x – t 4 J 2n–3 λ x – t y(t) dt = f(x), n =1, 2, ...
a 2
Solution:
2n–3 x √
1 2 2 d n cosh λ x – t
y(x)= √ √ f(t) dt.
π λ dx n a x – t
x
√
27. (x – t) –1/4 J –1/2 λ x – t y(t) dt = f(x).
a
Solution:
x √
λ d cosh λ x – t
y(x)= √ f(t) dt.
2π dx a x – t
x
√

28. (x – t) ν/2 J 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 I 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 I n–ν–2 λ x – t f t (t) dt.
λ
a
x √

2 2 –1/4 2 2
29. x – t J –1/2 λ x – t y(t) dt = f(x).
0
Solution: √
x 2 2
2λ d cosh λ x – t
y(x)= t √ f(t) dt.
π dx 0 x – t 2
2
•
Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 82