Definition:Integral Equation

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Definition

An integral equation is a mathematical equation in which an unknown function appears under an integral sign.


First Kind

An integral equation of the first kind is an integral equation of the form:

$\map f x = \lambda \ds \int_{\map a x}^{\map b x} \map K {x, y} \map g y \rd x$

where:

$\map f x$ and $\map K {x, y}$ are known functions
$\map a x$ and $\map b x$ are known functions of $x$, or constant
$\map g x$ is an unknown function.


Second Kind

An integral equation of the second kind is an integral equation of the form:

$\map g x = \map f x + \lambda \ds \int_{\map a x}^{\map b x} \map K {x, y} \map g y \rd x$

where:

$\map f x$ and $\map K {x, y}$ are known functions
$\map a x$ and $\map b x$ are known functions of $x$, or constant
$\map g x$ is an unknown function.


Third Kind

An integral equation of the third kind is an integral equation of the form:

$\map u x \map g x = \map f x + \lambda \ds \int_{\map a x}^{\map b x} \map K {x, y} \map g y \rd x$

where:

$\map u x$, $\map f x$ and $\map K {x, y}$ are known functions
$\map a x$ and $\map b x$ are known functions of $x$, or constant
$\map g x$ is an unknown function.


Parts of Integral Equation

Kernel

The function $\map K {x, y}$ is known as the kernel of the integral equation.


Parameter

The number $\lambda$ is known as the parameter of the integral equation.


Also see

  • Results about integral equations can be found here.


Historical Note

The first problem to be solved by an integral equation was the Tautochrone Problem, which Niels Henrik Abel solved using what is now known as Abel"s Integral Equation in $1823$.


Sources

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