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An [[Integral equation|integral equation]] that, in the one-dimensional case, has the form
 
An [[Integral equation|integral equation]] that, in the one-dimensional case, has 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/m/m064/m064220/m0642201.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\phi ( x) - \lambda
 +
\int\limits _ { a } ^ { b }  K ( x, s) \phi ( s)  ds -
 +
\lambda
 +
\sum _ {j = 1 } ^ { m }
 +
K _ {1} ( x, s _ {j} )
 +
\phi ( s _ {j} ) =
 +
$$
  
<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/m/m064/m064220/m0642202.png" /></td> </tr></table>
+
$$
 +
= \
 +
f ( x),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064220/m0642203.png" /> is the unknown and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064220/m0642204.png" /> is a given continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064220/m0642205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064220/m0642206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064220/m0642207.png" />, are given points, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064220/m0642208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064220/m0642209.png" /> are given continuous functions on the rectangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064220/m06422010.png" />. If
+
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
  
<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/m/m064/m064220/m06422011.png" /></td> </tr></table>
+
$$
 +
K _ {1} ( x, s _ {j} )  = a _ {j} K ( x, s _ {j} ),
 +
$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064220/m06422012.png" /> are positive constants, then (1) can be written as
+
where the $  a _ {j} $
 +
are positive constants, then (1) can be written 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/m/m064/m064220/m06422013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\phi ( x) - \lambda
 +
{}  ^ {*} \int\limits _ { a } ^ { b }  K ( x, s) \phi ( s)  ds  = \
 +
f ( x),\ \
 +
x \in [ a, b],
 +
$$
  
where the new integration symbol, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064220/m06422014.png" /> an arbitrary finite integrable function, is defined by (see [[#References|[1]]]):
+
where the new integration symbol, with $  \psi $
 +
an arbitrary finite integrable function, is defined by (see [[#References|[1]]]):
  
<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/m/m064/m064220/m06422015.png" /></td> </tr></table>
+
$$
 +
{}  ^ {*} \int\limits _ { a } ^ { b }  \psi ( s)  ds  = \
 +
\int\limits _ { a } ^ { b }  \psi ( s)  ds +
 +
\sum _ {j = 1 } ^ { m }  a _ {j} \psi ( s _ {j} ).
 +
$$
  
 
The theory of Fredholm equations (cf. [[Fredholm equation|Fredholm equation]]) and, in the case of a symmetric kernel, the theory of integral equations with symmetric kernel (cf. [[Integral equation with symmetric kernel|Integral equation with symmetric kernel]]), is valid for equation (2).
 
The theory of Fredholm equations (cf. [[Fredholm equation|Fredholm equation]]) and, in the case of a symmetric kernel, the theory of integral equations with symmetric kernel (cf. [[Integral equation with symmetric kernel|Integral equation with symmetric kernel]]), is valid for equation (2).
Line 21: Line 64:
 
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
 
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
  
<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/m/m064/m064220/m06422016.png" /></td> </tr></table>
+
$$
 +
\phi ( x) - \lambda
 +
{\int\limits \int\limits } _ { D } K _ {1} ( x, y) \phi ( y)  d \sigma _ {y} +
 +
\lambda \int\limits _  \Gamma  K _ {2} ( x, y) \phi ( y)  ds _ {y} +
 +
$$
  
<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/m/m064/m064220/m06422017.png" /></td> </tr></table>
+
$$
 +
+
 +
\lambda \sum _ {j = 1 } ^ { m }  K _ {3} ( x, y _ {j} ) \phi ( y _ {j} )  = f ( x),\  x \in D,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064220/m06422018.png" /> is some domain in the plane, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064220/m06422019.png" /> is its boundary, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064220/m06422020.png" /> are fixed points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064220/m06422021.png" />. This equation may also be written as
+
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
  
<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/m/m064/m064220/m06422022.png" /></td> </tr></table>
+
$$
 +
\phi ( x) - \lambda {\int\limits \int\limits } _ {D \cup \Gamma } K ( x, y)
 +
\phi ( y)  d \omega _ {y}  = f ( x),
 +
$$
  
if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064220/m06422023.png" /> and the volume element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064220/m06422024.png" /> are correspondingly defined. In this case, moreover, the theory of Fredholm integral equations remains valid.
+
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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Kneser,  "Belastete Integralgleichungen"  ''Rend. Circolo Mat. Palermo'' , '''37'''  (1914)  pp. 169–197</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Lichtenstein,  "Bemerkungen über belastete Integralgleichungen"  ''Studia Math.'' , '''3'''  (1931)  pp. 212–225</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.M. Gunter,  "Sur le problème des  "Belastete Integralgleichungen" "  ''Studia Math.'' , '''4'''  (1933)  pp. 8–14</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Kneser,  "Belastete Integralgleichungen"  ''Rend. Circolo Mat. Palermo'' , '''37'''  (1914)  pp. 169–197</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Lichtenstein,  "Bemerkungen über belastete Integralgleichungen"  ''Studia Math.'' , '''3'''  (1931)  pp. 212–225</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.M. Gunter,  "Sur le problème des  "Belastete Integralgleichungen" "  ''Studia Math.'' , '''4'''  (1933)  pp. 8–14</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR></table>

Latest revision as of 08:01, 6 June 2020


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