Carleman's equation

In mathematics, Carleman's equation is a Fredholm integral equation of the first kind with a logarithmic kernel. Its solution was first given by Torsten Carleman in 1922. The equation is

\int_a^b \ln|x-t| \, y(t) \, dt = f(x)

The solution for b − a ≠ 4 is

y(x) = \frac{1}{\pi^2 \sqrt{(x-a)(b-x)}} \left[ \int_a^b \frac{\sqrt{(t-a)(b-t)} f'_t(t) \, dt}{t-x} +\frac{1}{\ln \left[ \frac{1}{4} (b-a) \right]} \int_a^b \frac{f(t) \, dt}{\sqrt{(t-a)(b-t)}} \right]

If b − a = 4 then the equation is solvable only if the following condition is satisfied

\int_a^b \frac{f(t) \, dt}{\sqrt{(t-a)(b-t)}} = 0

In this case the solution has the form

y(x) = \frac{1}{\pi^2 \sqrt{(x-a)(b-x)}} \left[ \int_a^b \frac{\sqrt{(t-a)(b-t)} f'_t(t) \, dt}{t-x} +C \right]

where C is an arbitrary constant.

For the special case f(t) = 1 (in which case it is necessary to have b − a ≠ 4), useful in some applications, we get

y(x) = \frac{1}{\pi \ln \left[ \frac{1}{4} (b-a) \right]} \frac{1}{\sqrt{(x-a)(b-x)}}

References

CARLEMAN, T. (1922) Uber die Abelsche Integralgleichung mit konstanten Integrationsgrenzen. Math. Z., 15, 111–120
Gakhov, F. D., Boundary Value Problems [in Russian], Nauka, Moscow, 1977
A.D. Polyanin and A.V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4

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