Integro-differential equation

Builds on Ordinary differential equation Riemann integral Volterra integral equation

Definition

Let $I = [t_0, t_0 + T]$ with $T > 0$ , let $n \ge 1$ , and let $f \colon I \times \R^n \to \R^n$ and $K \colon I \times I \times \R^n \to \R^n$ be continuous. A (Volterra type) integro-differential equation is an equation $x'(t) = f(t, x(t)) + \int_{t_0}^{t} K(t, s, x(s)) \, ds, \quad t \in I,$ for an unknown continuously differentiable $x \colon I \to \R^n$ . More generally, an integro-differential equation is any equation with the unknown both under a derivative and under an integral. When the integral is over a fixed interval, the equation is of Fredholm type.

Remarks

Example: $x'(t) = -\int_0^t x(s) \, ds$ , $x(0) = 1$ . Differentiating gives $x'' = -x$ with $x(0) = 1$ and $x'(0) = 0$ , so $x = \cos t$ . Check: $-\int_0^t \cos s \, ds = -\sin t = x'(t)$ .

Non-example: $x'(t) = -x(t)$ contains no integral of the unknown, so it is an ordinary differential equation. Also $x(t) = 1 + \int_0^t x(s) \, ds$ contains no derivative; it is a Volterra integral equation.

Integrating the Volterra form from $t_0$ to $t$ turns the problem with $x(t_0) = x_0$ into a Volterra integral equation for $x$ . For a convolution kernel $K(t, s, x) = k(t - s) x$ , the Laplace transform turns the linear equation into an algebraic one.

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Ordinary differential equation concept
Riemann integral concept
Volterra integral equation concept
Integro-differential equation definition

Tag	`4NFHBBNZBK`
URI	`m2://4NFHBBNZBK`
Kind	definition
Writing	W0 stub
MSC	45J05
Defines	Integro-differential equation
Uses	Ordinary differential equation, Riemann integral, Volterra integral equation
Changed	2 November 2026

Integro-differential equation

Definition

Remarks

Reading path

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