Page 248 - Handbook Of Integral Equations

a 1
sinh A

3. ln 1 2 y(t) dt = f(x), –a ≤ x ≤ a.
–a 2 sinh 2 |x – t|
Solution with 0 < a < A:
1 d a
y(x)= w(t, a)f(t) dt w(x, a)
2M (a) da –a

a ξ
1 d 1 d

– w(x, ξ) w(t, ξ)f(t) dt dξ
2 |x| dξ M (ξ) dξ –ξ

1 d a w(x, ξ) ξ
– w(t, ξ) df(t) dξ,
2 dx |x| M (ξ) –ξ

where the prime stands for the derivative with respect to the argument and
–1
1
1
sinh 2 A cosh 2 x M(ξ)
M(ξ)= ln 1 , w(x, ξ)= √ .
sinh ξ π 2 cosh ξ – 2 cosh x
2
•
Reference: I. C. Gohberg and M. G. Krein (1967).
b

4. ln tanh(λx) – tanh(λt) y(t) dt = f(x).

a
This is a special case of equation 1.8.9 with g(x) = tanh(λx).
a
1
5. ln coth |x – t| y(t) dt = f(x), –a ≤ x ≤ a.
4
–a
Solution: a
1 d
y(x)= w(t, a)f(t) dt w(x, a)
2M (a) da –a

a ξ
1 d 1 d

– w(x, ξ) w(t, ξ)f(t) dt dξ
2 |x| dξ M (ξ) dξ –ξ

1 d a w(x, ξ) ξ
– w(t, ξ) df(t) dξ,
2 dx |x| M (ξ) –ξ

where the prime stands for the derivative with respect to the argument and
P –1/2 (cosh ξ) 1
M(ξ)= , w(x, ξ)= √ ,
Q –1/2 (cosh ξ) πQ –1/2 (cosh ξ) 2 cosh ξ – 2 cosh x
and P –1/2 (cosh ξ) and Q –1/2 (cosh ξ) are the Legendre functions of the first and second kind,
respectively.
•
Reference: I. C. Gohberg and M. G. Krein (1967).

3.6-2. Kernels Containing Logarithmic and Trigonometric Functions

b

6. ln cos(λx) – cos(λt) y(t) dt = f(x).

a
This is a special case of equation 1.8.9 with g(x) = cos(λx).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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