Kernel of an integral equation

Definition

Let $a < b$ , let $K \colon [a, b] \times [a, b] \to \C$ and $f \colon [a, b] \to \C$ be given functions, and let $\lambda \in \C$ . Consider an integral equation of one of the forms $\int_a^b K(x, s) \varphi(s) \, ds = f(x), \qquad \varphi(x) - \lambda \int_a^b K(x, s) \varphi(s) \, ds = f(x),$ or the same forms with the upper limit $b$ replaced by $x$ . The function $K$ is the kernel of the equation. The map $\varphi \mapsto \int_a^b K(\cdot, s) \varphi(s) \, ds$ is the integral operator with kernel $K$ .

Remarks

Example: in $\varphi(x) - \int_0^1 e^{x - s} \varphi(s) \, ds = x$ the kernel is $K(x, s) = e^{x - s}$ and $\lambda = 1$ .

Non-example: in $\varphi(x) = x + \int_0^1 \varphi(s)^2 \, ds$ the unknown does not enter linearly under the integral, so the equation has no kernel in this sense. For a nonlinear equation $\varphi(x) = f(x) + \int_a^b H(x, s, \varphi(s)) \, ds$ , the function $H$ of three variables is sometimes also called the kernel.

For a Volterra equation, with upper limit $x$ , only the values of $K$ with $s \le x$ matter, and one may set $K(x, s) = 0$ for $s > x$ .

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Function concept
Integral equation concept
Kernel of an integral equation definition

Tag	`1CRA2F3TY7`
URI	`m2://1CRA2F3TY7`
Kind	definition
Writing	W0 stub
MSC	45A05, 45P05
Defines	Kernel of an integral equation
Uses	Integral equation, Function

Kernel of an integral equation

Definition

Remarks

Reading path

Further reading

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