Dr. Zarina Bibi Ibrahim

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

Publications

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

3 Modification of Newton Method in Two Points Block Differentiation Formula

Authors: Zarina Bibi Ibrahim, Khairil Iskandar Othman, Nadhirah Kamal

Abstract:

Block methods for solving stiff systems of ordinary differential equations (ODEs) are based on backward differential formulas (BDF) with PE(CE)2 and Newton method. In this paper, we introduce Modified Newton as a new strategy to get more efficient result. The derivation of BBDF using modified block Newton method is presented. This new block method with predictor-corrector gives more accurate result when compared to the existing BBDF.

Keywords: stiff, modified Newton, BBDF, Jacobian matrix

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2 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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1 The Development of a New Block Method for Solving Stiff ODEs

Authors: Zarina Bibi Ibrahim, Khairil I. Othman, Mahfuzah Mahayaddin

Abstract:

We develop and demonstrate a computationally efficient numerical technique to solve first order stiff differential equations. This technique is based on block method whereby three approximate points are calculated. The Cholistani of varied step sizes are presented in divided difference form. Stability regions of the formulae are briefly discussed in this paper. Numerical results show that this block method perform very well compared to existing methods.

Keywords: Computational, block method, divided difference, stiff

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