Definition:Integral Equation of the Second Kind/Homogeneous

Definition

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.

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

• Definition:Integral Equation

• Definition:Integral Equation of the First Kind
• Definition:Integral Equation of the Third Kind

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

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): integral equation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): integral equation