Developing Fourth Order Block Backward Differentiation Formulas for Solving Second Order Ordinary Differential Equations Directly

2013 ◽  
Vol 19 (8) ◽  
pp. 2481-2485 ◽  
Author(s):  
Z. B. Ibrahim ◽  
N. Zainuddin ◽  
M. B. Suleiman ◽  
K. I. Othman
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Second Order ◽  
Backward Differentiation Formulas ◽  
Differentiation Formulas

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 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 Block Backward Differentiation Formulas for solving second order Fuzzy Differential Equations

MATEMATIKA ◽  
2017 ◽  
Vol 33 (2) ◽  
pp. 215
Author(s):  
Tiaw Kah Fook ◽  
Zarina Bibi Ibrahim
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Numerical Method ◽  
Newton Iteration ◽  
Higher Order ◽  
Second Order ◽  
Numerical Results ◽  
Numerical Examples ◽  
Backward Differentiation Formulas ◽  
Differentiation Formulas

In this paper, we study the numerical method for solving second order Fuzzy Differential Equations (FDEs) using Block Backward Differential Formulas (BBDF) under generalized concept of higher-order fuzzy differentiability. Implementation of the method using Newton iteration is discussed. Numerical results obtained by BBDF are presented and compared with Backward Differential Formulas (BDF) and exact solutions. Several numerical examples are provided to illustrate our methods.

scholarly journals A Note on the Integration of Scalar Fourth-Order Ordinary Differential Equations with Four-Dimensional Symmetry Algebras

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Said Waqas Shah ◽  
F. M. Mahomed ◽  
H. Azad
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Second Order ◽  
Two Dimensional ◽  
Lie Point Symmetries ◽  
Complete Integration ◽  
Symmetry Algebras

The complete integration of scalar fourth-order ODEs with four-dimensional symmetry algebras is performed by utilizing Lie’s method which was invoked to integrate scalar second-order ODEs admitting two-dimensional symmetry algebras. We obtain a complete integration of all scalar fourth-order ODEs that possess four Lie point symmetries.

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.

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