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.

DERIVATION OF 2-POINT ZERO STABLENUMERICAL ALGORITHM OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR SOLVING FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 5 (2) ◽  
pp. 579-583
Author(s):  
Muhammad Abdullahi ◽  
Bashir Sule ◽  
Mustapha Isyaku
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Taylor Series ◽  
Taylor Series Expansion ◽  
Order Ordinary Differential Equation ◽  
Differentiation Formula ◽  
First Order ◽  
Backward Differentiation Formula

This paper is aimed at deriving a 2-point zero stable numerical algorithm of block backward differentiation formula using Taylor series expansion, for solving first order ordinary differential equation. The order and zero stability of the method are investigated and the derived method is found to be zero stable and of order 3. Hence, the method is suitable for solving first order ordinary differential equation. Implementation of the method has been considered

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.

scholarly journals A NEW SUPERCLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR STIFF ORDINARY DIFFERENTIAL EQUATIONS

2014 ◽  
Vol 07 (01) ◽  
pp. 1350034 ◽  
Author(s):  
M. B. Suleiman ◽  
H. Musa ◽  
F. Ismail ◽  
N. Senu ◽  
Z. B. Ibrahim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Results ◽  
New Method ◽  
Differentiation Formula ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formula ◽  
Error Constant

A superclass of block backward differentiation formula (BBDF) suitable for solving stiff ordinary differential equations is developed. The method is of order 3, with smaller error constant than the conventional BBDF. It is A-stable and generates two points at each step of the integration. A comparison is made between the new method, the 2-point block backward differentiation formula (2BBDF) and 1-point backward differentiation formula (1BDF). The numerical results show that the method developed outperformed the 2BBDF and 1BDF methods in terms of accuracy. It also reduces the integration steps when compared with the 1BDF method.

ORDER AND CONVERGENCE OF THE ENHANCED 3-POINT FULLY IMPLICIT SUPER CLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR SOLVING INITIAL VALUE PROBLEM

2021 ◽  
Vol 5 (2) ◽  
pp. 442-446
Author(s):  
Muhammad Abdullahi ◽  
Hamisu Musa
Keyword(s):  
Initial Value Problem ◽  
Initial Value Problems ◽  
Sufficient Conditions ◽  
Initial Value ◽  
Differentiation Formula ◽  
Stiff Initial Value Problems ◽  
First Order ◽  
Backward Differentiation Formula ◽  
Fully Implicit ◽  
Necessary And Sufficient

This paper studied an enhanced 3-point fully implicit super class of block backward differentiation formula for solving stiff initial value problems developed by Abdullahi & Musa and go further to established the necessary and sufficient conditions for the convergence of the method. The method is zero stable, A-stable and it is of order 5. The method is found to be suitable for solving first order stiff initial value problems

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 Singly Diagonally Implicit Block Backward Differentiation Formulas for HIV Infection of CD4+T Cells

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 625
Author(s):  
Saufianim Jana Aksah ◽  
Zarina Bibi Ibrahim
Keyword(s):  
T Cells ◽  
Hiv Infection ◽  
Approximate Solutions ◽  
Infection Model ◽  
Triangular Form ◽  
Differentiation Formula ◽  
Fully Implicit ◽  
Backward Differentiation Formulas ◽  
Hiv Infection Model ◽  
Differentiation Formulas

In this study, a singly diagonally implicit block backward differentiation formula (SDIBBDF) was proposed to approximate solutions for a dynamical HIV infection model of CD 4 + T cells. A SDIBBDF method was developed to overcome difficulty when implementing the fully implicit method by deriving the proposed method in lower triangular form with equal diagonal coefficients. A comparative analysis between the proposed method, BBDF, classical Euler, fourth-order Runge-Kutta (RK4) method, and a Matlab solver was conducted. The numerical results proved that the SDIBBDF method was more efficient in solving the model than the methods to be compared.

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