Page 34 - Handbook Of Integral Equations

1.1-7. Kernels Containing Arbitrary Powers

x

λ
42. (x – t) y(t) dt = f(x), f(a)=0, 0< λ <1.
a
Differentiating with respect to x, we arrive at the generalized Abel equation 1.1.46:
y(t) dt 1
x
= f (x).

x
(x – t) 1–λ λ
a
Solution:
d 2 x f(t) dt sin(πλ)
y(x)= k , k = .
dx 2 a (x – t) λ πλ
•
Reference: F. D. Gakhov (1977).
x
µ
43. (x – t) y(t) dt = f(x).
a
For µ =0, 1, 2, ... , see equations 1.1.1, 1.1.2, 1.1.4, 1.1.12, and 1.1.23. For –1< µ < 0, see
equation 1.1.42.
Set µ = n – λ, where n =1, 2, ... and 0 ≤ λ < 1, and f(a)= f (a)= ··· = f x (n–1) (a)=0.

x
On differentiating the equation n times, we arrive at an equation of the form 1.1.46:
x
y(t) dτ Γ(µ – n +1) (n)

= f x (x),
(x – t) λ Γ(µ +1)
a
where Γ(µ) is the gamma function.
β
Example. Set f(x)= Ax , where β ≥ 0, and let µ > –1 and µ – β ≠ 0, 1, 2, ... In this case, the solution has
A Γ(β +1)
the form y(x)= x β–µ–1 .
Γ(µ +1) Γ(β – µ)
•
Reference: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971).
x

µ
µ
44. (x – t )y(t) dt = f(x).
a
µ
This is a special case of equation 1.9.2 with g(x)= x .
1
Solution: y(x)= x 1–µ x
f (x) .
µ x
x

µ µ
45. Ax + Bt y(t) dt = f(x).
a
µ
For B = –A, see equation 1.1.44. This is a special case of equation 1.9.4 with g(x)= x .
1 d – Aµ x – Bµ

Solution: y(x)= x A+B t A+B f (t) dt .
t
A + B dx a
x
y(t) dt
46. = f(x), 0 < λ <1.
a (x – t) λ
The generalized Abel equation.
Solution:
sin(πλ) d x f(t) dt sin(πλ) f(a) x f (t) dt

t
y(x)= = + .
π dx a (x – t) 1–λ π (x – a) 1–λ a (x – t) 1–λ
•
Reference: E. T. Whittacker and G. N. Watson (1958).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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