Definition:Volterra Integral Equation

Definition

The Volterra integral equations are integral equations of the first and second kind such that the limits of integration are between a constant $a$ and the variable $x$:

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.

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 known as

The Volterra integral equations can also be seen in the form Volterra's integral equations.

Also see

• Results about Volterra integral equations can be found here.

Source of Name

This entry was named for Samuel Giuseppe Vito Volterra.

Sources

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