scholarly journals Multistage Bernstein collocation method for solving strongly nonlinear damped systems

2018 ◽  
Vol 25 (1) ◽  
pp. 122-131
Author(s):  
Ahmad Sami Bataineh ◽  
Abd Alkreem Al-Omari ◽  
Osman Rasit Isik ◽  
Ishak Hashim
Keyword(s):  
Collocation Method ◽  
Solution Method ◽  
Approximate Solutions ◽  
Step Function ◽  
Kutta Method ◽  
External Excitation ◽  
Strongly Nonlinear ◽  
Runge Kutta Method ◽  
Damped Systems ◽  
Approximate Solution Method

In this paper, we propose an approximate solution method, called multistage Bernstein collocation method (MBCM), to solve strongly nonlinear damped systems. The method is given with an error analysis. The systems investigated include step function external excitation and periodical function external excitation. We found that the MBCM gives more accurate approximate solutions than the standard collocation method. Since MBCM splits the domain of the problem [Formula: see text] into n parts, one can overcome oscillations arising from the standard collocation method by increasing n. The results obtained from MBCM compare favorably with that of the fourth-order Runge–Kutta method (RK4).

scholarly journals Analytical Study of Fractional-Order Multiple Chaotic FitzHugh-Nagumo Neurons Model Using Multistep Generalized Differential Transform Method

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Shaher Momani ◽  
Asad Freihat ◽  
Mohammed AL-Smadi
Keyword(s):  
Fractional Order ◽  
Fractional Derivatives ◽  
Analytical Study ◽  
Approximate Solutions ◽  
Differential Transform Method ◽  
Kutta Method ◽  
Transform Method ◽  
Runge Kutta Method ◽  
Differential Transform ◽  
Generalized Differential

The multistep generalized differential transform method is applied to solve the fractional-order multiple chaotic FitzHugh-Nagumo (FHN) neurons model. The algorithm is illustrated by studying the dynamics of three coupled chaotic FHN neurons equations with different gap junctions under external electrical stimulation. The fractional derivatives are described in the Caputo sense. Furthermore, we present figurative comparisons between the proposed scheme and the classical fourth-order Runge-Kutta method to demonstrate the accuracy and applicability of this method. The graphical results reveal that only few terms are required to deduce the approximate solutions which are found to be accurate and efficient.

An Optimal Homotopy Asymptotic Approach to a Damped Dynamical System of a Rotating Electrical Machine

2015 ◽  
Vol 801 ◽  
pp. 202-206 ◽  
Author(s):  
Nicolae Herisanu ◽  
Vasile Marinca
Keyword(s):  
Dynamical System ◽  
Analytical Approach ◽  
Conservative System ◽  
Electrical Machine ◽  
Kutta Method ◽  
Asymptotic Approach ◽  
Runge Kutta Method ◽  
Optimal Homotopy Asymptotic Method ◽  
Damped Systems ◽  
Specific Working

In this paper, the non-conservative system of a rotating electrical machine is analytically investigated by means of an effective analytical approach, namely the Optimal Homotopy Asymptotic Method (OHAM). Besides analytical developments, in order to prove the efficiency and accuracy of the proposed approach, numerical simulations are performed for a specific working regime. Comparison between approximate analytical results obtained by OHAM and numerical integration results obtained by a fourth-order Runge-Kutta method emphasize that OHAM is reliable and easy to use in finding analytical solutions to damped systems.

scholarly journals Accurate Numerical Method for Singular Initial-Value Problems

2021 ◽  
Vol 08 (03) ◽  
pp. 01-11
Author(s):  
Tesfaye Aga Bullo ◽  
Gemechis File Duressa ◽  
Gashu Gadisa Kiltu
Keyword(s):  
Numerical Method ◽  
Initial Value Problems ◽  
Approximate Solutions ◽  
Stability And Convergence ◽  
Order System ◽  
Kutta Method ◽  
Initial Value ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Fifth Order

In this paper, an accurate numerical method is presented to find the numerical solution of the singular initial value problems. The second-order singular initial value problem under consideration is transferred into a first-order system of initial value problems, and then it can be solved by using the fifth-order Runge Kutta method. The stability and convergence analysis is studied. The effectiveness of the proposed methods is confirmed by solving three model examples, and the obtained approximate solutions are compared with the existing methods in the literature. Thus, the fifth-order Runge-Kutta method is an accurate numerical method for solving the singular initial value problems.

