Definition:Integral Equation/Second Kind

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Definition

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.


Thus an integral equation of the second kind is an example of an integral equation of the third kind:

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

where $\map u x \equiv 1$.


Homogeneous

An integral equation of the second kind

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

is described as homogeneous if and only if $\map f x \equiv 0$.


That is, if it is of the form:

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

where:

$\map K {x, y}$ is a known function
$\map a x$ 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.


Examples

Fredholm Integral Equation of the Second Kind

A Fredholm integral equation of the second kind is an integral equation of the form:

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


Volterra Integral Equation of the Second Kind

A Volterra integral equation of the second kind is an integral equation of the form:

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

where $g$ is an unknown real function.


Also see

  • Results about integral equations of the second kind can be found here.


Sources

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