Implicit block method for solving neutral delay differential equations

Abstract

Most neutral delay differential equations (NDDEs) that appear in real life phenomena cannot be solved analytically. Therefore, scientists and engineers resort for numerical solutions that can be made as accurately as possible. Numerical methods for NDDEs have been of great interests, since the existing methods for ordinary differential equations cannot directly solve the delay terms in NDDEs. Most numerical methods have been developed to approximate only one new value in a single integration step and using a fixed stepsize. Variable stepsize algorithm requires extra computation cost since the coefficients of the method need to be recalculated whenever the stepsize changes. Hence, this study aims to develop an implicit block method for solving systems of first order NDDEs using variable stepsize. Formulae of three-point one-block method using variable stepsize are derived. The performance of the three-point one-block method is compared with that of the two-point one-block method. The maximum and average errors of the two methods are comparable and the three-point one-block method requires less number of total steps taken. The numerical results indicates that the three-point one-block method can approximate the solution for NDDEs as accurate and efficient as possible when compared with two-point one-block method.