Carleman's equation - Alchetron, The Free Social Encyclopedia

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

∫ a b ln ⁡ | x − t | y ( t ) d t = f ( x )

The solution for b − a ≠ 4 is

y ( x ) = 1 π 2 ( x − a ) ( b − x ) [ ∫ a b ( t − a ) ( b − t ) f t ′ ( t ) d t t − x + 1 ln ⁡ [ 1 4 ( b − a ) ] ∫ a b f ( t ) d t ( t − a ) ( b − t ) ]

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

∫ a b f ( t ) d t ( t − a ) ( b − t ) = 0

In this case the solution has the form

y ( x ) = 1 π 2 ( x − a ) ( b − x ) [ ∫ a b ( t − a ) ( b − t ) f t ′ ( t ) d t t − x + C ]

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 ) = 1 π ln ⁡ [ 1 4 ( b − a ) ] 1 ( x − a ) ( b − x )

References

Carleman's equation Wikipedia

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