Integral equation

Definition

Let $a < b$ and let $\varphi \colon [a, b] \to \C$ be an unknown function. For each $x \in [a, b]$ , let $I_x \subseteq [a, b]$ be an interval of integration, either the fixed interval $[a, b]$ or the interval $[a, x]$ . An integral equation for $\varphi$ is an equation of the form $F\left(x, \varphi(x), \int_{I_x} H\left(x, s, \varphi(s)\right) ds\right) = 0 \quad \text{for all } x \in [a, b],$ where $F$ and $H$ are given functions. A solution is a function $\varphi$ for which the integrals exist and the equation holds for every $x \in [a, b]$ .

Remarks

Example: $\varphi(x) = x + \int_0^1 x s \, \varphi(s) \, ds$ on $[0, 1]$ is an integral equation with $I_x = [0, 1]$ . Try $\varphi(x) = c x$ : the integral is $c x / 3$ , so $c = 1 + c/3$ and $c = 3/2$ . So $\varphi(x) = 3x/2$ is a solution.

Non-example: $\varphi'(x) = \varphi(x)$ is a differential equation, not an integral equation, since $\varphi$ does not appear under an integral. Its integral form $\varphi(x) = \varphi(0) + \int_0^x \varphi(s) \, ds$ is an integral equation.

The same definition is used with a region of $\R^m$ in place of $[a, b]$ , and with vector-valued unknowns. When $I_x = [a, b]$ the equation is of Fredholm type; when $I_x = [a, x]$ it is of Volterra type.

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Function concept
Riemann integral concept
Integral equation definition

Tag	`FZYDPWMS46`
URI	`m2://FZYDPWMS46`
Kind	definition
Writing	W0 stub
MSC	45A05, 45G10
Defines	Integral equation
Uses	Function, Riemann integral

Integral equation

Definition

Remarks

Reading path

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