Volterra integral equation

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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

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

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

The function 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 with , 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]

Conversion of Volterra equation of the first kind to the second kind[edit]

A linear Volterra equation of the first kind can always be reduced to a linear Volterra equation of the second kind, assuming that . Taking the derivative of the first kind Volterra equation gives us:

Dividing through by yields:
Defining and completes the transformation of the first kind equation into a linear Volterra equation of the second kind.

Numerical solution using trapezoidal rule[edit]

A standard method for computing the numerical solution of a linear Volterra equation of the second kind is the trapezoidal rule, which for equally-spaced subintervals is given by:

Assuming equal spacing for the subintervals, the integral component of the Volterra equation may be approximated by:
Defining , , and , we have the system of linear equations:
This is equivalent to the matrix equation:
For well-behaved kernels, the trapezoidal rule tends to work well.

Application: Ruin theory[edit]

One area where Volterra integral equations appear is in ruin theory, the study of the risk of insolvency in actuarial science. The objective is to quantify the probability of ruin , where is the initial surplus and is the time of ruin. In the classical model of ruin theory, the net cash position is a function of the initial surplus, premium income earned at rate , and outgoing claims :

where is a Poisson process for the number of claims with intensity . Under these circumstances, the ruin probability may be represented by a Volterra integral equation of the form[6]:
where is the survival function of the claims distribution.

See also[edit]


  1. ^ Polyanin, Andrei D.; Manzhirov, Alexander V. (2008). Handbook of Integral Equations (2nd ed.). Boca Raton, FL: Chapman and Hall/CRC. ISBN 978-1584885078.
  2. ^ Inaba, Hisashi (2017). "The Stable Population Model". Age-Structured Population Dynamics in Demography and Epidemiology. Singapore: Springer. pp. 1–74. doi:10.1007/978-981-10-0188-8_1. ISBN 978-981-10-0187-1.
  3. ^ Brunner, Hermann (2017). Volterra Integral Equations: An Introduction to Theory and Applications. Cambridge Monographs on Applied and Computational Mathematics. Cambridge, UK: Cambridge University Press. ISBN 978-1107098725.
  4. ^ Daddi-Moussa-Ider, A.; Vilfan, A.; Golestanian, R. (6 April 2022). "Diffusiophoretic propulsion of an isotropic active colloidal particle near a finite-sized disk embedded in a planar fluid–fluid interface". Journal of Fluid Mechanics. 940: A12. arXiv:2109.14437. doi:10.1017/jfm.2022.232.
  5. ^ Daddi-Moussa-Ider, A.; Lisicki, M.; Löwen, H.; Menzel, A. M. (5 February 2020). "Dynamics of a microswimmer–microplatelet composite". Physics of Fluids. 32 (2): 021902. arXiv:2001.06646. doi:10.1063/1.5142054.
  6. ^ "Lecture Notes on Risk Theory" (PDF). School of Mathematics, Statistics and Actuarial Science. University of Kent. February 20, 2010. pp. 17–22.

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