# Integro-differential equation

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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 (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=47385
This article was adapted from an original article by V.A. Trenogin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article