NUMERICAL SOLUTION OF DELAY DIFFERENTIAL EQUATIONS BY USING RUNGE KUTTA METHOD

Abstract

The main purpose of this Project was to find a numerical solution of delay differential equations by using Runge-kutta type methods. Specifically, numerical solution of first order delay differential equations (DDEs) with a single delay has been discussed. This equation cannot be easily evaluated analytically. As a result, an efficient numerical technique has been applied to find the solution which is indeed an approximate solution. In this aspect, DDE23 solver was employed to find the approximate solution of DDEs. This solver integrates the differential equations with the explicit Runge-Kutta (2,3) pair. In addition to this, it has been also tried to see 2-stage 4th order pseudo Runge-Kutta method to find the numerical solution of DDEs. This method has been adapted the 2-stage 4th order PRKM of ODEs to 2-stage 4th order PRKM of DDEs. The DDE23 solver and 2-stage 4th order pseudo Runge-Kutta methods were used. The Hermite interpolation formula to approximate the delay argument. Newton forward divided difference formula has been used to get the coefficients of Hermite interpolation as these coefficients are very important during the approximation of the delay term by Hermite interpolation. Different test problems were tested. The numerical results that were obtained by DDE23 were close enough to the analytical solution. As a consequence, using DDE23 solver to find the numerical solution of DDEs is suggested.