Approximate solution of system of nonlinear Volterra integro-differential equations by using Bernstein collocation method

This paper presents a numerical matrix method based on Bernstein polynomials (BPs) for approximate the solution of a system of m-th order nonlinear Volterra integro-differential equations under initial conditions. The approach is based on operational matrices of BPs. Using the collocation points,this approach reduces the systems of Volterra integro-differential equations associated with the given conditions, to a system of nonlinear algebraic equations. By solving such arising non linear system, the Bernstein coefficients can be determined to obtain the finite Bernstein series approach. Numerical examples are tested and the resultes are incorporated to demonstrate the validity and applicability of the approach. Comparisons with a number of conventional methods are made in order to verify the nature of accuracy and the applicability of the proposed approach. Keywords: Systems of nonlinear Volterra integro-differential equations; The Bernstein polyno- mials and series; Collocation points. ۲۰۱۰ AMS Subject Classication: ۳۴A۱۲, ۳۴A۳۴, ۴۵D۰۵, ۴۵G۱۵, ۴۵J۰۵, ۶۵R۲۰.

Keywords:

‎Systems of nonlinear Volterra integro-differential equations , The Bernstein polynomials and series , Operational matrices , Numerical matrix method , Collocation points