Damped Mathieu Equation with a Modulation Property of the Homotopy Perturbation Method

In this article,the main objective is to employ the homotopy perturbation method(HPM)as an alternative to classical perturbation methods for solving nonlinear equations having periodic coefficients.As a simple example,the nonlinear damping Mathieu equation has been investigated.In this investigation,two nonlinear solvability conditions are imposed.One of them was imposed in the first-order homotopy perturbation and used to study the stability behavior at resonance and non-resonance cases.The next level of the perturbation approaches another solvability condition and is applied to obtain the unknowns become clear in the solution for the firstorder solvability condition.The approach assumed here is so significant for solving many parametric nonlinear equations that arise within the engineering and nonlinear science.

He’s Homotopy Perturbation Method and Fractional Complex Transform for Analysis Time Fractional Fornberg-Whitham Equation

In this article,time fractional Fornberg-Whitham equation of He’s fractional derivative is studied.To transform the fractional model into its equivalent differential equation,the fractional complex transform is used and He’s homotopy perturbation method is implemented to get the approximate analytical solutions of the fractional-order problems.The graphs are plotted to analysis the fractional-order mathematical modeling.

A New Approximation to the Linear Matrix Equation AX = B by Modification of He’s Homotopy Perturbation Method

It is well known that the matrix equations play a significant role in engineering and applicable sciences. In this research article, a new modification of the homotopy perturbation method (HPM) will be proposed to obtain the approximated solution of the matrix equation in the form AX = B. Moreover, the conditions are deduced to check the convergence of the homotopy series. Numerical implementations are adapted to illustrate the properties of the modified method.

Application of homotopy perturbation method to the radial thrust problem

The dynamics of a spacecraft propelled by a continuous radial thrust resembles that of a nonlinear oscillator.This is analyzed in this work with a novel method that combines the definition of a suitable homotopy with a classical perturbation approach,in which the low thrust is assumed to be a perturbation of the nominal Keplerian motion.The homotopy perturbation method provides the analytical(approximate)solution of the dynamical equations in polar form to estimate the corresponding spacecraft propelled trajectory with a short computational time.The accuracy of the analytical results was tested in an orbital-targeting mission scenario.

Unsteady time-dependent incompressible Newtonian fluid flow between two parallel plates by homotopy analysis method(HAM),homotopy perturbation method(HPM)and collocation method(CM)

Analytical and numerical analyses have performed to study the problem of the flow of incompressible Newtonian fluid between two parallel plates approaching or receding from each other symmetrically.The Navier–Stokes equations have been transformed into an ordinary differential equation using a similarity transformation.The powerful analytical methods called collocation method(CM),the homotopy perturbation method(HPM),and the homotopy analysis method(HAM)have been used to solve nonlinear differential equations.It has been attempted to show the capabilities and wide-range applications of the proposed methods in comparison with a type of numerical analysis as fourth-order Runge–Kutta numerical method in solving this problem.Also,velocity fields have been computed and shown graphically for various values of physical parameters.The objective of the present work is to investigate the effect of Reynolds number and suction or injection characteristic parameter on the velocity field.

Nonlinear Vibration Analysis of Functionally GradedNanobeam Using Homotopy Perturbation Method

In this paper, He’s homotopy perturbation method is utilized to obtainthe analytical solution for the nonlinear natural frequency of functionally gradednanobeam. The functionally graded nanobeam is modeled using the Eringen’s nonlocalelasticity theory based on Euler-Bernoulli beam theory with von Karman nonlinearityrelation. The boundary conditions of problem are considered with both sidessimply supported and simply supported-clamped. The Galerkin’s method is utilizedto decrease the nonlinear partial differential equation to a nonlinear second-order ordinarydifferential equation. Based on numerical results, homotopy perturbationmethodconvergence is illustrated. According to obtained results, it is seen that the second termof the homotopy perturbation method gives extremely precise solution.

Application of Modified Couple Stress Theory and Homotopy Perturbation Method in Investigation of Electromechanical Behaviors of Carbon Nanotubes

The paper presents the size-dependant behaviors of the carbon nanotubes under electrostatic actuation using the modified couple stress theory and homotopy perturbation method.Due to the less accuracy of the classical elasticity theorems,the modified couple stress theory is applied in order to capture the size-dependant properties of the carbon nanotubes.Both of the static and dynamic behaviors under static DC and step DC voltages are discussed.The effects of various dimensions and boundary conditions on the deflection and pull-in voltages of the carbon nanotubes are to be investigated in detail via application of the homotopy perturbation method to solve the nonlinear governing equations semi-analytically.

