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)
How to Cite This Entry:
Mixed integral equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mixed_integral_equation&oldid=47862
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article