scholarly journals Half Step Numerical Method for Solution of Second Order Initial Value Problems

2021 ◽  
pp. 77-81
Author(s):  
Adeniran Adebayo O. ◽  
Edaogbogun Kikelomo
Keyword(s):  
Numerical Method ◽  
Initial Value Problems ◽  
Second Order ◽  
Variable Step Size ◽  
Initial Value ◽  
Step Size ◽  
Stable Order ◽  
Variable Step ◽  
Interpolation Technique ◽  
Order Four

This paper presents a half step numerical method for solving directly general second order initial value problems. The scheme is developed via collocation and interpolation technique invoked on power series polynomial. The proposed method is consistent, zero stable, order four and three. This method can estimate the approximate solution at both step and off step points simultaneously by using variable step size. Numerical results are given to show the efficiency of the proposed scheme over some existing schemes of same and higher order.

scholarly journals An efficient hybrid numerical scheme for solvinggeneral second order initial value problems (IVPs)

2015 ◽  
Vol 4 (2) ◽  
pp. 411 ◽  
Author(s):  
Oluwadare Adeniran ◽  
Babatunde Ogundare
Keyword(s):  
Initial Value Problems ◽  
Numerical Scheme ◽  
Second Order ◽  
Variable Step Size ◽  
Initial Value ◽  
Step Size ◽  
Interpolation Technique ◽  
One Step ◽  
Grid Points ◽  
Order Four

<p>The paper presents a one step hybrid numerical scheme with two off grid points for solving directly the general second order initial value problems of ordinary differential equations. The scheme is developed using collocation and interpolation technique. The proposed scheme is consistent, zero stable and of order four. This scheme can estimate the approximate solution at both step and off step points simultaneously by using variable step size. Numerical results are given to show the efficiency of the proposed scheme over the existing schemes.</p>

scholarly journals Bernstein polynomial induced two step hybrid numerical scheme for solution of second order initial value problems

2021 ◽  
Vol 2 (1) ◽  
pp. 15-25
Author(s):  
A. O. Adeniran ◽  
Longe Idowu O. ◽  
Edaogbogun Kikelomo
Keyword(s):  
Initial Value Problems ◽  
Numerical Scheme ◽  
Bernstein Polynomial ◽  
Grid Point ◽  
Approximate Solutions ◽  
Second Order ◽  
Variable Step Size ◽  
Initial Value ◽  
Step Size ◽  
Order Four

This paper presents a two-step hybrid numerical scheme with one off-grid point for the numerical solution of general second-order initial value problems without reducing to two systems of the first order. The scheme is developed using the collocation and interpolation technique invoked on Bernstein polynomial. The proposed scheme is consistent, zero stable, and is of order four($4$). The developed scheme can estimate the approximate solutions at both steps and off-step points simultaneously using variable step size. Numerical results obtained in this paper show the efficiency of the proposed scheme over some existing methods of the same and higher orders.

Solving Directly Second Order Initial Value Problems with Lucas Polynomial

2019 ◽  
pp. 1-7
Author(s):  
A. O. Adeniran ◽  
I. O. Longe
Keyword(s):  
Initial Value Problems ◽  
Numerical Scheme ◽  
Second Order ◽  
Numerical Implementation ◽  
Hybrid Scheme ◽  
Initial Value ◽  
Variable Step ◽  
Interpolation Technique ◽  
One Step ◽  
Order Four

Aims/ Objectives: This paper presents a one step hybrid numerical scheme with one o gridpoints for solving directly the general second order initial value problems.Study Design: Section one which is the introduction, give a brief about initial value problem.In the next section derivation of one step hybrid scheme is considered. Section Three providesthe analysis of the scheme, while numerical implementation of the scheme and conclusion are inSections four and ve respectively.Methodology: The scheme is developed using collocation and interpolation technique invokedon Lucas polynomial.Results: The proposed scheme is consistent, zero stable and of order four and can estimate theapproximate solution at both step and o step points simultaneously by using variable step size.Conclusion: Numerical results are given to show the eciency of the proposed scheme over someexisting schemes of same and higher order[ [1],[2], [3],[4], [5], [6]].

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.

scholarly journals A 4-Point Block Method for Solving Higher Order Ordinary Differential Equations Directly

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Nazreen Waeleh ◽  
Zanariah Abdul Majid
Keyword(s):  
Initial Value Problems ◽  
Adaptive Strategy ◽  
Variable Step Size ◽  
Block Method ◽  
Initial Value ◽  
Step Size ◽  
First Order ◽  
Variable Step ◽  
Improved Performance ◽  
Fifth Order

An alternative block method for solving fifth-order initial value problems (IVPs) is proposed with an adaptive strategy of implementing variable step size. The derived method is designed to compute four solutions simultaneously without reducing the problem to a system of first-order IVPs. To validate the proposed method, the consistency and zero stability are also discussed. The improved performance of the developed method is demonstrated by comparing it with the existing methods and the results showed that the 4-point block method is suitable for solving fifth-order IVPs.

L-stable Explicit Nonlinear Method with Constant and Variable Step-size Formulation for Solving Initial Value Problems

2018 ◽  
Vol 19 (7-8) ◽  
pp. 741-751 ◽  
Author(s):  
Sania Qureshi ◽  
Higinio Ramos
Keyword(s):  
Initial Value Problems ◽  
Blow Up ◽  
Singular Solutions ◽  
Variable Step Size ◽  
Explicit Method ◽  
Nonlinear Method ◽  
Initial Value ◽  
Step Size ◽  
Third Order ◽  
Variable Step

AbstractIn this work, we develop a nonlinear explicit method suitable for both autonomous and non-autonomous type of initial value problems in Ordinary Differential Equations (ODEs). The method is found to be third order accurate having L-stability. It is shown that if a variable step-size strategy is employed then the performance of the proposed method is further improved in comparison with other methods of same nature and order. The method is shown to be working well for initial value problems having singular solutions, singularly perturbed and stiff problems, and blow-up ODE problems, which is illustrated using a few numerical experiments.

scholarly journals An Order Four Continuous Numerical Method for Solving General Second Order Ordinary Differential Equations

2021 ◽  
pp. 42-47
Author(s):  
Friday Obarhua ◽  
Oluwasemire John Adegboro
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Initial Value Problems ◽  
Hybrid Methods ◽  
Second Order ◽  
Initial Value ◽  
Hybrid Numerical Method ◽  
Order Four ◽  
Finite Domains

Continuous hybrid methods are now recognized as efficient numerical methods for problems whose solutions have finite domains or cannot be solved analytically. In this work, the continuous hybrid numerical method for the solution of general second order initial value problems of ordinary differential equations is considered. The method of collocation of the differential system arising from the approximate solution to the problem is adopted using the power series as a basis function. The method is zero stable, consistent, convergent. It is suitable for both non-stiff and mildly-stiff problems and results were found to compete favorably with the existing methods in terms of accuracy.

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