Homotopy Perturbation Method for Time-Fractional Shock Wave Equation

A scheme is developed to study numerical solution of the time-fractional shock wave equation and wave equation under initial conditions by the homotopy perturbation method(HPM).The fractional derivatives are taken in the Caputo sense.The solutions are given in the form of series with easily computable terms.Numerical results are illustrated through the graph.

Solution of analytical model for fuel spray penetration via homotopy perturbation method

The present paper attempts to solve equations in the initial stage and the two-phase flow regime of fuel spray penetration using the HPM-Padétechnique,which is a combination of the homotopy perturbation method(HPM)and Padéapproximation.At the initial stage,the effects of the droplet drag and the air entrainment were explained while in the two-phase flow stage,the spray droplets had the same velocities as the entrained air.The results for various injection pressures and ambient densities are presented graphically and then discussed upon.The obtained results for these two stages show a good agreement with previously obtained expressions via successive approximations in the available literature.The numerical result indicates that the proposed method is straight forward to implement,efficient and accurate for solving nonlinear equations of fuel spray.

Nonlinear vibration analysis of a rotor supported by magnetic bearings using homotopy perturbation method

In this paper,the effects of nonlinear forces due to the electromagnetic field of bearing and the unbalancing force on nonlinear vibration behavior of a rotor is investigated.The rotor is modeled as a rigid body that is supported by two magnetic bearings with eightpolar structures.The governing dynamics equations of the system that are coupled nonlinear second order ordinary differential equations(ODEs)are derived,and for solving these equations,the homotopy perturbation method(HPM)is used.By applying HPM,the possibility of presenting a harmonic semi-analytical solution,is provided.In fact,with equality the coefficient of auxiliary parameter(p),the system of coupled nonlinear second order and non-homogenous differential equations are obtained so that consists of unbalancing effects.By considering some initial condition for displacement and velocity in the horizontal and vertical directions,free vibration analysis is done and next,the forced vibration analysis under the effect of harmonic forces also is investigated.Likewise,various parameters on the vibration behavior of rotor are studied.Changes in amplitude and response phase per excitation frequency are investigated.Results show that by increasing excitation frequency,the motion amplitude is also increases and by passing the critical speed,it decreases.Also it shows that the magnetic bearing system performance is in stable maintenance of rotor.The parameters affecting on vibration behavior,has been studied and by comparison the results with the other references,which have a good precision up to 2nd order of embedding parameter,it implies the accuracy of this method in current research.

Approximate solutions to fractional differential equations

In this paper, the time-fractional coupled viscous Burgers' equation(CVBE)and Drinfeld-Sokolov-Wilson equation(DSWE) are solved by the Sawi transform coupled homotopy perturbation method(HPM). The approximate series solutions to these two equations are obtained. Meanwhile, the absolute error between the approximate solution given in this paper and the exact solution given in the literature is analyzed. By comparison of the graphs of the function when the fractional order α takes different values, the properties of the equations are given as a conclusion.

Modified integral equation combined with the decomposition method for time fractional differential equations with variable coefficients

In this paper,the modified integral equation,namely,Elzaki transformation coupled with the Adomian decomposition method called Elzaki Adomian decomposition method(EADM)is used to investigate the solution of time-fractional fourth-order parabolic partial differential equations(PDEs)with variable coefficients.The introduced method is used to solve two models of the proposed problem,the analytical and approximate solutions of the models are obtained.The outcomes illustrate that the proposed technique is a highly accurate,and facilitates the process of solving differential equations by comparing it,with the exact solution and those obtained by the variation iteration method(VIM)and Laplace homotopy perturbation method(LHPM).

用有效的解析和数值方法估算船舶在随机横浪中的横摇运动

A steady-state roll motion of ships with nonlinear damping and restoring moments for all times is modeled by a second-order nonlinear differential equation.Analytical expressions for the roll angle,velocity,acceleration,and damping and restoring moments are derived using a modified approach of homotopy perturbation method(HPM).Also,the operational matrix of derivatives of ultraspherical wavelets is used to obtain a numerical solution of the governing equation.Illustrative examples are provided to examine the applicability and accuracy of the proposed methods when compared with a highly accurate numerical scheme.

Dynamical Behaviors of Nonlinear Coronavirus (COVID-19) Model with Numerical Studies

The development of mathematical modeling of infectious diseases is a key research area in various elds including ecology and epidemiology.One aim of these models is to understand the dynamics of behavior in infectious diseases.For the new strain of coronavirus(COVID-19),there is no vaccine to protect people and to prevent its spread so far.Instead,control strategies associated with health care,such as social distancing,quarantine,travel restrictions,can be adopted to control the pandemic of COVID-19.This article sheds light on the dynamical behaviors of nonlinear COVID-19 models based on two methods:the homotopy perturbation method(HPM)and the modied reduced differential transform method(MRDTM).We invoke a novel signal ow graph that is used to describe the COVID-19 model.Through our mathematical studies,it is revealed that social distancing between potentially infected individuals who are carrying the virus and healthy individuals can decrease or interrupt the spread of the virus.The numerical simulation results are in reasonable agreement with the study predictions.The free equilibrium and stability point for the COVID-19 model are investigated.Also,the existence of a uniformly stable solution is proved.

