Definition:Integral Equation/First Kind
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Definition
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.
Thus an integral equation of the first 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 $\map u x \equiv 0$.
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 First Kind
A Fredholm integral equation of the first kind is an integral equation of the form:
- $\ds \map f x = \lambda \int_a^b \map K {x, y} \map g y \rd y$
where $g$ is an unknown real function.
Volterra Integral Equation of the First Kind
A Volterra integral equation of the first kind is an integral equation of the form:
- $\ds \map f x = \lambda \int_a^x \map K {x, y} \map g y \rd y$
where $g$ is an unknown real function.
Also see
- Results about integral equations of the first 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