A NEW CLASS OF L-STABLE HYBRID ONE-STEP METHODS FOR THE NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS

2015 ◽  
Vol 1 (2) ◽  
pp. 39-44
Author(s):  
Mohammad Mehdizadeh ◽  
Maryam Molayi
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Hybrid Methods ◽  
Akzo Nobel ◽  
New Class ◽  
Numerical Studies ◽  
One Step ◽  
Judicious Choice ◽  
Stable Hybrid

In this paper, a class of one-step hybrid methods for the numerical solution of ordinary differential equations (ODEs) are considered. The accuracy and stability properties of these methods are investigated. By judicious choice of the coefficients in these formulae a class of method is derived which is shown to be L-stable and so is appropriate for the solution of certain ordinary differential and stiff differential equations. We apply the new method for numerical integration of some famous stiff chemical problems such chemical Akzo-Nobel problem, ROBER problem (suggested by Robertson) and some others which are very popular in numerical studies.

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 L-Stable Block Hybrid Numerical Algorithm for First-Order Ordinary Differential Equations

2020 ◽  
pp. 160-165
Author(s):  
B. I. Akinnukawe ◽  
K. O. Muka
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hybrid Methods ◽  
Initial Conditions ◽  
Multistep Method ◽  
Stability And Convergence ◽  
First Order ◽  
One Step ◽  
Grid Points ◽  
The Stability

In this work, a one-step L-stable Block Hybrid Multistep Method (BHMM) of order five was developed. The method is constructed for solving first order Ordinary Differential Equations with given initial conditions. Interpolation and collocation techniques, with power series as a basis function, are employed for the derivation of the continuous form of the hybrid methods. The discrete scheme and its second derivative are derived by evaluating at the specific grid and off-grid points to form the main and additional methods respectively. Both hybrid methods generated are composed in matrix form and implemented as a block method. The stability and convergence properties of BHMM are discussed and presented. The numerical results of BHMM have proven its efficiency when compared to some existing methods.

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