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An equation containing the unknown function under the sign of both differential and integral operations. Integral equations and differential equations are also integro-differential equations.
 
An equation containing the unknown function under the sign of both differential and integral operations. Integral equations and differential equations are also integro-differential equations.
  
 
==Linear integro-differential equations.==
 
==Linear integro-differential equations.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051780/i0517801.png" /> be a given function of one variable, let
+
Let $  f $
 +
be a given function of one variable, let
  
<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/i/i051/i051780/i0517802.png" /></td> </tr></table>
+
$$
 +
L _ {x} [ U ]  \equiv \
 +
\sum_{i=0}^ { l }  p _ {i} ( x) U  ^ {(i)} ( x) ,
 +
$$
  
<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/i/i051/i051780/i0517803.png" /></td> </tr></table>
+
$$
 +
M _ {y} [ U ]  \equiv  \sum_{j=0}^ { m }  q _ {i} ( y) U  ^ {(j)} ( y)
 +
$$
  
be differential expressions with sufficiently smooth coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051780/i0517804.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051780/i0517805.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051780/i0517806.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051780/i0517807.png" /> be a known function that is sufficiently smooth on the square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051780/i0517808.png" />. An equation of the form
+
be differential expressions with sufficiently smooth coefficients $  p _ {i} $
 +
and $  q _ {i} $
 +
on $  [ a , b ] $,  
 +
and let $  K $
 +
be a known function that is sufficiently smooth on the square $  [ a , b ] \times [ a , b] $.  
 +
An equation of 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/i/i051/i051780/i0517809.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
L _ {x} [ U ]  = \lambda
 +
\int\limits _ { a } ^ { b }  K ( x , y ) M _ {y} [ U ]  d y + f ( x)
 +
$$
  
is called a linear integro-differential equation; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051780/i05178010.png" /> is a parameter. If in (1) the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051780/i05178011.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051780/i05178012.png" />, then (1) is called an integro-differential equation with variable integration limits; it can be written in the form
+
is called a linear integro-differential equation; $  \lambda $
 +
is a parameter. If in (1) the function $  K ( x , y ) \equiv 0 $
 +
for $  y > x $,  
 +
then (1) is called an integro-differential equation with variable integration limits; it can be written 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/i/i051/i051780/i05178013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
L _ {x} [ U ]  = \lambda
 +
\int\limits _ { 0 } ^ { x }  K ( x , y ) M _ {y} [ U ]  d y + f ( x ) .
 +
$$
  
For (1) and (2) one may pose the [[Cauchy problem|Cauchy problem]] (find the solution satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051780/i05178014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051780/i05178015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051780/i05178016.png" /> are given numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051780/i05178017.png" /> is the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051780/i05178018.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051780/i05178019.png" />), as well as various boundary value problems (e.g., the problem of periodic solutions). In a number of cases (cf. [[#References|[3]]], [[#References|[4]]]), problems for (1) and (2) can be simplified, or even reduced, to, respectively, Fredholm integral equations of the second kind or Volterra equations (cf. also [[Fredholm equation|Fredholm equation]]; [[Volterra equation|Volterra equation]]). At the same time, a number of specific phenomena arise for integro-differential equations that are not characteristic for differential or integral equations.
+
For (1) and (2) one may pose the [[Cauchy problem|Cauchy problem]] (find the solution satisfying $  U  ^ {(i)} ( \alpha ) = c _ {i} $,  
 +
$  i = 0 \dots l - 1 $,  
 +
where $  c _ {i} $
 +
are given numbers, $  l $
 +
is the order of $  L _ {x} [ U ] $,  
 +
and $  \alpha \in [ a , b ] $),  
 +
as well as various boundary value problems (e.g., the problem of periodic solutions). In a number of cases (cf. [[#References|[3]]], [[#References|[4]]]), problems for (1) and (2) can be simplified, or even reduced, to, respectively, Fredholm integral equations of the second kind or Volterra equations (cf. also [[Fredholm equation|Fredholm equation]]; [[Volterra equation|Volterra equation]]). At the same time, a number of specific phenomena arise for integro-differential equations that are not characteristic for differential or integral equations.
  
 
The simplest non-linear integro-differential equation has the form
 
The simplest non-linear integro-differential equation 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/i/i051/i051780/i05178020.png" /></td> </tr></table>
+
$$
 +
U ( x)  = \lambda
 +
\int\limits _ { a } ^ { b }  F ( x , y , U ( y) \dots U  ^ {(m)}
 +
( y) )  d y + f ( x) .
 +
$$
  
 
The [[Contracting-mapping principle|contracting-mapping principle]], the [[Schauder method|Schauder method]], as well as other methods of non-linear functional analysis, are applied in investigations of this equation.
 
