Volterra integral equation

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 Volterra equation of the first kind is

:$f(x)\; =\; int\_a^x\; K(x,t),phi(t),dt.$

A Volterra equation of the second kind is

:$phi(x)\; =\; f(x)\; +\; lambda\; int\_a^x\; K(x,t),phi(t),dt.$

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

The Volterra integral equations were 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.

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.
* [http://mathworld.wolfram.com/VolterraIntegralEquationoftheFirstKind.html Volterra Integral Equation of the First Kind at MathWorld]
* [http://mathworld.wolfram.com/VolterraIntegralEquationoftheSecondKind.html Volterra Integral Equation of the Second Kind at MathWorld]
* [http://eqworld.ipmnet.ru/en/solutions/ie.htm Integral Equations: Exact Solutions] at EqWorld: The World of Mathematical Equations