Page 340 - Handbook Of Integral Equations

where

∞
R(x, t)= R + (x – t)+ R – (t – x)+ R + (x – s)R – (t – s) ds,
0
and the functions R + (x) and R – (x) satisfy the conditions R + (x) = 0 and R – (x) = 0 for x <0
and are uniquely defined by their Fourier transforms as follows:

∞ 1 1 ∞ ln Ω(t)
1+ R ± (t)e ±iut dt =exp – ln Ω(u) ∓ dt .
0 2 2πi –∞ t – u
Alternatively, R + (x) and R – (x) can be obtained by constructing the solutions of the
equations

∞
R + (x)+ K(x – t)R + (t) dt = K(x), 0 ≤ x ≤ ∞,
0

∞
R – (x)+ K(t – x)R – (t) dt = K(–x), 0 ≤ x ≤ ∞.
0
2 . Solution with ν >0:
◦
ν ν
m–1 –x ∞ m–1 –t
◦
y(x)= f(x)+ C m x e + R (x, t) f(t)+ C m t e dt,
m=1 0 m=1
where the C m are arbitrary constants,

∞
(0)
(1)
(0)
(1)
◦
R (x, t)= R (x – t)+ R (t – x)+ R (x – s)R (t – s) ds,
+
–
–
+
0
(0) (1)
and the functions R + (x) and R (x) are uniquely defined by their Fourier transforms:
–
ν

∞ u – i ∞
(1)
(0)
1+ R (t)e ±iut dt = 1+ R (t)e ±iut dt ,
± ±
0 u + i 0

∞ 1 1 ∞ ln Ω (t)
◦
(0) ±iut
1+ R (t)e dt =exp – ln Ω (u) ∓ dt ,
◦
±
0 2 2πi –∞ t – u
ν
ν
◦
Ω (u)(u + i) = Ω(u)(u – i) .
3 .For ν < 0, the solution exists only if the conditions
◦

∞
f(x)ψ m (x) dx =0, m =1, 2, ... , –ν,
0
are satisfied. Here ψ 1 (x), ... , ψ ν (x) is the system of linearly independent solutions of the
transposed homogeneous equation

∞
ψ(x) – K(t – x)ψ(t) dt =0.
0
Then
∞
y(x)= f(x)+ R (x, t)f(t) dt,
∗
0
where

∞
(0)
(1)
(0)
(1)
∗
R (x, t)= R (x – t)+ R (t – x)+ R (x – s)R (t – s) ds,
+ – + –
0
(1) (0)
◦
and the functions R + (x) and R (x) are uniquely defined in item 2 by their Fourier trans-
–
forms.
•
References: V. I. Smirnov (1974), F. D. Gakhov and Yu. I. Cherskii (1978), I. M. Vinogradov (1979).
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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