Page 223 - Handbook Of Integral Equations

3.1-4. Kernels Containing Square Roots

a

√ √
18. x – t y(t) dt = f(x), 0 < a < ∞.

0
√
This is a special case of equation 3.8.3 with g(x)= x.
Solution:
d √

y(x)= xf (x) .
x
dx
The right-hand side f(x) of the equation must satisfy certain conditions. The general
form of the right-hand side is

f(x)= F(x)+ Ax + B, A = –F (a), B = 1 aF (a) – F(a) – F(0) ,

x 2 x
where F(x) is an arbitrary bounded twice differentiable function (with bounded first deriva-
tive).
a

√ √
19. x – β t y(t) dt = f(x), β >0.

0
√ √
This is a special case of equation 3.8.4 with g(x)= x and β = λ.
a

√
20. x – t y(t) dt = f(x).

0
√
◦
This is a special case of equation 3.8.5 with g(x)= x (see item 3 of 3.8.5).
a

√
21. x – t y(t) dt = f(x).

0
√
This is a special case of equation 3.8.6 with g(t)= t (see item 3 of 3.8.6).
◦
a
y(t)
22. √ dt = f(x), 0 < a ≤ ∞.
0 |x – t|
1
This is a special case of equation 3.1.29 with k = .
2
Solution:
A d a dt t f(s) ds 1
y(x)= – , A = √ .
2
x 1/4 dx x (t – x) 1/4 0 s 1/4 (t – s) 1/4 8π Γ (3/4)
∞ y(t)

23. √ dt = f(x).
–∞ |x – t|
1
This is a special case of equation 3.1.34 with λ = .
2
Solution:
1 ∞ f(x) – f(t)
y(x)= dt.
4π |x – t| 3/2
–∞

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 202