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.

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 Optimization of One-Step Block Method for Solving Second-Order Fuzzy Initial Value Problems

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-25
Author(s):  
Safa Al-Refai ◽  
Muhammed I. Syam ◽  
Mohammed Al-Refai
Keyword(s):  
Initial Value Problems ◽  
Second Order ◽  
Numerical Results ◽  
Stability And Convergence ◽  
Block Method ◽  
Initial Value ◽  
Convergence Results ◽  
One Step ◽  
The Stability

In this article, we present a one-step hybrid block method for approximating the solutions of second-order fuzzy initial value problems. We prove the stability and convergence results of the method and present several examples to illustrate the efficiency and accuracy of the proposed method. The numerical results are compared with the existing ones in the literature.

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 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 Numerical Simulation of One Step Block Method for Treatment of Second Order Forced Motions in Mass-Spring Systems

2021 ◽  
pp. 1-11
Author(s):  
J. Sabo ◽  
T. Y. Kyagya ◽  
W. J. Vashawa
Keyword(s):  
Numerical Simulation ◽  
Initial Value Problems ◽  
Second Order ◽  
Block Method ◽  
Initial Value ◽  
Mathematical Problems ◽  
One Step ◽  
Grid Points ◽  
The One ◽  
Mass Spring

This paper discuss the numerical simulation of one step block method for treatment of second order forced motions in mass-spring systems of initial value problems. The one step block method has been developed with the introduction of off-mesh point at both grid and off- grid points using interpolation and collocation procedure to increase computational burden which may jeopardize the accuracy of the method in terms of error. The basic properties of the one step block method was established and numerical analysis shown that the one step block method was found to be consistent, convergent and zero-stable. The one step block method was simulated on three highly stiff mathematical problems to validate the accuracy of the block method without reduction, and obviously the results shown are more accurate over the existing method in literature.

scholarly journals Functionally Fitted Block Method for Solving the General Oscillatory Second-Order Initial Value Problems and Hyperbolic Partial Differential Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-14
Author(s):  
S. N. Jator ◽  
F. F. Ngwane ◽  
N. O. Kirby
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Initial Value Problems ◽  
Second Order ◽  
Hyperbolic Partial Differential Equations ◽  
Fixed Frequency ◽  
Block Method ◽  
Initial Value ◽  
Partial Differential ◽  
Predictor Corrector

We present a block hybrid functionally fitted Runge–Kutta–Nyström method (BHFNM) which is dependent on the stepsize and a fixed frequency. Since the method is implemented in a block-by-block fashion, the method does not require starting values and predictors inherent to other predictor-corrector methods. Upon deriving our method, stability is illustrated, and it is used to numerically solve the general second-order initial value problems as well as hyperbolic partial differential equations. In doing so, we demonstrate the method’s relative accuracy and efficiency.

scholarly journals Solving the Telegraph and Oscillatory Differential Equations by a Block Hybrid Trigonometrically Fitted Algorithm

2015 ◽  
Vol 2015 ◽  
pp. 1-15 ◽  
Author(s):  
F. F. Ngwane ◽  
S. N. Jator
Keyword(s):  
Differential Equations ◽  
Stability Property ◽  
Telegraph Equation ◽  
Hyperbolic Partial Differential Equations ◽  
Initial Value ◽  
Step Size ◽  
Grid Points ◽  
The Stability ◽  
Predictor Corrector ◽  
Trigonometrically Fitted Method

We propose a block hybrid trigonometrically fitted (BHT) method, whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including systems arising from the semidiscretization of hyperbolic Partial Differential Equations (PDEs), such as the Telegraph equation. The BHT is formulated from eight discrete hybrid formulas which are provided by a continuous two-step hybrid trigonometrically fitted method with two off-grid points. The BHT is implemented 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 BHT is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.

scholarly journals Hybrid Third Derivative Block Method for the Solution of General Second Order Initial Value Problems with Generalized One Step Point

2019 ◽  
Vol 12 (3) ◽  
pp. 1199-1214
Author(s):  
Ra'ft Abdelrahim ◽  
Z. Omar ◽  
O. Ala’yed ◽  
B. Batiha
Keyword(s):  
Power Series ◽  
Convergence Analysis ◽  
Initial Value Problem ◽  
Initial Value Problems ◽  
Numerical Procedure ◽  
Second Order ◽  
Block Method ◽  
Initial Value ◽  
One Step ◽  
Third Derivative

This paper deals with two-step hybrid block method with one generalized off-step points for solving second order initial value problem. In derivation of this method, power series of order nine are interpolated at the first two step points while its second and third derivatives are collocated at all point in the selected interval. The new developed method is employed to solve several problems of second order initial value problems. Convergence analysis of the new method alongside numerical procedure is established. The performance of the proposed method is found to be more accurate than existing method available in the literature when solving the same problems.

Sign in / Sign up

Export Citation Format

Share Document