scholarly journals Classical and Higher Order Runge-Kutta Methods with Nonlinear Shooting Technique for Solving Van der Pol (VdP) Equation

2018 ◽  
Vol 66 (1) ◽  
pp. 43-47
Author(s):  
Farhana Ahmed Simi ◽  
Goutam Saha
Keyword(s):  
Exact Solution ◽  
Perturbation Method ◽  
Numerical Solution ◽  
Accurate Result ◽  
Research Work ◽  
Single Step ◽  
The Other ◽  
Van Der Pol ◽  
Shooting Technique ◽  
Runge Kutta

The goal of the research work is to examine the improvement of numerical solution of VdP equation. The well-known VdP equation is governed by the second order nonlinear ODE and then solved numerically using the classical Runge-Kutta (RK) method, RK-Fehlberg method of order five, Verner method of order eight and Cash-Karp method of order six with nonlinear shooting technique. In this work, numerical simulations have been carried out using NVdP code which is written in MATHEMATICA. Also, the accuracy and efficiency of the solution of VdP equation using different RK methods with nonlinear shooting technique has been investigated. For analysis of accuracy, the approximate exact solution obtained by perturbation method is used for the comparison. It is observed that all the different RK methods give accurate result of the VdP equation. But the classical RK method shows slightly better performance than the other single step techniques. Dhaka Univ. J. Sci. 66(1): 43-47, 2018 (January)

scholarly journals Solutions of the Force-Free Duffing-van der Pol Oscillator Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Najeeb Alam Khan ◽  
Muhammad Jamil ◽  
Syed Anwar Ali ◽  
Nadeem Alam Khan
Keyword(s):  
Approximate Solution ◽  
Numerical Solution ◽  
Approximate Method ◽  
Governing Equation ◽  
Laplace Transformation ◽  
Large Time ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Alternative Framework ◽  
Runge Kutta

A new approximate method for solving the nonlinear Duffing-van der pol oscillator equation is proposed. The proposed scheme depends only on the two components of homotopy series, the Laplace transformation and, the Padé approximants. The proposed method introduces an alternative framework designed to overcome the difficulty of capturing the behavior of the solution and give a good approximation to the solution for a large time. The Runge-Kutta algorithm was used to solve the governing equation via numerical solution. Finally, to demonstrate the validity of the proposed method, the response of the oscillator, which was obtained from approximate solution, has been shown graphically and compared with that of numerical solution.

scholarly journals Implicit Methods for Numerical Solution of Singular Initial Value Problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Md. Rahaman Habibur ◽  
M. Kamrul Hasan ◽  
Md. Ayub Ali ◽  
M. Shamsul Alam
Keyword(s):  
Singular Point ◽  
Numerical Solution ◽  
Initial Value Problems ◽  
Accurate Result ◽  
Implicit Method ◽  
Implicit Methods ◽  
Initial Value ◽  
Runge Kutta

AbstractVarious order of implicit method has been formulated for solving initial value problems having an initial singular point. The method provides better result than those obtained by used implicit formulae developed based on Euler and Runge-Kutta methods. Romberg scheme has been used for obtaining more accurate result.

scholarly journals THE REPRODUCING KERNEL ALGORITHM FOR NUMERICAL SOLUTION OF VAN DER POL DAMPING MODEL IN VIEW OF THE ATANGANA–BALEANU FRACTIONAL APPROACH

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040010 ◽  
Author(s):  
SHAHER MOMANI ◽  
BANAN MAAYAH ◽  
OMAR ABU ARQUB
Keyword(s):  
Numerical Simulations ◽  
Numerical Solution ◽  
Reproducing Kernel ◽  
Computational Algorithm ◽  
Mathematical Structure ◽  
The Other ◽  
Modern Trend ◽  
Van Der Pol ◽  
New Approach ◽  
Damping Model

The aim of this paper is to propose the Atangana–Baleanu fractional methodology for fathoming the Van der Pol damping model by using the reproducing kernel algorithm. To this end, we discuss the mathematical structure of this new approach and some other numerical properties of solutions. Furthermore, all needed requirements for characterizing solutions by applying the reproducing kernel algorithm are debated. In this orientation, modern trend and new computational algorithm in terms of analytic and approximate Atangana–Baleanu fractional solutions are proposed. Finally, numerical simulations in fractional emotion is constructed one next to the other with tabulated data and graphical portrayals.

