Zanariah Abdul Majid

Implementation of four-point fully implicit block method for solving ordinary differential equations

This paper describes the development of a four-point fully implicit block method for solving first order ordinary differential equations (ODEs) using variable step size. This method will estimate the solutions of initial value problems (IVPs) at four points simultaneously. The method developed is suitable for the numerical integration of non-stiff and mildly stiff differential systems. The performances of the four-point block method are compared in terms of maximum error, total number of steps and execution times to the non-block method 1PVSO in [Z. Omar, Developing parallel block methods for solving higher order ODEs directly, Ph.D. Thesis, University Putra Malaysia, Malaysia, 1999].

Direct Two-Point Block One-Step Method for Solving General Second-Order Ordinary Differential Equations

A direct two-point block one-step method for solving general second-order ordinary differential equations (ODEs) directly is presented in this paper. The one-step block method will solve the second-order ODEs without reducing to first-order equations. The direct solutions of the general second-order ODEs will be calculated at two points simultaneously using variable step size. The method is formulated using the linear multistep method, but the new method possesses the desirable feature of the one-step method. The implementation is based on the predictor and corrector formulas in the mode. The stability and precision of this method will also be analyzed and deliberated. Numerical results are given to show the efficiency of the proposed method and will be compared with the existing method.

A variable step implicit block multistep method for solving first-order ODEs

A new four-point implicit block multistep method is developed for solving systems of first-order ordinary differential equations with variable step size. The method computes the numerical solution at four equally spaced points simultaneously. The stability of the proposed method is investigated. The Gauss-Seidel approach is used for the implementation of the proposed method in the PE(CE)^m mode. The method is presented in a simple form of Adams type and all coefficients are stored in the code in order to avoid the calculation of divided difference and integration coefficients. Numerical examples are given to illustrate the efficiency of the proposed method.

Solving delay differential equations using coupled block method

A variable step size variable order coupled block method for the numerical solution of delay differential equation is described. The approximation of the delay term is calculated using divided difference interpolation. The numerical results are presented and it shows that the proposed code is suitable for solving delay differential equations.

A new approach for solving dual fuzzy nonlinear equations using Broyden's and Newton's methods

We present a new approach for solving dual fuzzy nonlinear equations. In this approach, we use Newtons method for initial iteration and Broydens method for the rest of the iterations. The fuzzy coefficients are presented in parametric form. Numerical results on well-known benchmark fuzzy nonlinear equations are reported to authenticate the effectiveness and efficiency of the approach.

The Four Point-EDGMSOR Iterative Method for Solution of 2D Helmholtz Equations

This paper examines the use of the Four Point-Explicit Decoupled Group (EDG) iterative method together with MSOR iterative method, namely 4 Point-EDGMSOR. The efficiency of this method will be investigated to solve two-dimensional Helmholtz equation by using half-sweep finite difference approximation. In fact, the formulation of and implementation of the proposed method are also presented. Some numerical examples are presented to point out efficiency of the proposed method.

Solving Second-Order Delay Differential Equations by Direct Adams-Moulton Method

This paper will consider the implementation of fifth-order direct method in the form of Adams-Moulton method for solving directly second-order delay differential equations (DDEs). The proposed direct method approximates the solutions using constant step size. The delay differential equations will be treated in their original forms without being reduced to systems of first-order ordinary differential equations (ODEs). Numerical results are presented to show that the proposed direct method is suitable for solving second-order delay differential equations.

Diagonally Implicit Block Backward Differentiation Formulas for Solving Ordinary Differential Equations

This paper focuses on the derivation of diagonally implicit two-point block backward differentiation formulas (DI2BBDF) for solving first-order initial value problem (IVP) with two fixed points. The method approximates the solution at two points simultaneously. The implementation and the stability of the proposed method are also discussed. A performance of the DI2BBDF is compared with the existing methods.

Study of Quadrupole Mass Filter Supplied with a New Impulsional Potential

The paper reports on some theoretical studies concerning the impulsional mode of a quadrupole mass filter (QMF) supplied with a new periodic impulsional radio frequency voltage of the form Vaccos(ωt)⌊1 + kcos(⅓ωt)⌋/1 – √kcos(2ωt) with 0 ≤ k < 1 (⌊.⌋ means floor function) and eventually compares it to the classical sinusoidal case, k = 0. The physical properties of the confined ions in the r and z directions are illustrated and the fractional mass resolutions m/Δm of the confined ions in the first stability regions of both potential were analyzed for hydrogen isotopes and presented.

Parallel block codes for solving large systems of ordinary differential equations

The two-point fully implicit block methods are developed for solving large systems of ordinary differential equations (ODEs) using variable step size on a parallel shared memory computer. The method calculates the numerical solution at two equally spaced points simultaneously within a block. The sequential and parallel performances of these methods are compared with the performances of the two-point block method using variable step size and order developed earlier by Omar (1999). For large problems, the parallel implementation produced a good speedup with respect to the sequential timing and hence better efficiency for the methods developed. The stability of the methods is also investigated.