An approach for approximate solution of fractional-order smoking model with relapse class

2018 ◽  
Vol 11 (06) ◽  
pp. 1850077 ◽  
Author(s):  
Anwar Zeb ◽  
Vedat Suat Erturk ◽  
Umar Khan ◽  
Gul Zaman ◽  
Shaher Momani
Keyword(s):  
Approximate Solution ◽  
Comparative Study ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Approximate Solutions ◽  
Kutta Method ◽  
Unique Positive Solution ◽  
Runge Kutta Method ◽  
Proposed Model ◽  
Definition Of

In this paper, we develop a fractional-order smoking model by considering relapse class. First, we formulate the model and find the unique positive solution for the proposed model. Then we apply the Grünwald–Letnikov approximation in the place of maintaining a general quadrature formula approach to the Riemann–Liouville integral definition of the fractional derivative. Building on this foundation avoids the need for domain transformations, contour integration or involved theory to compute accurate approximate solutions of fractional-order giving up smoking model. A comparative study between Grünwald–Letnikov method and Runge–Kutta method is presented in the case of integer-order derivative. Finally, we present the obtained results graphically.

Implicit Order 3/2 Strong Method for Numerical Solutions of Stratonovich Stochastic Differential Equations

2019 ◽  
Vol 16 (8) ◽  
pp. 3137-3140
Author(s):  
Yazid Alhojilan
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Approximation Method ◽  
Taylor Expansion ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Higher Order ◽  
Implicit Method ◽  
Kutta Method ◽  
Runge Kutta Method

Due to that the explicit methods in solving stochastic differential equations give instability and inaccurate results, the aim of this paper is to derive an effective implicit method gives higher-order approximate solutions for a stiff stochastic differential equations by using Runge-Kutta method. It relies on the Stratonovich-Taylor expansion and uses the notion of perturbation and coupling to carry out the method. The validity of this new approximation method is shown by implementing in MATLAB and, showing the convergence of the method graphically.

scholarly journals Numerical Treatments for Volterra Delay Integro-differential Equations

2009 ◽  
Vol 9 (3) ◽  
pp. 292-318 ◽  
Author(s):  
F. A. Rihan ◽  
E. H. Doha ◽  
M. I. Hassan ◽  
N. M. Kamel
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Quadrature Rule ◽  
Kutta Method ◽  
Physical Sciences ◽  
Stability Properties ◽  
Runge Kutta Method ◽  
A New Technique ◽  
Numerical Treatments ◽  
Differential Part

AbstractThis paper presents a new technique for numerical treatments of Volterra delay integro-differential equations that have many applications in biological and physical sciences. The technique is based on the mono-implicit Runge — Kutta method for treating the differential part and the collocation method (using Boole’s quadrature rule) for treating the integral part. The efficiency and stability properties of this technique have been studied. Numerical results are presented to demonstrate the effectiveness of the methodology.

scholarly journals Some Novel Complex Dynamic Behaviors of a Class of Four-Dimensional Chaotic or Hyperchaotic Systems Based on a Meshless Collocation Method

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-15
Author(s):  
Du Mingjing ◽  
Yulan Wang
Keyword(s):  
Collocation Method ◽  
Fourth Order ◽  
Hyperchaotic System ◽  
Kutta Method ◽  
New Approach ◽  
Runge Kutta Method ◽  
Dynamical Behaviors ◽  
Meshless Collocation ◽  
Runge Kutta ◽  
Hyperchaotic Systems

In the field of complex systems, there is a need for better methods of knowledge discovery due to their nonlinear dynamics. The numerical simulation of chaotic or hyperchaotic system is mainly performed by the fourth-order Runge–Kutta method, and other methods are rarely reported in previous work. A new method, which divides the entire intervals into N equal subintervals based on a meshless collocation method, has been constructed in this paper. Some new complex dynamical behaviors are shown by using this new approach, and the results are in good agreement with those obtained by the fourth-order Runge–Kutta method.

On the Harmonic Oscillations for the Motion of a Dynamical System

2017 ◽  
Vol 13 (2) ◽  
pp. 4657-4670
Author(s):  
W. S. Amer
Keyword(s):  
Dynamical System ◽  
Numerical Solutions ◽  
Equation Of Motion ◽  
Parameter Method ◽  
Kutta Method ◽  
Small Parameter Method ◽  
Harmonic Oscillations ◽  
Pivot Point ◽  
Runge Kutta Method ◽  
High Consistency

This work touches two important cases for the motion of a pendulum called Sub and Ultra-harmonic cases. The small parameter method is used to obtain the approximate analytic periodic solutions of the equation of motion when the pivot point of the pendulum moves in an elliptic path. Moreover, the fourth order Runge-Kutta method is used to investigate the numerical solutions of the considered model. The comparison between both the analytical solution and the numerical ones shows high consistency between them.

Sign in / Sign up

Export Citation Format

Share Document