Page 61 - Handbook Of Integral Equations

1.4. Equations Whose Kernels Contain Logarithmic
Functions

1.4-1. Kernels Containing Logarithmic Functions

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

xx
x
x

2. ln(x – t)y(t) dt = f(x).
0
Solution:
x
z –Cz
z –Cz
∞ (x – t) e ∞ x e
y(x)= – f (t) dt dz – f (0) dz,

tt x
0 0 Γ(z +1) 0 Γ(z +1)
1 1

where C = lim 1+ + ··· + – ln k = 0.5772 ... is the Euler constant and Γ(z)is
k→∞ 2 k +1
the gamma function.
•
References: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971), A. G. Butkovskii (1979).
x
3. [ln(x – t)+ A]y(t) dt = f(x).
a
Solution:
x ∞ z (A–C)z

d d x e
y(x)= – ν A (x – t)f(t) dt, ν A (x)= dz,
dx dx Γ(z +1)
a 0
where C = 0.5772 ... is the Euler constant and Γ(z) is the gamma function.
For a = 0, the solution can be written in the form
x
z (A–C)z
z (A–C)z
∞ (x – t) e ∞ x e
y(x)= – f (t) dt dz – f (0) dz,

tt x
0 0 Γ(z +1) 0 Γ(z +1)
•
Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).
x

4. (A ln x + B ln t)y(t) dt = f(x).
a
This is a special case of equation 1.9.4 with g(x)=ln x.For B = –A, see equation 1.4.1.
Solution:
sign(ln x) d – A x – B

y(x)= ln x A+B ln t A+B f (t) dt .
t
A + B dx a
x

5. (A ln x + B ln t + C)y(t) dt = f(x).
a
This is a special case of equation 1.9.5 with g(x)= x.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 39