The [[Contracting-mapping principle|contracting-mapping principle]], the [[Schauder method|Schauder method]], as well as other methods of non-linear functional analysis, are applied in investigations of this equation.
Line 28: Line 70:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V. Volterra,  "Leçons sur les équations intégrales et les équations intégro-différentielles" , Gauthier-Villars  (1913)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V. Volterra,  "Una teoria matematica sulla lotta per l'esistenza"  ''Scienta'' , '''41'''  (1927)  pp. 85–102</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Ya.V. Bykov,  "On some problems in the theory of integro-differential equations" , Frunze  (1957)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.M. Vainberg,  "Integro-differential equations"  ''Itogi Nauk. Mat. Anal. Teor. Veroyatnost. Regulirovanie 1962''  (1964)  pp. 5–37  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.N. Filatov,  "Asymptotic methods in the theory of differential and integro-differential equations" , Tashkent  (1974)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V. Volterra,  "Leçons sur les équations intégrales et les équations intégro-différentielles" , Gauthier-Villars  (1913)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V. Volterra,  "Una teoria matematica sulla lotta per l'esistenza"  ''Scienta'' , '''41'''  (1927)  pp. 85–102</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Ya.V. Bykov,  "On some problems in the theory of integro-differential equations" , Frunze  (1957)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.M. Vainberg,  "Integro-differential equations"  ''Itogi Nauk. Mat. Anal. Teor. Veroyatnost. Regulirovanie 1962''  (1964)  pp. 5–37  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.N. Filatov,  "Asymptotic methods in the theory of differential and integro-differential equations" , Tashkent  (1974)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 19:38, 12 January 2024


An equation containing the unknown function under the sign of both differential and integral operations. Integral equations and differential equations are also integro-differential equations.

Linear integro-differential equations.

Let $ f $ be a given function of one variable, let

$$ L _ {x} [ U ] \equiv \ \sum_{i=0}^ { l } p _ {i} ( x) U ^ {(i)} ( x) , $$

$$ M _ {y} [ U ] \equiv \sum_{j=0}^ { m } q _ {i} ( y) U ^ {(j)} ( y) $$

be differential expressions with sufficiently smooth coefficients $ p _ {i} $ and $ q _ {i} $ on $ [ a , b ] $, and let $ K $ be a known function that is sufficiently smooth on the square $ [ a , b ] \times [ a , b] $. An equation of the form

$$ \tag{1 } L _ {x} [ U ] = \lambda \int\limits _ { a } ^ { b } K ( x , y ) M _ {y} [ U ] d y + f ( x) $$

is called a linear integro-differential equation; $ \lambda $ is a parameter. If in (1) the function $ K ( x , y ) \equiv 0 $ for $ y > x $, then (1) is called an integro-differential equation with variable integration limits; it can be written in the form

$$ \tag{2 } L _ {x} [ U ] = \lambda \int\limits _ { 0 } ^ { x } K ( x , y ) M _ {y} [ U ] d y + f ( x ) . $$

For (1) and (2) one may pose the Cauchy problem (find the solution satisfying $ U ^ {(i)} ( \alpha ) = c _ {i} $, $ i = 0 \dots l - 1 $, where $ c _ {i} $ are given numbers, $ l $ is the order of $ L _ {x} [ U ] $, and $ \alpha \in [ a , b ] $), as well as various boundary value problems (e.g., the problem of periodic solutions). In a number of cases (cf. [3], [4]), problems for (1) and (2) can be simplified, or even reduced, to, respectively, Fredholm integral equations of the second kind or Volterra equations (cf. also Fredholm equation; Volterra equation). At the same time, a number of specific phenomena arise for integro-differential equations that are not characteristic for differential or integral equations.

The simplest non-linear integro-differential equation has the form

$$ U ( x) = \lambda \int\limits _ { a } ^ { b } F ( x , y , U ( y) \dots U ^ {(m)} ( y) ) d y + f ( x) . $$

The contracting-mapping principle, the Schauder method, as well as other methods of non-linear functional analysis, are applied in investigations of this equation.

Questions of stability of solutions, eigen-function expansions, asymptotic expansions in a small parameter, etc., can be studied for integro-differential equations. Partial integro-differential and integro-differential equations with multiple integrals are often encountered in practice. The Boltzmann and Kolmogorov–Feller equations are examples of these.

References

[1] V. Volterra, "Leçons sur les équations intégrales et les équations intégro-différentielles" , Gauthier-Villars (1913)
[2] V. Volterra, "Una teoria matematica sulla lotta per l'esistenza" Scienta , 41 (1927) pp. 85–102
[3] Ya.V. Bykov, "On some problems in the theory of integro-differential equations" , Frunze (1957) (In Russian)
[4] M.M. Vainberg, "Integro-differential equations" Itogi Nauk. Mat. Anal. Teor. Veroyatnost. Regulirovanie 1962 (1964) pp. 5–37 (In Russian)
[5] A.N. Filatov, "Asymptotic methods in the theory of differential and integro-differential equations" , Tashkent (1974) (In Russian)

Comments

Ordinary integro-differential equations are of interest e.g. in population dynamics ([a2]). Also, partial integro-differential equations, i.e., equations for functions of several variables which appear as arguments both of integral and of partial differential operators, are of interest e.g. in continuum mechanics ([a1], [a3]).

References

[a1] F. Bloom, "Ill-posed problems for integrodifferential equations in mechanics and electromagnetic theory" , SIAM (1981)
[a2] J.M. Cushing, "Integrodifferential equations and delay models in population dynamics" , Springer (1977)
[a3] H. Grabmüller, "Singular perturbation techniques applied to integro-differential equations" , Pitman (1978)
How to Cite This Entry:
Integro-differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integro-differential_equation&oldid=15621
This article was adapted from an original article by V.A. Trenogin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article