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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.

==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|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

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

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.

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====
<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====
Ordinary integro-differential equations are of interest e.g. in population dynamics ([[#References|[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 ([[#References|[a1]]], [[#References|[a3]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Bloom,   "Ill-posed problems for integrodifferential equations in mechanics and electromagnetic theory" , SIAM  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.M. Cushing,   "Integrodifferential equations and delay models in population dynamics" , Springer  (1977)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Grabmüller,   "Singular perturbation techniques applied to integro-differential equations" , Pitman  (1978)</TD></TR></table>