Difference between revisions of "Mixed integral equation"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
||
Line 1: | Line 1: | ||
+ | <!-- | ||
+ | m0642201.png | ||
+ | $#A+1 = 24 n = 0 | ||
+ | $#C+1 = 24 : ~/encyclopedia/old_files/data/M064/M.0604220 Mixed integral equation | ||
+ | Automatically converted into TeX, above some diagnostics. | ||
+ | Please remove this comment and the {{TEX|auto}} line below, | ||
+ | if TeX found to be correct. | ||
+ | --> | ||
+ | |||
+ | {{TEX|auto}} | ||
+ | {{TEX|done}} | ||
+ | |||
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 | ||
− | + | $$ \tag{1 } | |
+ | \phi ( x) - \lambda | ||
+ | \int\limits _ { a } ^ { b } K ( x, s) \phi ( s) ds - | ||
+ | \lambda | ||
+ | \sum _ {j = 1 } ^ { m } | ||
+ | K _ {1} ( x, s _ {j} ) | ||
+ | \phi ( s _ {j} ) = | ||
+ | $$ | ||
− | + | $$ | |
+ | = \ | ||
+ | f ( x), | ||
+ | $$ | ||
− | where | + | 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 | + | 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 | + | where the new integration symbol, with $ \psi $ |
+ | an arbitrary finite integrable function, is defined by (see [[#References|[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|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 | ||
− | + | $$ | |
+ | \phi ( x) - \lambda | ||
+ | {\int\limits \int\limits } _ { D } K _ {1} ( x, y) \phi ( y) d \sigma _ {y} + | ||
+ | \lambda \int\limits _ \Gamma K _ {2} ( x, y) \phi ( y) ds _ {y} + | ||
+ | $$ | ||
− | + | $$ | |
+ | + | ||
+ | \lambda \sum _ {j = 1 } ^ { m } K _ {3} ( x, y _ {j} ) \phi ( y _ {j} ) = f ( x),\ x \in D, | ||
+ | $$ | ||
− | where | + | 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 | + | 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) |
Mixed integral equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mixed_integral_equation&oldid=18355