Volterra Integro-Differential Equations

Abstract

Volterra studied the hereditary influences when he was examining a population growth model. The research work resulted in a specific topic, where both differential and integral operators appeared together in the same equation. This new type of equations was termed as Volterra integro-differential equations [1–4], given in the form

$${u^{\left( n \right)}}\left( x \right) = f\left( x \right) + \lambda \int_0^{\left( x \right)} {K\left( {x,t} \right)u\left( t \right)dt,} $$

(5.1)

Where ${u^{\left( n \right)}}\left( x \right) = \frac{{{d^n}u}}{{d{x^n}}}$. Because the resulted equation in (5.1) combines the differential operator and the integral operator, then it is necessary to define initial conditions u(0), u′ (0), , u ⁽ⁿ⁻¹⁾(0) for the determination of the particular solution u(x) of the Volterra integro-differential equation (5.1). Any Volterra integro-differential equation is characterized by the existence of one or more of the derivatives u′ (x), u″ (x), outside the integral sign. The Volterra integro-differential equations may be observed when we convert an initial value problem to an integral equation by using Leibnitz rule.