Numerical solution of fourth-order stiff ordinary differential equations by implicit high order block backward differentiation formula

2021 ◽  
Author(s):  
Asma Izzati Asnor ◽  
Siti Ainor Mohd Yatim ◽  
Zarina Bibi Ibrahim
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
High Order ◽  
Differentiation Formula ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formula

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.

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 Uniform Order Eleven of Eight-Step Hybrid Block Backward Differentiation Formulae for the Solution of Stiff Ordinary Differential Equations

2021 ◽  
Vol 09 (08) ◽  
Author(s):  
Pius Tumba ◽  
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Grid Point ◽  
Stiff Systems ◽  
Differentiation Formula ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formula ◽  
First Derivative ◽  
Backward Differentiation Formulae ◽  
Continuous Formulation

In this research, we developed a uniform order eleven of eight step Second derivative hybrid block backward differentiation formula for integration of stiff systems in ordinary differential equations. The single continuous formulation developed is evaluated at some grid point of x=x_(n+j),j=0,1,2,3,4,5 and6 and its first derivative was also evaluated at off-grid point x=x_(n+j),j=15/2 and grid point x=x_(n+j),j=8. The method is suitable for the solution of stiff ordinary differential equations and the accuracy and stability properties of the newly constructed method are investigated and are shown to be A-stable. Our numerical results obtained are compared with the theoretical solutions as well as ODE23 solver.

scholarly journals A Numerical Algorithm for Solving Stiff Ordinary Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
S. A. M. Yatim ◽  
Z. B. Ibrahim ◽  
K. I. Othman ◽  
M. B. Suleiman
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Computation Time ◽  
Test Problems ◽  
Variable Order ◽  
Step Size ◽  
Differentiation Formula ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formula ◽  
The Stability

An advanced method using block backward differentiation formula (BBDF) is introduced with efficient strategy in choosing the step size and order of the method. Variable step and variable order block backward differentiation formula (VSVO-BBDF) approach is applied throughout the numerical computation. The stability regions of the VSVO-BBDF method are investigated and presented in distinct graphs. The improved performances in terms of accuracy and computation time are presented in the numerical results with different sets of test problems. Comparisons are made between the proposed method and MATLAB’s suite of ordinary differential equations (ODEs) solvers, namely, ode15s and ode23s.

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

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

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