The Vibrational Motion of a Dynamical System Using Homotopy Perturbation Technique

This paper outlines the vibrational motion of a nonlinear system with a spring of linear stiffness. Homotopy perturbation technique (HPT) is used to obtain the asymptotic solution of the governing equation of motion. The numerical solution of this equation is obtained using the fourth order Runge-Kutta method (RKM). The comparison between both solutions reveals high consistency between them which confirms that, the accuracy of the obtained solution using aforementioned perturbation technique. The time history of the attained solution is represented through some plots to reveal the good effect of the different parameters of the considered system on the motion at any instant. The conditions of the stability of the attained solution are presented and discussed.

A MAPLE PACKAGE OF AUTOMATED DERIVATION OF HOMOTOPY ANALYSIS SOLUTION FOR PERIODIC NONLINEAR OSCILLATIONS

Based on the homotopy analysis method, a general analytic technique for strongly nonlinear problems, a Maple package of automated derivation （ADHO） for periodic nonlinear oscillation systems is presented. This Maple package is valid for periodic oscillation systems in rather general, and can automatically deliver the accurate approximations of the frequency co and the mean of motion δof a nonlinear periodic oscillator. Based on the homotopy analysis method which is valid even for highly nonlinear problems, this Maple package can give accurate approximate expressions even for nonlinear oscillation systems with strong nonlinearity. Besides, the package is user-friendly： One just needs to input a governing equation and initial conditions, and then gets satisfied analytic approximations in few seconds. Several different types of examples are given in this paper to illustrate the validity of this Maple package. Such kind of package provides us a helpful and easy-to-use tool in science and engineering to analyze periodic of this Maple package from the is published publicly. nonlinear oscillations. And it is free address http：//numericaltank.sjtu to download the electronic version edu.cn/sjliao.htm once the paper

An Approximate Numerical Methods for Mathematical and Physical Studies for Covid-19 Models

The advancement in numerical models of serious resistant illnesses is a key research territory in different fields including the nature and the study of disease transmission.One of the aims of these models is to comprehend the elements of conduction of these infections.For the new strain of Covid-19(Coronavirus),there has been no immunization to protect individuals from the virus and to forestall its spread so far.All things being equal,control procedures related to medical services,for example,social distancing or separation,isolation,and travel limitations can be adjusted to control this pandemic.This article reveals some insights into the dynamic practices of nonlinear Coronavirus models dependent on the homotopy annoyance strategy(HPM).We summon a novel sign stream chart that is utilized to depict the Coronavirus model.Through the numerical investigations,it is uncovered that social separation of the possibly tainted people who might be conveying the infection and the healthy virus-free people can diminish or interrupt the spread of the infection.The mathematical simulation results are highly concurrent with the statistical forecasts.The free balance and dependability focus for the Coronavirus model is discussed and the presence of a consistently steady arrangement is demonstrated.

An efficient analytical technique for fractional partial differential equations occurring in ion acoustic waves in plasma

In this work,we apply an efficient analytical algorithm namely homotopy perturbation Sumudu transform method(HPSTM)to find the exact and approximate solutions of linear and nonlinear time-fractional regularized long wave(RLW)equations.The RLW equations describe the nature of ion acoustic waves in plasma and shallow water waves in oceans.The derived results are very significant and imperative for explaining various physical phenomenons.The suggested method basically demonstrates how two efficient techniques,the Sumudu transform scheme and the homotopy perturbation technique can be integrated and applied to find exact and approximate solutions of linear and nonlinear time-fractional RLW equations.The nonlinear expressions can be simply managed by application of He’s polynomials.The result shows that the HPSTM is very powerful,efficient,and simple and it eliminates the round-off errors.It has been observed that the proposed technique can be widely employed to examine other real world problems.

Optimal analytical and numerical approximations to the(un)forced(un)damped parametric pendulum oscillator

The(un)forced(un)damped parametric pendulum oscillator(PPO)is analyzed analytically and numerically using some simple,effective,and more accurate techniques.In the first technique,the ansatz method is employed for analyzing the unforced damped PPO and for deriving some optimal and accurate analytical approximations in the form of angular Mathieu functions.In the second approach,some approximations to(un)forced damped PPO are obtained in the form of trigonometric functions using the ansatz method.In the third approach,He’s frequency-amplitude principle is applied for deriving some approximations to the(un)damped PPO.In the forth approach,He’s homotopy technique is employed for analyzing the forced(un)damped PPO numerically.In the fifth approach,the p-solution Method,which is constructed based on Krylov–Bogoliúbov Mitropolsky method,is introduced for deriving an approximation to the forced damped PPO.In the final approach,the hybrid Padé-finite difference method is carried out for analyzing the damped PPO numerically.All proposed techniques are compared to the fourth-order Runge–Kutta(RK4)numerical solution.Moreover,the global maximum residual distance error is estimated for checking the accuracy of the obtained approximations.The proposed methodologies and approximations can help many researchers in studying and investigating several nonlinear phenomena related to the oscillations that can arise in various branches of science,e.g.waves and oscillations in plasma physics.