Page 63 - Handbook Of Integral Equations

x

µ µ
14. ln (λx) – ln (λt) y(t) dt = f(x).
a
µ
This is a special case of equation 1.9.2 with g(x)=ln (λx).
1 d 1–µ

Solution: y(x)= x ln (λx)f (x) .
x
µ dx
x

β γ
15. A ln (λx)+ B ln (µt)+ C y(t) dt = f(x).
a
β
γ
This is a special case of equation 1.9.6 with g(x)= A ln (λx) and h(t)= B ln (µt)+ C.
x

λ
16. [ln(x/t)] y(t) dt = f(x), 0 < λ <1.
a
Solution:
2 x
k d f(t) dt sin(πλ)

y(x)= x , k = .
x dx t[ln(x/t)] λ πλ
a
x y(t) dt
17. = f(x), 0 < λ <1.
a [ln(x/t)] λ
This is a special case of equation 1.9.42 with g(x)=ln x and h(x) ≡ 1.
Solution: x
sin(πλ) d f(t) dt
y(x)= .
π dx t[ln(x/t)] 1–λ
a
1.4-2. Kernels Containing Power-Law and Logarithmic Functions
x

18. (x – t) ln(x – t)+ A y(t) dt = f(x).
a
Solution:
z (A–C)z
d 2 x d ∞ x e
y(x)= – ν A (x – t)f(t) dt, ν A (x)= dz,
dx 2 dx Γ(z +1)
a 0
where C = 0.5772 ... is the Euler constant and Γ(z) is the gamma function.
•
Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).
x ln(x – t)+ A
19. λ y(t) dt = f(x), 0 < λ <1.
a (x – t)
Solution:

sin(πλ) d x F(t) dt x
y(x)= – , F(x)= ν h (x – t)f(t) dt,
π dx a (x – t) 1–λ a
z hz
d ∞ x e
ν h (x)= dz, h = A + ψ(1 – λ),
dx 0 Γ(z +1)

where Γ(z) is the gamma function and ψ(z)= Γ(z) is the logarithmic derivative of the
z
gamma function.
•
Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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