Fuziyah Ishak

Numerical Solution and Stability of Block Method for Solving Functional Differential Equations

In this article, we describe the development of a two-point block method for solving functional differential equations. The block method, implemented in variable stepsize technique produces two approximations simultaneously using the same back values. The grid-point formulae for the variable steps are derived, calculated and stored at the start of the program for greater efficiency. The delay solutions for the unknown function and its derivative at earlier times are interpolated using the previous computed values. Stability regions for the block method are illustrated. Numerical results are given to demonstrate the accuracy and efficiency of the block method.

Block method with Hermite interpolation for numerical solution of delay differential equations

In this paper, we consider a two-point implicit block method for solving delay differential equations. For greater efficiency, the block method is implemented in variable stepsize technique. The most optimal stepsize is taken while achieving the desired accuracy. The implicit method is solved using predictor-corrector scheme where the corrector is iterated until convergence. Grid point formulae are derived using a predictor and a corrector of order five. The formulae produce two new values in a single integration step. Delay solutions are approximated using Hermite interpolation of order five. The advantage of using Hermite interpolation is that it requires less support points than the existing interpolation technique in order to achieve the overall accuracy. Numerical results indicate that the two-point block method with Hermite interpolation technique is efficient and reliable in solving a wide range of delay differential equations.

Efficient technique for solving stiff delay differential equations in variable stepsize variable order

In this paper, we describe the development of predictor corrector variable stepsize variable order method based on backward differentiation formula. The formula is represented in divided difference form where the coefficients of differentiation are computed by a simple recurrence relation. This representation will reduce the computational cost of recalculating the differentiation coefficients. Numerical results show that this method is reliable, efficient and accurate in solving a wide range of stiff delay differential equations.

Parallel method using MPI for solving large systems of delay differential equations

In this paper, we describe a parallel algorithm for solving large systems of first order delay differential equations. The algorithm is based on a variable stepsize variable order block method. The method produces two new approximations in a single integration step. The formulae derivation permits concurrent computation between two processors. The parallel algorithm is implemented by calling the Message Passing Interface (MPI) library. The performance of the sequential and parallel block method is compared with a sequential non-block method. Moreover, the performance of the parallel algorithm is assessed in terms of speedup and efficiency. It is shown from the numerical results that the overall performance of the block method is increased by parallelizing each point in a block.

Solving delay differential equations by predictor-corrector method using lagrange and hermite interpolations

Delay differential equations (DDEs) appear naturally in modeling many real life phenomena. DDEs differ from ordinary differential equations since the derivative of the unknown function contains the expression of the unknown function at earlier and present states as well. DDEs that cannot be solved analytically are solved numerically. In this work, we solve DDEs using predictor-corrector multistep method where the corrector is iterated until convergence. The predictor uses the Adams-Bashforth four-step explicit method and the corrector uses Adams-Moulton three-step implicit method. Two types of interpolation polynomials which are Lagrange and Hermite interpolations are used to approximate the delay solutions. The accuracy of the adapted Adams-Bashforth-Moulton methods using these two polynomials is compared.