Degenerate integral equation

A linear Fredholm integral equation with a degenerate kernel. Its general form is

$$ \tag{1 } \lambda x ( P) - \int\limits _ { D } \sum _ {i = 1 } ^ { N } \phi _ {i} ( P) \psi _ {i} ( Q) x ( Q) dQ = f ( P). $$

The integration is effected over a domain $ D $ (usually an $ n $-dimensional domain) of a Euclidean space, $ P $ and $ Q $ are points of $ D $, $ \lambda $ is a real or complex parameter, and the functions appearing in (1) are square-integrable on $ D $. One looks for the solution of a degenerate integral equation in the form

$$ x ( P) = \ { \frac{1} \lambda } f ( P) + \sum _ {i = 1 } ^ { N } c _ {i} \phi _ {i} ( P) , $$

where the coefficients $ c _ {i} $ are found from the system of linear algebraic equations

$$ \tag{2 } \lambda c _ {j} - \sum _ {i = 1 } ^ { N } c _ {i} \int\limits _ { D } \psi _ {j} ( Q) \phi _ {i} ( Q) dQ = { \frac{1} \lambda } \int\limits _ { D } f ( P) \psi _ {j} ( P) dP. $$

If, for a given $ \lambda $, system (2) has a unique solution, equation (1) is uniquely solvable as well. The values $ \lambda \neq 0 $ (there are not more than $ N $ such values) for which the determinant of the system (2) is zero are eigen values. The conditions for solvability of a degenerate integral equation (1) are given by the Fredholm alternative. If $ \lambda = 0 $, a degenerate integral equation (1) is a Fredholm equation of the first kind; in order that it is solvable it is necessary and sufficient that the function $ f $ is representable as a linear combination of the functions $ \phi _ {i} $. Equation (1) will then have a solution representable as

$$ x ( P) = \ \sum _ {j = 1 } ^ { N } d _ {j} \psi _ {j} ( P) + \psi ( P) $$

where the coefficients $ d _ {j} $ are uniquely defined, while $ \psi $ is any function satisfying the conditions

$$ \int\limits _ { D } \psi ( P) \psi _ {j} ( P) dP = 0,\ \ j = 1 \dots N. $$

The importance of degenerate integral equations in the general theory of Fredholm equations is based on the fact that the solution of any Fredholm equation of the second kind can be approximated by solutions of degenerate integral equations in the mean-square (and certain other) metrics to any degree of accuracy. Their degenerate kernels approximate the kernel of the initial equation in one sense or another.

An abstract analogue and a generalization of a degenerate integral equation is a linear operator equation of the type

$$ \lambda x - Ax = f, $$

where $ x $ and $ f $ belong to a Banach space $ E $, and where the operator $ A $ has finite-dimensional range. The properties of such equations are similar to those of the degenerate integral equation (1).

References

[1]	S.G. Mikhlin, "Linear integral equations" , Hindushtan Publ. Comp. , Delhi (1960) (Translated from Russian)

Comments

For approximation by equations with degenerate kernels see Degenerate kernels, method of.

References

[a1]	B.L. Moiseiwitsch, "Integral equations" , Longman (1977)
[a2]	H. Hochstadt, "Integral equations" , Wiley (1973)
[a3]	L.V. Kantorovich, V.I. Krylov, "Approximate methods of higher analysis" , Noordhoff (1958) (Translated from Russian)
[a4]	P.P. Zabreiko (ed.) A.I. Koshelev (ed.) M.A. Krasnoselskii (ed.) S.G. Mikhlin (ed.) L.S. Rakovshchik (ed.) V.Ya. Stet'senko (ed.) T.O. Shaposhnikova (ed.) R.S. Anderssen (ed.) , Integral equations - a reference text , Noordhoff (1958) (Translated from Russian)
[a5]	I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1981)