Numerical Solution of volterra Integral Equations with Delay Using Block Methods

Abstract

In the last thirty years there has been a great deal of work in the field of differential equations with modified argument. These equations arise in a wide variety of scientific and technical applications, including the modeling of problems from the natural and social sciences such as physics, biological sciences and economics [7]. A special class is represented by the differential equations with affine modification of the argument which can be delay differential equation (DDE) or differential equations with linear modification of the argument. Many results concerning these equations are given in the papers [1]-[4]. These equations are equivalent to the following integral equation Where h, g and k are given continuous functions, is a scalar parameter (we will take equal to one), and f(x) is unknown function to be determined. Equation (1) is called Volterra integral equation with delay when b(x)=x and it is called fredholm integral equation with delay when b(x)=b ,where b is constant, moreover it is called of the first kind if h(x)=0 and of the second kind if h(x)=1,also if g(x)=0 the equation (1) is called homogenous and called nonhomogenous if g(x) 0 [6]. In this paper we consider the Block method applied to the Volterra integral equation with constant delay and with given continuous functio