Fredholm equation of the first kind

Builds on Kernel of an integral equation Fredholm integral equation

Definition

Let $a < b$ be real numbers. Let $K \colon [a, b] \times [a, b] \to \C$ be a kernel, for example continuous or square-integrable, and let $f \colon [a, b] \to \C$ be a given function. Let $\varphi$ denote an unknown function on $[a, b]$ . The Fredholm equation of the first kind with kernel $K$ and right-hand side $f$ is $\int_a^b K(x, y) \, \varphi(y) \, dy = f(x), \qquad x \in [a, b].$ A solution is a function $\varphi$ in the chosen function space for which the integral exists and the equality holds. In $L^2$ the equality is required for almost every $x$ , and otherwise for every $x$ .

Remarks

Example: on $[0, 1]$ , take $K(x, y) = xy$ and $f(x) = x / 2$ . The constant function $\varphi = 1$ is a solution, because $\int_0^1 x y \, dy = x/2$ for every $x$ . It is not the only one: $\varphi(y) = 1 + (2 - 3y)$ also works, since $\int_0^1 y (2 - 3y) \, dy = 1 - 1 = 0$ . With the same kernel and $f(x) = 1$ , there is no solution, since the left side is always a multiple of $x$ and vanishes at $x = 0$ .

Non-example: the equation $\varphi(x) - \int_a^b K(x, y) \varphi(y) \, dy = f(x)$ is not of the first kind, because $\varphi$ also appears outside the integral; it is a Fredholm equation of the second kind.

The equation is a special Fredholm integral equation: the limits of integration are the fixed numbers $a$ and $b$ . In operator notation it reads $T_K \varphi = f$ , where $(T_K \varphi)(x) = \int_a^b K(x, y) \varphi(y) \, dy$ .

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Fredholm integral equation concept
Kernel of an integral equation concept
Fredholm equation of the first kind definition

Tag	`V1FFDDHCH2`
URI	`m2://V1FFDDHCH2`
Kind	definition
Writing	W0 stub
MSC	45B05, 45A05
Defines	Fredholm equation of the first kind
Uses	Kernel of an integral equation, Fredholm integral equation

Fredholm equation of the first kind

Definition

Remarks

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