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A linear Fredholm integral equation with a [[Degenerate kernel|degenerate kernel]]. Its general form is
 
A linear Fredholm integral equation with a [[Degenerate kernel|degenerate kernel]]. Its general form is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d0308101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\lambda x ( P) -
 +
\int\limits _ { D }
 +
\sum _ {i = 1 } ^ { N }
 +
\phi _ {i} ( P) \psi _ {i} ( Q) x ( Q)  dQ  = f ( P).
 +
$$
  
The integration is effected over a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d0308102.png" /> (usually an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d0308103.png" />-dimensional domain) of a Euclidean space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d0308104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d0308105.png" /> are points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d0308106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d0308107.png" /> is a real or complex parameter, and the functions appearing in (1) are square-integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d0308108.png" />. One looks for the solution of a degenerate integral equation in the form
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d0308109.png" /></td> </tr></table>
+
$$
 +
x ( P)  = \
 +
{
 +
\frac{1} \lambda
 +
}
 +
f ( P) + \sum _ {i = 1 } ^ { N }
 +
c _ {i} \phi _ {i} ( P) ,
 +
$$
  
where the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d03081010.png" /> are found from the system of linear algebraic equations
+
where the coefficients $  c _ {i} $
 +
are found from the system of linear algebraic equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d03081011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\lambda c _ {j} -
 +
\sum _ {i = 1 } ^ { N }
 +
c _ {i} \int\limits _ { D } \psi _ {j} ( Q) \phi _ {i} ( Q)  dQ
 +
= {
 +
\frac{1} \lambda
 +
}
 +
\int\limits _ { D } f ( P) \psi _ {j} ( P) dP.
 +
$$
  
If, for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d03081012.png" />, system (2) has a unique solution, equation (1) is uniquely solvable as well. The values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d03081013.png" /> (there are not more than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d03081014.png" /> 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|Fredholm alternative]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d03081015.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d03081016.png" /> is representable as a linear combination of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d03081017.png" />. Equation (1) will then have a solution representable as
+
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|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d03081018.png" /></td> </tr></table>
+
$$
 +
x ( P)  = \
 +
\sum _ {j = 1 } ^ { N }
 +
d _ {j} \psi _ {j} ( P) + \psi ( P)
 +
$$
  
where the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d03081019.png" /> are uniquely defined, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d03081020.png" /> is any function satisfying the conditions
+
where the coefficients d _ {j} $
 +
are uniquely defined, while $  \psi $
 +
is any function satisfying the conditions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d03081021.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { D } \psi ( P) \psi _ {j} ( P)  dP  = 0,\ \
 +
j = 1 \dots N.
 +
$$
  
 
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.
 
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.
Line 23: Line 75:
 
An abstract analogue and a generalization of a degenerate integral equation is a linear operator equation of the type
 
An abstract analogue and a generalization of a degenerate integral equation is a linear operator equation of the type
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d03081022.png" /></td> </tr></table>
+
$$
 +
\lambda x - Ax  = f,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d03081023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d03081024.png" /> belong to a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d03081025.png" />, and where the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030810/d03081026.png" /> has finite-dimensional range. The properties of such equations are similar to those of the degenerate integral equation (1).
+
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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.G. Mikhlin,  "Linear integral equations" , Hindushtan Publ. Comp. , Delhi  (1960)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.G. Mikhlin,  "Linear integral equations" , Hindushtan Publ. Comp. , Delhi  (1960)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 06:16, 24 February 2022


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)
How to Cite This Entry:
Degenerate integral equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Degenerate_integral_equation&oldid=12411
This article was adapted from an original article by A.B. Bakushinskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article