Parallel implementation of fourth order block backward differentiation formulas for solving system of stiff ordinary differential equations

2013 ◽  
Author(s):  
Zarina Bibi Ibrahim ◽  
Nor Ain Azeany Mohd Nasir ◽  
Khairil Iskandar Othman ◽  
Mohamed Suleiman
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Parallel Implementation ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formulas ◽  
Differentiation Formulas

scholarly journals Fixed Coefficient A ( α ) Stable Block Backward Differentiation Formulas for Stiff Ordinary Differential Equations

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 846 ◽  
Author(s):  
Zarina Ibrahim ◽  
Nursyazwani Mohd Noor ◽  
Khairil Othman
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Execution Time ◽  
Stability Region ◽  
Scientific Literature ◽  
Necessary Conditions ◽  
Computational Time ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formulas ◽  
Differentiation Formulas

The main contribution in this paper is to construct an implicit fixed coefficient Block Backward Differentiation Formulas denoted as A ( α ) -BBDF with equal intervals for solving stiff ordinary differential equations (ODEs). To avoid calculating the differentiation coefficients at each step of the integration, the coefficients of the formulas will be stored, with the intention of optimizing the performance in terms of precision and computational time. The plots of their A ( α ) stability region are provided, and the order of the method is also verified. The necessary conditions for convergence, such as the consistency and zero stability of the method, are also discussed. The numerical results clearly showed the efficiency of the method in terms of accuracy and execution time as compared to other existing methods in the scientific literature.

scholarly journals A Class of Hybrid Multistep Block Methods with A–Stability for the Numerical Solution of Stiff Ordinary Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 914
Author(s):  
Zarina Bibi Ibrahim ◽  
Amiratul Ashikin Nasarudin
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Stability And Convergence ◽  
Stiff Differential Equations ◽  
Stiff Ordinary Differential Equations ◽  
Block Methods ◽  
Backward Differentiation Formulas ◽  
The Stability ◽  
Differentiation Formulas

Recently, block backward differentiation formulas (BBDFs) are used successfully for solving stiff differential equations. In this article, a class of hybrid block backward differentiation formulas (HBBDFs) methods that possessed A –stability are constructed by reformulating the BBDFs for the numerical solution of stiff ordinary differential equations (ODEs). The stability and convergence of the new method are investigated. The methods are found to be zero-stable and consistent, hence the method is convergent. Comparisons between the proposed method with exact solutions and existing methods of similar type show that the new extension of the BBDFs improved the stability with acceptable degree of accuracy.

scholarly journals Stability Analysis of Singly Diagonally Implicit Block Backward Differentiation Formulas for Stiff Ordinary Differential Equations

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 211 ◽  
Author(s):  
Saufianim Jana Aksah ◽  
Zarina Ibrahim ◽  
Iskandar Mohd Zawawi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Triangular Matrix ◽  
Practical Significance ◽  
Computational Time ◽  
Step Size ◽  
Time Step ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formulas ◽  
Differentiation Formulas

In this research, a singly diagonally implicit block backward differentiation formulas (SDIBBDF) for solving stiff ordinary differential equations (ODEs) is proposed. The formula reduced a fully implicit method to lower triangular matrix with equal diagonal elements which will results in only one evaluation of the Jacobian and one LU decomposition for each time step. For the SDIBBDF method to have practical significance in solving stiff problems, its stability region must at least cover almost the whole of the negative half plane. Step size restriction of the proposed method have to be considered in order to ensure stability of the method in computing numerical results. Efficiency of the SDIBBDF method in solving stiff ODEs is justified when it managed to outperform the existing methods for both accuracy and computational time.

scholarly journals On the Integration of Stiff ODEs Using Block Backward Differentiation Formulas of Order Six

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 952
Author(s):  
Amiratul Ashikin Nasarudin ◽  
Zarina Bibi Ibrahim ◽  
Haliza Rosali
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Similar Type ◽  
Differentiation Formula ◽  
First Order ◽  
Backward Differentiation Formula ◽  
Fully Implicit ◽  
Stiff Odes ◽  
Backward Differentiation Formulas ◽  
Differentiation Formulas