Couple stress nanofluid flow through a bifurcated artery—Application of catheterization process

In this article,we are exploring the hemodynamics of nanofluid,flowing through a bifurcated artery with atherosclerosis in the presence of a catheter.For treating obstruction in the artery,one can use the catheter whose outer surface is carrying the drug coated with nano-particles.The resultant solvent is considered as blood nano-fluid.Blood being a complex fluid,is modeled by couple stress fluid.In the presence of nano-particles,the temperature and the concentration distribution are understood in a bifurcated stenotic artery.The concluded mathematical model is governed by coupled non-linear equations,and are solved by using the homotopy perturbation method.Consequently,we have explored is the effects of fluid and the embedded geometric parameters on the hemodynamics characteristics.It is also realized that high wall shear stress exists for couple stress nano-fluid when compared to Newtonian nano-fluid.which is computed at a location corresponding to maximum constriction(z=12.5)of the artery.

Fractional Analysis of Thin Film Flow of Non-Newtonian Fluid

Modeling and analysis of thin film flow with respect to magneto hydro dynamical effect has been an important theme in the field of fluid dynamics,due to its vast industrial applications.The analysis involves studying the behavior and response of governing equations on the basis of various parameters such as thickness of the film,film surface profile,shear stress,liquid velocity,volumetric flux,vorticity,gravity,viscosity among others,along with different boundary conditions.In this article,we extend this analysis in fractional space using a homotopy based scheme,considering the case of a Non-Newtonian Pseudo-Plastic fluid for lifting and drainage on a vertical wall.After applying similarity transformations,the given problems are reduced to highly non-linear and inhomogeneous ordinary differential equations.Moreover,fractional differential equations are obtained using basic definitions of fractional calculus.The Homotopy Perturbation Method(HPM),along with fractional calculus is used for obtaining approximate solutions.Physical quantities such as the velocity profile,volume flux and average velocity respectively for lift and drainage cases have been calculated.To the best of our knowledge,the given problems have not been attempted before in fractional space.Validity and convergence of the obtained solutions are confirmed by finding residual errors.From a physical perspective,a comprehensive study of the effects of various parameters on the velocity profile is also performed.Study reveals that Stokes number St,non-Newtonian parameterand magnetic parameter M have inverse relationship with fluid velocity in lifting case.In the drainage case,Stokes number St and non-Newtonian parameterhave direct relationship with fluid velocity,but magnetic parameter M shows inverse relationship with velocity.The investigation also shows that the fractional parameterhas direct relationship with the fluid velocity in lifting case,while it has inverse relationship with velocity in the drainage case.

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.

Nonlinear Transversal Vibration Analysis of an Axially Moving Viscoelastic Yarn in Tufting Process

The nonlinear transversal vibration of axially moving yarn with time-dependent tension is investigated. Yarn material is modeled as Kelvin model. A partial differential equation governing the transversal vibration is derived from the Newton's second law.Galerkin method is used to truncate the governing nonlinear differential equation,and thus the first-order ordinary differential equation is obtained. The periodic vibration equation and the natural frequency of moving yarn are received by applying homotopy perturbation method. As a result,the condition which should be avoided during the tufting process for resonance is obtained.

Numerical Solution of Fractional Differential Equations

In this article, two numerical techniques, namely, the homotopy perturbation and the matrix approach methods have been proposed and implemented to obtain an approximate solution of the linear fractional differential equation. To test the effectiveness of these methods, two numerical examples with known exact solution are illustrated. Numerical experiments show that the accuracy of these methods is in a good agreement with the exact solution. However, a comparison between these methods shows that the matrix approach method provides more accurate results.

A Theoretical Model of pH-Based Potentiometric Biosensor Based on Immobilized Enzyme Membrane

A theoretical model for the non steady-state response of a pH-based potentiometric biosensor immobilizing organophosphorus hydrolase (OPH) is discussed. The model is based on a system of five coupled nonlinear reaction-diffusion equations under non steady-state conditions for enzyme reactions occurring in potentiometric biosensor that describes the concentration of substrate and hydrolysis products within the membrane. New approximate analytical expressions for the concentration of the substrate (organophosphorus pesticides (OPs)) and products are derived for all values of Thiele modulus and buffer concentration using new approach of homotopy perturbation method. The analytical results are also compared with numerical ones and a good agreement is obtained. The obtained results are valid for the whole solution domain.