Volterra integro-differential equation

Builds on Integro-differential equation Volterra integral equation Initial value problem

Definition

Let $t_0 \in \R$ , $T > 0$ , $I = [t_0, t_0 + T]$ and $n \ge 1$ . Let $f \colon I \times \R^n \to \R^n$ and $K \colon I \times I \times \R^n \to \R^n$ be continuous, and let $x_0 \in \R^n$ . A Volterra integro-differential equation with initial value $x_0$ is the problem of finding a continuously differentiable $x \colon I \to \R^n$ with $x'(t) = f\big(t, x(t)\big) + \int_{t_0}^{t} K\big(t, s, x(s)\big) \, ds \quad \text{for } t \in I, \qquad x(t_0) = x_0.$ It is linear when $f(t, x) = A(t) x + b(t)$ and $K(t, s, x) = B(t, s) x$ for continuous matrix functions $A$ , $B$ and a continuous $b$ . Equations with higher derivatives of $x$ , or with derivatives of $x$ inside the integral, are included in the same name.

Remarks

Example: $x'(t) = \int_0^t x(s) \, ds$ , $x(0) = 1$ , with $f = 0$ and $K(t, s, x) = x$ . Differentiating gives $x'' = x$ with $x(0) = 1$ and $x'(0) = 0$ , so $x(t) = \cosh t$ . Check: $\int_0^t \cosh s \, ds = \sinh t = x'(t)$ .

Non-example: $x'(t) = \int_0^1 x(s) \, ds$ for $t \in [0, 1]$ has an integral over the fixed interval $[0, 1]$ , including future times, so it is an integro-differential equation of Fredholm type, not of Volterra type.

Integrating from $t_0$ to $t$ turns the problem into the Volterra integral equation $x(t) = x_0 + \int_{t_0}^{t} \left[f\big(s, x(s)\big) + \int_{t_0}^{s} K\big(s, r, x(r)\big) \, dr\right] ds.$ If $f$ and $K$ are Lipschitz in their last variable, uniformly in the others, successive approximations converge on all of $I$ and give a unique solution. This generalizes the initial value problem for ordinary differential equations, which is the case $K = 0$ .

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Initial value problem concept
Integro-differential equation concept
Volterra integral equation concept
Volterra integro-differential equation definition

Tag	`6ZQRVRD8KP`
URI	`m2://6ZQRVRD8KP`
Kind	definition
Writing	W0 stub
MSC	45J05, 45D05
Defines	Volterra integro-differential equation
Uses	Integro-differential equation, Volterra integral equation, Initial value problem

Volterra integro-differential equation

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here