In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Volterra integral equations are a special type of integral equation

Integral equation

In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...

s. They are divided into two groups referred to as the first and the second kind.

A linear Volterra equation of the first kind is


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


In operator theory

Operator theory

In mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....

, and in Fredholm theory

Fredholm theory

In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given in terms of the spectral theory of Fredholm...

, the corresponding equations are called the Volterra operator

Volterra operator

In mathematics, in the area of functional analysis and operator theory, the Volterra operator, named after Vito Volterra, represents the operation of indefinite integration, viewed as a bounded linear operator on the space L2 of complex-valued square integrable functions on the interval...

.

A linear Volterra integral equation is a convolution

Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...

 equation if


The function in the integral is often called the kernel

Kernel (mathematics)

In mathematics, the word kernel has several meanings. Kernel may mean a subset associated with a mapping:* The kernel of a mapping is the set of elements that map to the zero element , as in kernel of a linear operator and kernel of a matrix...

.
Such equations can be analysed and solved by means of Laplace transform techniques.

The Volterra integral equations were introduced by Vito Volterra

Vito Volterra

Vito Volterra was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations....

 and then studied by Traian Lalescu

Traian Lalescu

Traian Lalescu was a Romanian mathematician. His main focus was on integral equations and he contributed to work in the areas of functional equations, trigonometric series, mathematical physics, geometry, mechanics, algebra, and the history of mathematics....

 in his 1908 thesis, Sur les équations de Volterra, written under the direction of Émile Picard

Charles Émile Picard

Charles Émile Picard FRS was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie Française in 1924.- Biography :...

. In 1911, Lalescu wrote the first book ever on integral equations.

Volterra integral equations find application in demography, the study of viscoelastic materials,
and in insurance mathematics through the renewal equation.