Integral equation of the second kind

Builds on Linear integral equation Kernel of an integral equation

Definition

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

Remarks

Example: $\varphi(x) = 1 + \lambda \int_0^1 \varphi(s) \, ds$ on $[0, 1]$ , with $K = 1$ . Any solution is a constant $c$ with $c = 1 + \lambda c$ , so for $\lambda \ne 1$ the unique solution is $\varphi = 1/(1 - \lambda)$ . For $\lambda = 1$ there is no solution.

Non-example: $\int_0^1 \varphi(s) \, ds = 1$ is not of the second kind, since $\varphi$ does not appear outside the integral; it is of the first kind.

In operator notation, $(I - \lambda K) \varphi = f$ , where $I$ is the identity and $K$ is the integral operator. Some authors write $\varphi(x) - \lambda \int K \varphi = f(x)$ or put $\lambda$ into the kernel.

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Kernel of an integral equation concept
Linear integral equation concept
Integral equation of the second kind definition

Tag	`2EEKCBNYS0`
URI	`m2://2EEKCBNYS0`
Kind	definition
Writing	W0 stub
MSC	45A05, 45B05
Defines	Integral equation of the second kind
Uses	Linear integral equation, Kernel of an integral equation

Integral equation of the second kind

Definition

Remarks

Reading path

Further reading

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