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]

A linear Volterra equation of the first kind can always be reduced to a linear Volterra equation of the second kind, assuming that ${\displaystyle K(t,t)\neq 0}$. Taking the derivative of the first kind Volterra equation gives us:$${\displaystyle {df \over {dt}}=\int _{a}^{t}{\partial K \over {\partial t}}x(s)ds+K(t,t)x(t)}$$Dividing through by ${\displaystyle K(t,t)}$ yields:$${\displaystyle x(t)={1 \over {K(t,t)}}{df \over {dt}}-\int _{a}^{t}{1 \over {K(t,t)}}{\partial K \over {\partial t}}x(s)ds}$$Defining ${\textstyle {\widetilde {f}}(t)={1 \over {K(t,t)}}{df \over {dt}}}$ and ${\textstyle {\widetilde {K}}(t,s)=-{1 \over {K(t,t)}}{\partial K \over {\partial t}}}$ completes the transformation of the first kind equation into a linear Volterra equation of the second kind.

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 ${\displaystyle \Delta x}$ is given by:$${\displaystyle \int _{a}^{b}f(x)dx\approx {\Delta x \over {2}}\left[f(x_{0})+2\sum _{i=1}^{n-1}f(x_{i})+f(x_{n})\right]}$$Assuming equal spacing for the subintervals, the integral component of the Volterra equation may be approximated by:$${\displaystyle \int _{a}^{t}K(t,s)x(s)ds\approx {\Delta s \over {2}}\left[K(t,s_{0})x(s_{0})+2K(t,s_{1})x(s_{1})+\cdots +2K(t,s_{n-1})x(s_{n-1})+K(t,s_{n})x(s_{n})\right]}$$Defining ${\displaystyle x_{i}=x(s_{i})}$, ${\displaystyle f_{i}=f(t_{i})}$, and ${\displaystyle K_{ij}=K(t_{i},s_{j})}$, we have the system of linear equations:$${\displaystyle {\begin{aligned}x_{0}&=f_{0}\\x_{1}&=f_{1}+{\Delta s \over {2}}\left(K_{10}x_{0}+K_{11}x_{1}\right)\\x_{2}&=f_{2}+{\Delta s \over {2}}\left(K_{20}x_{0}+2K_{21}x_{1}+K_{22}x_{2}\right)\\&\vdots \\x_{n}&=f_{n}+{\Delta s \over {2}}\left(K_{n0}x_{0}+2K_{n1}x_{1}+\cdots +2K_{n,n-1}x_{n-1}+K_{nn}x_{n}\right)\end{aligned}}}$$This is equivalent to the matrix equation:$${\displaystyle x=f+Mx\implies x=(I-M)^{-1}f}$$For well-behaved kernels, the trapezoidal rule tends to work well.

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 ${\displaystyle \psi (u)=\mathbb {P} [\tau (u)<\infty ]}$, where ${\displaystyle u}$ is the initial surplus and ${\displaystyle \tau (u)}$ is the time of ruin. In the classical model of ruin theory, the net cash position ${\displaystyle X_{t}}$ is a function of the initial surplus, premium income earned at rate ${\displaystyle c}$, and outgoing claims ${\displaystyle \xi }$:$${\displaystyle X_{t}=u+ct-\sum _{i=1}^{N_{t}}\xi _{i},\quad t\geq 0}$$where ${\displaystyle N_{t}}$ is a Poisson process for the number of claims with intensity ${\displaystyle \lambda }$. Under these circumstances, the ruin probability may be represented by a Volterra integral equation of the form^[6]:$${\displaystyle \psi (u)={\lambda \over {c}}\int _{u}^{\infty }S(x)dx+{\lambda \over {c}}\int _{0}^{u}\psi (u-x)S(x)dx}$$where ${\displaystyle S(\cdot )}$ is the survival function of the claims distribution.

• Fredholm integral equation
• Integral equation
• Integro-differential equation

• Traian Lalescu, Introduction à la théorie des équations intégrales. Avec une préface de É. Picard, Paris: A. Hermann et Fils, 1912. VII + 152 pp.
• "Volterra equation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Volterra Integral Equation of the First Kind". MathWorld.
• Weisstein, Eric W. "Volterra Integral Equation of the Second Kind". MathWorld.
• Integral Equations: Exact Solutions at EqWorld: The World of Mathematical Equations
• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 19.2. Volterra Equations". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.

• IntEQ: a Python package for numerically solving Volterra integral equations