Mixed integral equation

An integral equation that, in the one-dimensional case, has the form

$$ \tag{1 } \phi ( x) - \lambda \int\limits _ { a } ^ { b } K ( x, s) \phi ( s) ds - \lambda \sum _ {j = 1 } ^ { m } K _ {1} ( x, s _ {j} ) \phi ( s _ {j} ) = $$

$$ = \ f ( x), $$

where $ \phi $ is the unknown and $ f $ is a given continuous function on $ [ a, b] $, $ s _ {j} \in [ a, b] $, $ j = 1 \dots m $, are given points, and $ K $, $ K _ {1} $ are given continuous functions on the rectangle $ [ a, b] \times [ a, b] $. If

$$ K _ {1} ( x, s _ {j} ) = a _ {j} K ( x, s _ {j} ), $$

where the $ a _ {j} $ are positive constants, then (1) can be written as

$$ \tag{2 } \phi ( x) - \lambda {} ^ {*} \int\limits _ { a } ^ { b } K ( x, s) \phi ( s) ds = \ f ( x),\ \ x \in [ a, b], $$

where the new integration symbol, with $ \psi $ an arbitrary finite integrable function, is defined by (see [1]):

$$ {} ^ {*} \int\limits _ { a } ^ { b } \psi ( s) ds = \ \int\limits _ { a } ^ { b } \psi ( s) ds + \sum _ {j = 1 } ^ { m } a _ {j} \psi ( s _ {j} ). $$

The theory of Fredholm equations (cf. Fredholm equation) and, in the case of a symmetric kernel, the theory of integral equations with symmetric kernel (cf. Integral equation with symmetric kernel), is valid for equation (2).

In the case of multi-dimensional mixed integral equations, the unknown function can be part of the integrands of integrals over manifolds of different dimensions. For example, in the two-dimensional case the integral equation may have the form

$$ \phi ( x) - \lambda {\int\limits \int\limits } _ { D } K _ {1} ( x, y) \phi ( y) d \sigma _ {y} + \lambda \int\limits _ \Gamma K _ {2} ( x, y) \phi ( y) ds _ {y} + $$

$$ + \lambda \sum _ {j = 1 } ^ { m } K _ {3} ( x, y _ {j} ) \phi ( y _ {j} ) = f ( x),\ x \in D, $$

where $ D $ is some domain in the plane, $ \Gamma $ is its boundary, and $ y _ {j} $ are fixed points in $ D \cup \Gamma $. This equation may also be written as

$$ \phi ( x) - \lambda {\int\limits \int\limits } _ {D \cup \Gamma } K ( x, y) \phi ( y) d \omega _ {y} = f ( x), $$

if the function $ K $ and the volume element $ d \omega _ {y} $ are correspondingly defined. In this case, moreover, the theory of Fredholm integral equations remains valid.

References

[1]	A. Kneser, "Belastete Integralgleichungen" Rend. Circolo Mat. Palermo , 37 (1914) pp. 169–197
[2]	L. Lichtenstein, "Bemerkungen über belastete Integralgleichungen" Studia Math. , 3 (1931) pp. 212–225
[3]	N.M. Gunter, "Sur le problème des "Belastete Integralgleichungen" " Studia Math. , 4 (1933) pp. 8–14
[4]	V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian)