Volterra integral equation

In mathematics, the Volterra integral equations are a special type of integral equations.^[1] They are divided into two groups referred to as the first and the second kind.

A linear Volterra equation of the first kind is

${\displaystyle f(t)=\int _{a}^{t}K(t,s)\,x(s)\,ds}$

where f is a given function and x is an unknown function to be solved for. A linear Volterra equation of the second kind is

${\displaystyle x(t)=f(t)+\int _{a}^{t}K(t,s)x(s)\,ds.}$

In operator theory, and in Fredholm theory, the corresponding operators are called Volterra operators. A useful method to solve such equations, the Adomian decomposition method, is due to George Adomian.

A linear Volterra integral equation is a convolution equation if

${\displaystyle x(t)=f(t)+\int _{t_{0}}^{t}K(t-s)x(s)\,ds.}$

The function ${\displaystyle K}$ in the integral is called the kernel. Such equations can be analyzed and solved by means of Laplace transform techniques.

For a weakly singular kernel of the form ${\displaystyle K(t,s)=(t^{2}-s^{2})^{-\alpha }}$ with ${\displaystyle 0<\alpha <1}$, Volterra integral equation of the first kind can conveniently be transformed into a classical Abel integral equation.

The Volterra integral equations were introduced by Vito Volterra and then studied by Traian Lalescu in his 1908 thesis, Sur les équations de Volterra, written under the direction of Émile Picard. In 1911, Lalescu wrote the first book ever on integral equations.

Volterra integral equations find application in demography as Lotka's integral equation,^[2] the study of viscoelastic materials, in actuarial science through the renewal equation,^[3] and in fluid mechanics to describe the flow behavior near finite-sized boundaries.^[4][5]