Page 229 - Handbook Of Integral Equations

µ
1 t y(xt)
46. dt = f(x), 0 < λ <1.
0 (1 – t) λ
µ
The transformation ξ = xt, w(ξ)= ξ y(ξ) leads to the generalized Abel equation 1.1.46:
x
w(ξ) dξ 1+µ–λ

= x f(x).
(x – ξ) λ
0

∞ y(x + t) – y(x – t)
47. dt = f(x).
0 t
Solution:
1 ∞ f(x + t) – f(x – t)
y(x)= – dt.
π 2 0 t
•
Reference: V. A. Ditkin and A. P. Prudnikov (1965).
3.1-7. Singular Equations
In this subsection, all singular integrals are understood in the sense of the Cauchy principal value.
y(t) dt
∞

48. = f(x).
–∞ t – x
Solution:
1 ∞ f(t) dt
y(x)= – .
π 2 t – x
–∞
The integral equation and its solution form a Hilbert transform pair (in the asymmetric
form).
•
Reference: V. A. Ditkin and A. P. Prudnikov (1965).
b
y(t) dt
49. = f(x).
a t – x
This equation is encountered in hydrodynamics in solving the problem on the flow of an ideal
inviscid fluid around a thin profile (a ≤ x ≤ b). It is assumed that |a| + |b| < ∞.
◦
1 . The solution bounded at the endpoints is
1 b f(t) dt
y(x)= – (x – a)(b – x) √ ,
π 2 a (t – a)(b – t) t – x
provided that
b

f(t) dt
√ =0.
(t – a)(b – t)
a
◦
2 . The solution bounded at the endpoint x = a and unbounded at the endpoint x = b is
b
1 x – a b – t f(t)
y(x)= – dt.
π 2 b – x t – a t – x
a
◦
3 . The solution unbounded at the endpoints is
√
b

1 (t – a)(b – t)
y(x)= – √ f(t) dt + C ,
π 2 (x – a)(b – x) a t – x
b
where C is an arbitrary constant. The formula y(t) dt = C/π holds.
a
Solutions that have a singularity point x = s inside the interval [a, b] can be found in
Subsection 12.4-3.
•
Reference: F. D. Gakhov (1977).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 208