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 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 Diagonally Implicit Block Backward Differentiation Formula with Optimal Stability Properties for Stiff Ordinary Differential Equations

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1342 ◽  
Author(s):  
Hazizah Mohd Ijam ◽  
Zarina Bibi Ibrahim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Triangular Matrix ◽  
Differentiation Formula ◽  
Stability Properties ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formula ◽  
Optimal Stability ◽  
Best Value ◽  
Selection Of

This paper aims to select the best value of the parameter ρ from a general set of linear multistep formulae which have the potential for efficient implementation. The ρ -Diagonally Implicit Block Backward Differentiation Formula ( ρ -DIBBDF) was proposed to approximate the solution for stiff Ordinary Differential Equations (ODEs) to achieve the research objective. The selection of ρ for optimal stability properties in terms of zero stability, absolute stability, error constant and convergence are discussed. In the diagonally implicit formula that uses a lower triangular matrix with identical diagonal entries, allowing a maximum of one lower-upper (LU) decomposition per integration stage to be performed will result in substantial computing benefits to the user. The numerical results and plots of accuracy indicate that the ρ -DIBBDF method performs better than the existing fully and diagonally Block Backward Differentiation Formula (BBDF) methods.

scholarly journals A Quantitative Comparison of Numerical Method for Solving Stiff Ordinary Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
S. A. M. Yatim ◽  
Z. B. Ibrahim ◽  
K. I. Othman ◽  
M. B. Suleiman
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Computation Time ◽  
Quantitative Comparison ◽  
Initial Value ◽  
Step Size ◽  
Stiff Ordinary Differential Equations ◽  
Block Methods ◽  
Variable Step

We derive a variable step of the implicit block methods based on the backward differentiation formulae (BDF) for solving stiff initial value problems (IVPs). A simplified strategy in controlling the step size is proposed with the aim of optimizing the performance in terms of precision and computation time. The numerical results obtained support the enhancement of the method proposed as compared to MATLAB's suite of ordinary differential equations (ODEs) solvers, namely, ode15s and ode23s.

scholarly journals Qualitative behavior of stiff ODEs through a stochastic approach

2020 ◽  
Vol 10 (2) ◽  
pp. 181-187
Author(s):  
Hande Uslu ◽  
Murat Sari ◽  
Tahir Cosgun
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Real Life ◽  
Stochastic Approach ◽  
Computational Time ◽  
Qualitative Behavior ◽  
Computational Error ◽  
Linear Ordinary Differential Equations ◽  
Stiff Ordinary Differential Equations ◽  
Simulation Techniques

In the last few decades, stiff differential equations have attracted a great deal of interest from academic society, because much of the real life is covered by stiff behavior. In addition to importance of producing model equations, capturing an exact behavior of the problem by dealing with a solution method is also handling issue. Although there are many explicit and implicit numerical methods for solving them, those methods cannot be properly applied due to their computational time, computational error or effort spent for construction of a structure. Therefore, simulation techniques can be taken into account in capturing the stiff behavior. In this respect, this study aims at analyzing stiff processes through stochastic approaches. Thus, a Monte Carlo based algorithm has been presented for solving some stiff ordinary differential equations and system of stiff linear ordinary differential equations. The produced results have been qualitatively and quantitatively discussed.

scholarly journals BBDF-Alpha for solving stiff ordinary differential equations with oscillating solutions

2020 ◽  
Vol 51 (2) ◽  
pp. 123-136
Author(s):  
Iskandar Shah Mohd Zawawi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Experiments ◽  
Stability Region ◽  
Newton Iteration ◽  
Computational Time ◽  
First Order ◽  
Stiff Ordinary Differential Equations ◽  
Oscillating Solutions ◽  
The Stability

In this paper, the block backward differentiation α formulas (BBDF-α) is derived for solving first order stiff ordinary differential equations with oscillating solutions. The consistency and zero stability conditions are investigated to prove the convergence of the method. The stability region in the entire negative half plane shows that the derived method is A-stable for certain values of α. The implementation of the method using Newton iteration is also discussed. Several numerical experiments are conducted to demonstrate the performance of the method in terms of accuracy and computational time.

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