Page 238 - Handbook Of Integral Equations

3.3-4. Kernels Containing Hyperbolic Cotangent

b

21. coth(λx) – coth(λt) y(t) dt = f(x).

a
This is a special case of equation 3.8.3 with g(x) = coth(λx).
b

k
22. coth x – coth t y(t) dt = f(x), 0 < k <1.
k
0
k
This is a special case of equation 3.8.3 with g(x) = coth x.
3.4. Equations Whose Kernels Contain Logarithmic
Functions

3.4-1. Kernels Containing Logarithmic Functions

b

1. ln(x/t) y(t) dt = f(x).

a
This is a special case of equation 3.8.3 with g(x)=ln x.
Solution:
1 d

y(x)= xf (x) .
x
2 dx
◦
The right-hand side f(x) of the integral equation must satisfy certain relations (see item 2 of
equation 3.8.3).
b
2. ln |x – t| y(t) dt = f(x).
a
Carleman’s equation.
◦
1 . Solution with b – a ≠ 4:
√ b
b

1 (t – a)(b – t) f (t) dt 1 f(t) dt
t
y(x)= √ + √ .
π 2 (x – a)(b – x) a t – x π ln 1 4 (b – a) a (t – a)(b – t)
◦
2 .If b – a = 4, then for the equation to be solvable, the condition
b
f(t)(t – a) –1/2 (b – t) –1/2 dt =0
a
must be satisfied. In this case, the solution has the form
√
b

1 (t – a)(b – t) f (t) dt

t
y(x)= √ + C ,
π 2 (x – a)(b – x) a t – x
where C is an arbitrary constant.
•
Reference: F. D. Gakhov (1977).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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