Multistep block methods for solving higher order delay differential equations - Hoo, Yann Seong (2016)
Delay differential equations (DDEs) play an important role in the investigated system which depends on the position of the system in the past and current time. The analytical solution of DDEs is hard to be found. Numerical methods provide an alternative way of constructing solutions to the problems. This thesis describes the development of numerical algorithms for solving higher order DDEs. One-point and two-point multistep block method based on the Adam-Bashforth- Moulton methods for solving higher ordinary differential equation are adapted to solve the higher order DDEs. The proposed methods are based on constant step size and variable step size approach. Two types of DDEs are considered, namely retarded and neutral DDEs. Only the DDEs with constant delays and pantograph type are considered in this thesis. The delay term in DDEs with constant delays is approximated using Hermite interpolation. Linear and Hermite interpolators are used to approximate the delay terms in DDEs of pantograph type. The derivatives of the delay terms are approximated by using difference formula. The thesis discusses the stability of the method when applied to DDEs with constant delays and pantograph type. The region of the stability is presented. Several problems are considered for illustrative purposes and the numerical approximations of their solutions are obtained using C-language. Numerical results of the proposed methods are compared with the existing numerical methods. Comparison among the methods indicated that the proposed methods achieve the desired accuracy. Block method are efficient when compare with the non-block method as the total steps taken can be reduced.