Numerical simulation of second-order initial-value problems using a new class of variable coefficients and two-step semi-hybrid methods

SIMULATION ◽  
2021 ◽  
pp. 003754972098082
Author(s):  
Ali Shokri ◽  
Mohammad Mehdizadeh Khalsaraei ◽  
Hamid Mohammad-Sedighi ◽  
Ali Atashyar
Keyword(s):  
Numerical Simulation ◽  
Initial Value Problems ◽  
Hybrid Methods ◽  
Variable Coefficients ◽  
Second Order ◽  
Initial Value ◽  
Hybrid Schemes ◽  
New Class ◽  
New Family ◽  
Algebraic Order

In this paper, a new family of two-step semi-hybrid schemes of the 12th algebraic order is proposed for the numerical simulation of initial-value problems of second-order ordinary differential equations. The proposed methods are symmetric and belong to the family of multiderivative methods. Each method of the new family appears to be hybrid, but after implementing the hybrid terms, it will continue as a multiderivative method. Therefore, the designation semi-hybrid is used. The consistency, convergence, stability, and periodicity of the methods are investigated and analyzed. In order to show the accuracy, consistency, convergence, and stability of the proposed family, it was tested on some well-known problems, such as the undamped Duffing’s equation. The simulation results demonstrate the efficiency and advantages of the proposed method compared to the currently available methods.

scholarly journals Direct Solution of Second Order Ordinary Differential Equations Using a Class of Hybrid Block Methods

2020 ◽  
Vol 5 (2) ◽  
Author(s):  
Emmanuel A Areo ◽  
Nosimot O Adeyanju ◽  
Sunday J Kayode
Keyword(s):  
Initial Value Problems ◽  
Hybrid Methods ◽  
Multistep Method ◽  
Second Order ◽  
Block Method ◽  
Initial Value ◽  
Order Error ◽  
Block Methods ◽  
Grid Points ◽  
Step Number

This research proposes the derivation of a class of hybrid methods for solution of second order initial value problems (IVPs) in block mode. Continuous linear multistep method of two cases with step number k = 4 is developed by interpolating the basis function at certain grid points and collocating the differential system at both grid and off-grid points. The basic properties of the method including order, error constant, zero stability, consistency and convergence were investigated. In order to examine the accuracy of the methods, some differential problems of order two were solved and results generated show a better performance when comparison is made with some current methods.Keywords- Block Method, Hybrid Points, Initial Value Problems, Power Series, Second Order 

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 Optimized Hybrid Methods for Solving Oscillatory Second Order Initial Value Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
N. Senu ◽  
F. Ismail ◽  
S. Z. Ahmad ◽  
M. Suleiman
Keyword(s):  
Hybrid Method ◽  
Initial Value Problems ◽  
Scientific Literature ◽  
Hybrid Methods ◽  
Second Order ◽  
Numerical Results ◽  
Phase Lag ◽  
Initial Value ◽  
Oscillatory Initial Value Problems

Two-step optimized hybrid methods of order five and order six are developed for the integration of second order oscillatory initial value problems. The optimized hybrid method (OHMs) are based on the existing nonzero dissipative hybrid methods. Phase-lag, dissipation or amplification error, and the differentiation of the phase-lag relations are required to obtain the methods. Phase-fitted methods based on the same nonzero dissipative hybrid methods are also constructed. Numerical results show that OHMs are more accurate compared to the phase-fitted methods and some well-known methods appeared in the scientific literature in solving oscillating second order initial value problems. It is also found that the nonzero dissipative hybrid methods are more suitable to be optimized than phase-fitted methods.

scholarly journals A New Family of Phase-Fitted and Amplification-Fitted Runge-Kutta Type Methods for Oscillators

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Zhaoxia Chen ◽  
Xiong You ◽  
Xin Shu ◽  
Mei Zhang
Keyword(s):  
Initial Value Problems ◽  
Numerical Experiments ◽  
Order Conditions ◽  
Initial Value ◽  
Oscillatory Solutions ◽  
New Methods ◽  
New Family ◽  
Algebraic Order ◽  
Runge Kutta ◽  
Phase Fitting

In order to solve initial value problems of differential equations with oscillatory solutions, this paper improves traditional Runge-Kutta (RK) methods by introducing frequency-depending weights in the update. New practical RK integrators are obtained with the phase-fitting and amplification-fitting conditions and algebraic order conditions. Two of the new methods have updates that are also phase-fitted and amplification-fitted. The linear stability and phase properties of the new methods are examined. The results of numerical experiments on physical and biological problems show the robustness and competence of the new methods compared to some highly efficient integrators in the literature.

A FAMILY OF HYBRID EIGHTH ORDER METHODS WITH MINIMAL PHASE-LAG FOR THE NUMERICAL SOLUTION OF THE SCHRÖDINGER EQUATION AND RELATED PROBLEMS

2000 ◽  
Vol 11 (02) ◽  
pp. 415-437 ◽  
Author(s):  
G. AVDELAS ◽  
A. KONGUETSOF ◽  
T. E. SIMOS
Keyword(s):  
Numerical Solution ◽  
Initial Value Problems ◽  
Hybrid Methods ◽  
Phase Lag ◽  
Initial Value ◽  
Eighth Order ◽  
New Methods ◽  
Algebraic Order ◽  
Periodic Initial Value Problems ◽  
Minimal Phase

In this paper a family of hybrid methods with minimal phase-lag are developed for the numerical solution of periodic initial-value problems. The methods are of eighth algebraic order and have large intervals of periodicity. The efficiency of the new methods is presented by their application to the wave equation and to coupled differential equations of the Schrödinger type.

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.

scholarly journals A Family of Functionally-Fitted Third Derivative Block Falkner Methods for Solving Second-Order Initial-Value Problems with Oscillating Solutions

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 713
Author(s):  
Higinio Ramos ◽  
Ridwanulahi Abdulganiy ◽  
Ruth Olowe ◽  
Samuel Jator
Keyword(s):  
Initial Value Problems ◽  
Second Order ◽  
Initial Value ◽  
Oscillatory Solutions ◽  
Oscillating Solutions ◽  
Algebraic Order ◽  
Direct Numerical Solution ◽  
The Stability ◽  
Direct Numerical Integration ◽  
Third Derivative

One of the well-known schemes for the direct numerical integration of second-order initial-value problems is due to Falkner (Falkner, 1936. Phil. Mag. S. 7, 621). This paper focuses on the construction of a family of adapted block Falkner methods which are frequency dependent for the direct numerical solution of second-order initial value problems with oscillatory solutions. The techniques of collocation and interpolation are adopted here to derive the new methods. The study of the properties of the proposed adapted block Falkner methods reveals that they are consistent and zero-stable, and thus, convergent. Furthermore, the stability analysis and the algebraic order conditions of the proposed methods are established. As may be seen from the numerical results, the resulting family is efficient and competitive compared to some recent methods in the literature.

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