In this research, a six-order, fully implicit Block Backward Differentiation Formula with two off-step points (BBDFO(6)), for the integration of first-order ordinary differential equations (ODEs) that exhibit stiffness, is proposed. The order, consistency and stability properties of the method are discussed, and the method is found to be zero stable and consistent. Hence, the method is convergent. The numerical comparisons with the existing methods of a similar type are given to demonstrate the accuracy of the derived method.

scholarly journals Block Milne’s Implementation For Solving Fourth Order Ordinary Differential Equations

2018 ◽  
Vol 8 (3) ◽  
pp. 2943-2948
Author(s):  
J. G. Oghonyon ◽  
S. A. Okunuga ◽  
K. S. Eke ◽  
O. A. Odetunmibi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Interpolation Polynomial ◽  
Numerical Results ◽  
Convergence Criteria ◽  
Step Size ◽  
Stiff Ordinary Differential Equations ◽  
Grid Points ◽  
Predictor Corrector

Block predictor-corrector method for solving non-stiff ordinary differential equations (ODEs) started with Milne’s device. Milne’s device is an extension of the block predictor-corrector method providing further benefits and better results. This study considers Milne’s devise for solving fourth order ODEs. A combination of Newton’s backward difference interpolation polynomial and numerical integration method are applied and integrated at some selected grid points to formulate the block predictor-corrector method. Moreover, Milne’s devise advances the computational efficiency by applying the principal local truncation error (PLTE) of the block predictor-corrector method after establishing the order. The numerical results were exhibited to attest the functioning of Milne’s devise in solving fourth order ODEs. The complete results were obtained with the aid of Mathematica 9 kernel for Microsoft Windows. Numerical results showcase that Milne’s device is more effective than existent methods in terms of design new step size, determining the convergence criteria and maximizing errors at all examined convergence levels.

One Step Cosine–Taylorlike Method For Solving Stiff Equations

2012 ◽  
Author(s):  
Rokiah @ Rozita Ahmad ◽  
Nazeeruddin Yaacob
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Explicit Method ◽  
Stiff Ordinary Differential Equations ◽  
One Step ◽  
Runge Kutta ◽  
New Formula ◽  
Better Than

Makalah ini membincangkan penghasilan kaedah tak tersirat bak Cosine–Taylor untuk menyelesaikan persamaan pembezaan biasa kaku. Perumusannya menghasilkan pengenalan kepada satu rumus baru bagi penyelesaian berangka bagi persamaan pembezaan biasa kaku. Kaedah baru ini memerlukan penghitungan tambahan yakni melakukan beberapa terbitan bagi fungsi yang terlibat. Walau bagaimanapun, keputusan yang diperoleh adalah lebih baik berbanding hasil yang didapati apabila menggunakan kaedah tak tersirat Runge–Kutta peringkat–4 dan kaedah tersirat Adam–Bashfiorth–Moulton (ABM). Perbandingan yang dibuat dengan kaedah bak Sine–Taylor menunjukkan kejituan bagi kedua–dua kaedah adalah hampir setara. Kata kunci: Kaedah tak tersirat; persamaan pembezaan biasa kaku; Runge–Kutta; kaedah tersirat; Adam–Bashforth–Moulton; bak Sine–Taylor This paper discusses the derivation of an explicit Cosine–Taylorlike method for solving stiff ordinary differential equations. The formulation has resulted in the introduction of a new formula for the numerical solution of stiff ordinary differential equations. This new method needs an extra work in order to solve a number of differentiations of the function involved. However, the result produced is better than the results from the explicit classical fourth–order Runge–Kutta (RK4) and the implicit Adam–Bashforth–Moulton (ABM) methods. When compared with the previously derived Sine–Taylorlike method, the accuracy for both methods is almost equivalent. Key words: Explicit method; stiff ordinary differential equations; Runge–Kutta; implicit method; Adam–Bashforth–Moulton; Sine–Taylorlike

scholarly journals Diagonally Implicit Block Backward Differentiation Formulas for Solving Ordinary Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
I. S. M. Zawawi ◽  
Z. B. Ibrahim ◽  
F. Ismail ◽  
Z. A. Majid
Keyword(s):  
Differential Equations ◽  
Fixed Points ◽  
Ordinary Differential Equations ◽  
Initial Value Problem ◽  
Initial Value ◽  
First Order ◽  
Backward Differentiation Formulas ◽  
The Stability ◽  
Differentiation Formulas ◽  
A Performance

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.

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