Integro-differential equation

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)