Page 112 - Handbook Of Integral Equations

x

2
2
74. [AI (λx)+ BI (λt)]y(t) dt = f(x).
ν ν
a
Solution with B ≠ –A:
1 d – 2A x – 2B

y(x)= I ν (λx) A+B I ν (λt) A+B f (t) dt .
t
A + B dx a
x

k s
75. [AI (λx)+ BI (βt)]y(t) dt = f(x).
ν µ
a
s
k
This is a special case of equation 1.9.6 with g(x)= AI (λx) and h(t)= BI (βt).
ν µ
x

76. [K 0 (λx) – K 0 (λt)]y(t) dt = f(x).
a
d f (x)

x
Solution: y(x)= – .
dx λK 1 (λx)
x

77. [K ν (λx) – K ν (λt)]y(t) dt = f(x).
a
This is a special case of equation 1.9.2 with g(x)= K ν (λx).
x
78. [AK ν (λx)+ BK ν (λt)]y(t) dt = f(x).
a
Solution with B ≠ –A:
A x B
1 d – –

y(x)= K ν (λx) A+B K ν (λt) A+B f (t) dt .
t
A + B dx a
x

s
k
79. [At K ν (λx)+ Bx K µ (λt)]y(t) dt = f(x).
a
s
k
This is a special case of equation 1.9.15 with g 1 (x)= AK ν (λx), h 1 (t)= t , g 2 (x)= Bx , and
h 2 (t)= K µ (λt).
x
80. [AI ν (λx)K µ (βt)+ BI ν (λt)K µ (βx)]y(t) dt = f(x).
a
This is a special case of equation 1.9.15 with g 1 (x)= AI ν (λx), h 1 (t)= K µ (βt), g 2 (x)=
BK µ (βx), and h 2 (t)= I ν (λt).
1.8-3. Kernels Containing Associated Legendre Functions
x
x
2 –µ/2
2
81. (x – t ) P µ y(t) dt = f(x), 0 < a < ∞.
ν
a t
µ
Here P (x) is the associated Legendre function (see Supplement 10).
ν
Solution:
d n 1–µ x n+µ–2 –n 2–n–µ
t
2
2
y(x)= x n+µ–1 x (x – t ) 2 t P ν f(t) dt ,
dx n x
a
1
where µ <1, ν ≥ – ,and n =1, 2, ...
2
•
Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 90