Linear integral equation

Builds on Integral equation Kernel of an integral equation Continuous function

Definition

Let $a < b$ , let $a_0, f \colon [a, b] \to \C$ be given functions, let $K \colon [a, b] \times [a, b] \to \C$ be a given kernel, and let $\lambda \in \C$ . For each $x \in [a, b]$ , let $I_x$ be either $[a, b]$ or $[a, x]$ . A linear integral equation for the unknown function $\varphi \colon [a, b] \to \C$ is an integral equation of the form $a_0(x) \varphi(x) = f(x) + \lambda \int_{I_x} K(x, s) \varphi(s) \, ds \quad \text{for all } x \in [a, b].$

Remarks

Example: $\varphi(x) = \sin x + \int_0^{\pi} \cos(x - s) \varphi(s) \, ds$ is linear, with $a_0 = 1$ , $f = \sin$ , $K(x, s) = \cos(x - s)$ and $\lambda = 1$ .

Non-example: $\varphi(x) = 1 + \int_0^1 x s \, \varphi(s)^2 \, ds$ is not linear, because $\varphi$ enters through $\varphi(s)^2$ . If $\varphi$ solves it, then $2\varphi$ does not solve the equation with $f$ replaced by $2$ .

The equation is of the first kind when $a_0 = 0$ and of the second kind when $a_0 = 1$ . It is of the third kind when $a_0$ is a continuous function that vanishes somewhere but not everywhere. If $a_0$ has no zeros, dividing by $a_0$ gives an equation of the second kind. In operator notation the equation reads $a_0 \varphi = f + \lambda K \varphi$ , where $K$ also denotes the integral operator.

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Tag	`KGNCXDK4D5`
URI	`m2://KGNCXDK4D5`
Kind	definition
Writing	W0 stub
MSC	45A05
Defines	Linear integral equation
Uses	Integral equation, Kernel of an integral equation, Continuous function

Linear integral equation

Definition

Remarks

Reading path

Further reading

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