Helical Shift Mechanics of Rubber V-Belt Variators

2011 ◽  
Vol 133 (4) ◽  
Author(s):  
Francesco Sorge ◽  
Marco Cammalleri
Keyword(s):  
Theoretical Model ◽  
Numerical Solution ◽  
Closed Form ◽  
Operating Conditions ◽  
The Other ◽  
Experimental Comparison ◽  
Axial Thrust ◽  
Shooting Technique ◽  
Strongly Nonlinear ◽  
Tension Level

A very common configuration of V-belt variators for motorcycles considers the correction of the belt tensioning depending on the resistant torque by means of suitable helical-shaped tracks allowing the driven half-pulleys to close/open. The theoretical model for belt-pulley coupling is rather complex for this configuration, where one half-pulley may run in advance and the other one behind with respect to the belt, and requires the repeated numerical solution of a strongly nonlinear differential system by a sort of shooting technique, until all the operating conditions are fulfilled (angular contact extent, torque, and axial force). After solving the full equations, the present study develops closed-form approximations, which are characterized by an excellent correspondence with the numerical plots, and suggests a simple and practical formulary for the axial thrust as a function of the torque and of the tension level. Then, the results of a theoretical–experimental comparison are also reported, and they indicate a fine agreement between the model and the real operation.

scholarly journals Haar Wavelet Operational Matrix Method for Fractional Oscillation Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Umer Saeed ◽  
Mujeeb ur Rehman
Keyword(s):  
Exact Solution ◽  
Fractional Order ◽  
Nonlinear Oscillation ◽  
Matrix Method ◽  
Operational Matrix ◽  
Haar Wavelet ◽  
The Other ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Operational Matrix Method

We utilized the Haar wavelet operational matrix method for fractional order nonlinear oscillation equations and find the solutions of fractional order force-free and forced Duffing-Van der Pol oscillator and higher order fractional Duffing equation on large intervals. The results are compared with the results obtained by the other technique and with exact solution.

scholarly journals Numerical comparison by different methods (second order Runge Kutta methods, Heun method, fixed point method and Ralston method) to differential equations with initial condition

2020 ◽  
Vol 25 (2) ◽  
pp. 299-305
Author(s):  
Diana Marcela Devia Narváez ◽  
Fernando Mesa ◽  
German Correa-Vélez
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Numerical Solution ◽  
Taylor Series ◽  
Higher Order ◽  
The Other ◽  
Numerical Comparison ◽  
Numerical Solution Methods ◽  
Midpoint Method ◽  
Runge Kutta

This manuscript contains a detailed comparison between numerical solution methods of ordinary differential equations, which start from the Taylor series method of order 2, stating that this series hinders calculations for higher order derivatives of functions of several variables, so that the Runge Kutta methods of order 2 are implemented, which achieve the required purpose avoiding the cumbersome calculations of higher order derivatives. In this document, different variants of the Runge-Kutta methods of order 2 will be exposed from an introduction and demonstration of the connection of these with the Taylor series of order 2, these methods are: the method of Heun, the method of midpoint and the Ralston method. It will be observed from the solution of test differential equations its respective error with respect to the analytical solution, obtaining an error index dictated by the mean square error EMC. Through this document we will know the best numerical approximation to the analytical solution of the different PVI (initial value problems) raised, also fixing a solution pattern for certain problems, that is, the appropriate method for each type of problem will be stipulated. It was observed that the Ralston method presented greater accuracy followed by the midpoint method and the Heun method, in the other PVI it is observed that the midpoint method yields the best numerical solution since it has a very low EMC and difficult to reach by the other methods.  

Effect of Viscous Dissipation on MHD Williamson Nanofluid Flow in a Porous Medium

2019 ◽  
Vol 8 (3) ◽  
pp. 5795-5802 ◽  
Keyword(s):  
Porous Medium ◽  
Steady State ◽  
Numerical Solution ◽  
Viscous Dissipation ◽  
Flow Problem ◽  
Numerical Study ◽  
Steady State Flow ◽  
Nanofluid Flow ◽  
Shooting Technique ◽  
Order Four

The main objective of this paper is to focus on a numerical study of viscous dissipation effect on the steady state flow of MHD Williamson nanofluid. A mathematical modeled which resembles the physical flow problem has been developed. By using an appropriate transformation, we converted the system of dimensional PDEs (nonlinear) into coupled dimensionless ODEs. The numerical solution of these modeled ordinary differential equations (ODEs) is achieved by utilizing shooting technique together with Adams-Bashforth Moulton method of order four. Finally, the results of discussed for different parameters through graphs and tables.

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