Kernel density estimation

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

A linear Volterra equation of the first kind is

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

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

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

In operator theory, and in Fredholm theory, the corresponding equations are called the Volterra operator.

A linear Volterra integral equation is a convolution equation if

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

The function $K$ in the integral is often called the kernel. Such equations can be analysed and solved by means of Laplace transform techniques.

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, the study of viscoelastic materials, and in insurance mathematics through the renewal equation.

References

• Traian Lalescu, Introduction à la théorie des équations intégrales. Avec une préface de É. Picard, Paris: A. Hermann et Fils, 1912. VII + 152 pp.
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• Integral Equations: Exact Solutions at EqWorld: The World of Mathematical Equations
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