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.

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 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 A Trigonometrically Fitted Block Method for Solving Oscillatory Second-Order Initial Value Problems and Hamiltonian Systems

2017 ◽  
Vol 2017 ◽  
pp. 1-14 ◽  
Author(s):  
F. F. Ngwane ◽  
S. N. Jator
Keyword(s):  
Hamiltonian Systems ◽  
Initial Value Problems ◽  
Grid Point ◽  
Second Order ◽  
Initial Value ◽  
Step Size ◽  
Energy Conserving ◽  
One Step ◽  
The Stability ◽  
Predictor Corrector

In this paper, we present a block hybrid trigonometrically fitted Runge-Kutta-Nyström method (BHTRKNM), whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including Hamiltonian systems such as the energy conserving equations and systems arising from the semidiscretization of partial differential equations (PDEs). Four discrete hybrid formulas used to formulate the BHTRKNM are provided by a continuous one-step hybrid trigonometrically fitted method with an off-grid point. We implement BHTRKNM in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods. The stability property of the BHTRKNM is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.

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]].

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.

scholarly journals Efficient k-Step Linear Block Methods to Solve Second Order Initial Value Problems Directly

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1752
Author(s):  
Higinio Ramos ◽  
Samuel N. Jator ◽  
Mark I. Modebei
Keyword(s):  
Computational Efficiency ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Computing Time ◽  
Approximate Solutions ◽  
Second Order ◽  
Initial Value ◽  
Special Equation ◽  
Block Methods ◽  
Grid Points

There are dozens of block methods in literature intended for solving second order initial-value problems. This article aimed at the analysis of the efficiency of k-step block methods for directly solving general second-order initial-value problems. Each of these methods consists of a set of 2k multi-step formulas (although we will see that this number can be reduced to k+1 in case of a special equation) that provides approximate solutions at k grid points at once. The usual way to obtain these formulas is by using collocation and interpolation at different points, which are not all necessarily in the mesh (it may also be considered intra-step or off-step points). An important issue is that for each k, all of them are essentially the same method, although they can adopt different formulations. Nevertheless, the performance of those formulations is not the same. The analysis of the methods presented give some clues as how to select the most appropriate ones in terms of computational efficiency. The numerical experiments show that using the proposed formulations, the computing time can be reduced to less than half.

scholarly journals Three-Step Block Method for Solving Nonlinear Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Phang Pei See ◽  
Zanariah Abdul Majid ◽  
Mohamed Suleiman
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Second Order ◽  
Multiple Shooting ◽  
Variable Step Size ◽  
Nonlinear Boundary Value Problems ◽  
Block Method ◽  
Step Size ◽  
Multiple Shooting Techniques

We propose a three-step block method of Adam’s type to solve nonlinear second-order two-point boundary value problems of Dirichlet type and Neumann type directly. We also extend this method to solve the system of second-order boundary value problems which have the same or different two boundary conditions. The method will be implemented in predictor corrector mode and obtain the approximate solutions at three points simultaneously using variable step size strategy. The proposed block method will be adapted with multiple shooting techniques via the three-step iterative method. The boundary value problem will be solved without reducing to first-order equations. The numerical results are presented to demonstrate the effectiveness of the proposed method.

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.

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