# 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 be a given function of one variable, let

be differential expressions with sufficiently smooth coefficients and on , and let be a known function that is sufficiently smooth on the square . An equation of the form

(1) |

is called a linear integro-differential equation; is a parameter. If in (1) the function for , then (1) is called an integro-differential equation with variable integration limits; it can be written in the form

(2) |

For (1) and (2) one may pose the Cauchy problem (find the solution satisfying , , where are given numbers, is the order of , and ), 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

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. V.A. Trenogin (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Integro-differential_equation&oldid=15621