Page 113 - Handbook Of Integral Equations

x
t
2
2 –µ/2
82. (x – t ) P µ y(t) dt = f(x), 0 < a < ∞.
ν
a x
µ
Here P (x) is the associated Legendre function (see Supplement 10).
ν
Solution: x
x
d n 2 2 n+µ–2 2–n–µ
y(x)= (x – t ) 2 P ν f(t) dt,
dx n a t
1
where µ <1, ν ≥ – ,and n =1, 2, ...
2
•
Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).
b x
2 –µ/2
2
83. (t – x ) P µ y(t) dt = f(x), 0 < b < ∞.
ν
x t
µ
Here P (x) is the associated Legendre function (see Supplement 10).
ν
Solution:
d n 1–µ b 2 2 n+µ–2 –n 2–n–µ
t
n n+µ–1
y(x)=(–1) x x (t – x ) 2 t P ν f(t) dt ,
dx n x x
1
where µ <1, ν ≥ – ,and n =1, 2, ...
2
•
Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).
b
t
2 –µ/2
2
84. (t – x ) P µ y(t) dt = f(x), 0 < b < ∞.
ν
x x
µ
Here P (x) is the associated Legendre function (see Supplement 10).
ν
Solution:
x
d n b 2 2 n+µ–2 2–n–µ
n
y(x)=(–1) (t – x ) 2 P ν f(t) dt,
dx n x t
1
where µ <1, ν ≥ – ,and n =1, 2, ...
2
•
Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).
1.8-4. Kernels Containing Hypergeometric Functions
x

85. (x – t) b–1 Φ a, b; λ(x – t) y(t) dt = f(x).
s
Here Φ(a, b; z) is the degenerate hypergeometric function (see Supplement 10).
Solution:
d n x (x – t) n–b–1
y(x)= Φ –a, n – b; λ(x – t) f(t) dt,
dx n s Γ(b)Γ(n – b)
where 0 < b < n and n =1, 2, ...
If the right-hand side of the equation is differentiable sufficiently many times and the
conditions f(s)= f (s)= ··· = f (n–1) (s) = 0 are satisfied, then the solution of the integral

x x
equation can be written in the form
x (x – t) n–b–1

y(x)= Φ –a, n – b; λ(x – t) f t (n) (t) dt.
s Γ(b)Γ(n – b)
•
Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 91