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 NUMERICAL SOLUTION FOR SOLVING SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS USING BLOCK METHOD

2012 ◽  
Vol 09 ◽  
pp. 560-565 ◽  
Author(s):  
NUR ZAHIDAH MUKHTAR ◽  
ZANARIAH ABDUL MAJID ◽  
FUDZIAH ISMAIL ◽  
MOHAMED SULEIMAN
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fluid Dynamic ◽  
Electric Circuit ◽  
Second Order ◽  
Variable Step Size ◽  
Step Size ◽  
Step Method ◽  
Mathematical Problems ◽  
One Step

The purpose of this paper is to present a four point direct block one-step method for solving directly the general second order nonstiff initial value problems (IVPs) of ordinary differential equations (ODEs). The mathematical problems in real world can be written in the form of differential equations and arise in the fields of science and engineering such as fluid dynamic, electric circuit, motion of rocket or satellite and other area of application. The proposed method will estimate the approximation solutions at four points simultaneously by using variable step size. Numerical results are given to show the efficiency of the proposed method.

Variable-step-size second-order-derivative multistep method for solving first-order ordinary differential equations in system simulation

2020 ◽  
Vol 11 (01) ◽  
pp. 2050001
Author(s):  
Lei Zhang ◽  
Chaofeng Zhang ◽  
Mengya Liu
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Multistep Method ◽  
Second Order ◽  
Order Derivative ◽  
Variable Step Size ◽  
Variable Order ◽  
Step Size ◽  
Variable Step ◽  
The Stability

According to the relationship between truncation error and step size of two implicit second-order-derivative multistep formulas based on Hermite interpolation polynomial, a variable-order and variable-step-size numerical method for solving differential equations is designed. The stability properties of the formulas are discussed and the stability regions are analyzed. The deduced methods are applied to a simulation problem. The results show that the numerical method can satisfy calculation accuracy, reduce the number of calculation steps and accelerate calculation speed.

STABILITY ANALYSIS OF A PROPOSED SCHEME OF ORDER FIVE FOR FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 6 (2) ◽  
pp. 898
Author(s):  
Sunday Emmanuel Fadugba ◽  
Roseline Bosede Ogunrinde ◽  
Rowland Rotimi Ogunrinde
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Step Length ◽  
Step Method ◽  
First Order ◽  
Absolute Relative Error ◽  
One Step ◽  
Test Equation ◽  
The Stability

This paper presents the stability analysis of a proposed scheme of order five (FCM) for first order Ordinary Differential Equations (ODEs). The proposed FCM is derived by means of an interpolating function of polynomial and exponential forms. The properties of FCM were discussed extensively. The linear stability of FCM in the context of the Third Order One-Step Method (TCM) and Second Order One-Step Method (SCM) for the solution of initial value problems of first order differential equations is presented. The stability region of FCM, TCM and SCM is investigated using the Dahlquist’s test equation. The numerical results obtained via FCM are compared with TCM and SCM. Moreover, by varying the step length, the accuracy and convergence of the methods in terms of the final absolute relative error are measured. The results show that FCM converges faster and more stable than its counterparts.

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 The Treatment of Second Order Ordinary Differential Equations Using Equidistant One-step Block Hybrid

2019 ◽  
pp. 1-9
Author(s):  
Y. Skwame ◽  
J. Sabo ◽  
M. Mathew
Keyword(s):  
Numerical Analysis ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Second Order ◽  
Direct Solution ◽  
Block Method ◽  
Initial Value ◽  
One Step ◽  
Better Than

A general one-step hybrid block method with equidistant of order 6 has been successfully developed for the direct solution of second order IVPs in this article. Numerical analysis shows that the developed method is consistent and zero-stable which implies its convergence. The analysis of the new method is examined on two highly and mildly stiff second-order initial value problems to illustrate the efficiency of the method. It is obvious that our method performs better than the existing method within which we compare our result with. Hence, the approach is an adequate one for solving special second order IVPs.

scholarly journals Step Change Strategies for Multistep Methods

1985 ◽  
Vol 14 (196) ◽  
Author(s):  
Ole Østerby
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multistep Methods ◽  
Step Change ◽  
Change Strategies ◽  
Step Size ◽  
Step Method ◽  
Adams Methods

When a system of ordinary differential equations is solved using a step-by-step method it is often desirable to change the step size during the course of the integration. We show that the commonly used formulas for calculating the new step sizes are not correct for multistep methods and we derive correct formulas for Adams methods.

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.

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