scholarly journals A New L-Stable Third Derivative Hybrid Method for Solving First Order Ordinary Differential Equations

2021 ◽  
pp. 58-69
Author(s):  
Lawrence Osa Adoghe
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hybrid Method ◽  
Multistep Method ◽  
Stability And Convergence ◽  
Stiff Systems ◽  
Convergence Properties ◽  
Main Method ◽  
The Stability ◽  
Third Derivative

In this paper, an L-stable third derivative multistep method has been proposed for the solution of stiff systems of ordinary differential equations. The continuous hybrid method is derived using interpolation and collocation techniques of power series as the basis function for the approximate solution. The method consists of the main method and an additional method which are combined to form a block matrix and implemented simultaneously. The stability and convergence properties of the block were investigated and discussed. Numerical examples to show the efficiency and accuracy of the new method were presented.

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.

scholarly journals 3-Point Block Methods for Direct Integration of General Second-Order Ordinary Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
J. O. Ehigie ◽  
S. A. Okunuga ◽  
A. B. Sofoluwe
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Absolute Stability ◽  
Second Order ◽  
Stability And Convergence ◽  
Block Methods ◽  
Good Region ◽  
The Stability ◽  
The Individual ◽  
Region Of Absolute Stability

A Multistep collocation techniques is used in this paper to develop a 3-point explicit and implicit block methods, which are suitable for generating solutions of the general second-order ordinary differential equations of the form . The derivation of both explicit and implicit block schemes is given for the purpose of comparison of results. The Stability and Convergence of the individual methods of the block schemes are investigated, and the methods are found to be 0-stable with good region of absolute stability. The 3-point block schemes derived are tested on standard mechanical problems, and it is shown that the implicit block methods are superior to the explicit ones in terms of accuracy.

scholarly journals Direct Two-Point Block One-Step Method for Solving General Second-Order Ordinary Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Zanariah Abdul Majid ◽  
Nur Zahidah Mokhtar ◽  
Mohamed Suleiman
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multistep Method ◽  
Second Order ◽  
Variable Step Size ◽  
Step Size ◽  
Step Method ◽  
One Step ◽  
The Stability ◽  
The One

A direct two-point block one-step method for solving general second-order ordinary differential equations (ODEs) directly is presented in this paper. The one-step block method will solve the second-order ODEs without reducing to first-order equations. The direct solutions of the general second-order ODEs will be calculated at two points simultaneously using variable step size. The method is formulated using the linear multistep method, but the new method possesses the desirable feature of the one-step method. The implementation is based on the predictor and corrector formulas in thePE(CE)mmode. The stability and precision of this method will also be analyzed and deliberated. Numerical results are given to show the efficiency of the proposed method and will be compared with the existing method.

scholarly journals Explicit Fourth-Derivative Two-Step Linear Multistep Method for Ordinary Differential Equations

2017 ◽  
Vol 4 (1) ◽  
Author(s):  
Ashiribo Wusu ◽  
Moses Akanbi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Single Step ◽  
Multistep Method ◽  
Linear Multistep Method ◽  
Convergence Properties ◽  
Step Method ◽  
Science And Engineering ◽  
Linear Difference Operator ◽  
Fourth Derivative

Many problems from science and engineering are modeled by Ordinary Differential Equations (ODEs) whose solutions describe the temporal evolution of the modeled processes. In most cases however, the arising equations are too complex to be solved analytically. Consequently, their solutions have to be approximated by numerical methods. In this article, we propose an explicit fourth-derivative two-step linear multistep method (FD2LMM) for ordinary differential equations. The proposed method is constructed by using the maximal order criteria which is obtained through the associated linear difference operator. The starting values used by the proposed method are obtained by suitable single-step method. The order, consistency, linear stability, and the convergence properties of the method are discussed. Numerical experiments are performed and the results are compared with those of existing methods in the 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.

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