Prof. Dr. Mohamed Suleiman

University: University Putra Malaysia
Department: Department of Mathematics
Research Fields: Numerical Analysis

Publications

3 Parallel Direct Integration Variable Step Block Method for Solving Large System of Higher Order Ordinary Differential Equations

Authors: Mohamed Suleiman, Zanariah Abdul Majid

Abstract:

The aim of this paper is to investigate the performance of the developed two point block method designed for two processors for solving directly non stiff large systems of higher order ordinary differential equations (ODEs). The method calculates the numerical solution at two points simultaneously and produces two new equally spaced solution values within a block and it is possible to assign the computational tasks at each time step to a single processor. The algorithm of the method was developed in C language and the parallel computation was done on a parallel shared memory environment. Numerical results are given to compare the efficiency of the developed method to the sequential timing. For large problems, the parallel implementation produced 1.95 speed-up and 98% efficiency for the two processors.

Keywords: Numerical Methods, block method, parallel method, higher order ODEs

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2 Parallel Block Backward Differentiation Formulas for Solving Ordinary Differential Equations

Authors: Mohamed Suleiman, Zarina Bibi Ibrahim, Khairil Iskandar Othman

Abstract:

A parallel block method based on Backward Differentiation Formulas (BDF) is developed for the parallel solution of stiff Ordinary Differential Equations (ODEs). Most common methods for solving stiff systems of ODEs are based on implicit formulae and solved using Newton iteration which requires repeated solution of systems of linear equations with coefficient matrix, I - hβJ . Here, J is the Jacobian matrix of the problem. In this paper, the matrix operations is paralleled in order to reduce the cost of the iterations. Numerical results are given to compare the speedup and efficiency of parallel algorithm and that of sequential algorithm.

Keywords: block, Backward Differentiation Formula, ordinarydifferential equations

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1 Direct Block Backward Differentiation Formulas for Solving Second Order Ordinary Differential Equations

Authors: Mohamed Suleiman, Zarina Bibi Ibrahim, Khairil Iskandar Othman

Abstract:

In this paper, a direct method based on variable step size Block Backward Differentiation Formula which is referred as BBDF2 for solving second order Ordinary Differential Equations (ODEs) is developed. The advantages of the BBDF2 method over the corresponding sequential variable step variable order Backward Differentiation Formula (BDFVS) when used to solve the same problem as a first order system are pointed out. Numerical results are given to validate the method.

Keywords: block, Backward Differentiation Formula, secondorder

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Abstracts

1 Development of Variable Order Block Multistep Method for Solving Ordinary Differential Equations

Authors: Mohamed Suleiman, Zarina Bibi Ibrahim, Khairil Iskandar Othman, Nor Ain Azeany

Abstract:

In this paper, a class of variable order fully implicit multistep Block Backward Differentiation Formulas (VOBBDF) using uniform step size for the numerical solution of stiff ordinary differential equations (ODEs) is developed. The code will combine three multistep block methods of order four, five and six. The order selection is based on approximation of the local errors with specific tolerance. These methods are constructed to produce two approximate solutions simultaneously at each iteration in order to further increase the efficiency. The proposed VOBBDF is validated through numerical results on some standard problems found in the literature and comparisons are made with single order Block Backward Differentiation Formula (BBDF). Numerical results shows the advantage of using VOBBDF for solving ODEs.

Keywords: Ordinary differential equations, block backward differentiation formulas, uniform step size

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