Approximate solution of the magneto-hydrodynamic flow over a nonlinear stretching sheet

The approximate solution of the magneto-hydrodynamic (MHD) boundary layer flow over a nonlinear stretching sheet is obtained by combining the Lie symmetry method with the homotopy perturbation method. The approximate solution is tabulated, plotted for the values of various parameters and compared with the known solutions. It is found that the approximate solution agrees very well with the known numerical solutions, showing the reliability and validity of the present work.     

Approximate solution of the flow over a nonlinear magneto-hydrodynamic stretching sheet

The approximate solution of the magneto-hydrodynamic(MHD) boundary layer flow over a nonlinear stretching sheet is obtained by combining the Lie symmetry method with the homotopy perturbation method.The approximate solution is tabulated,plotted for the values of various parameters and compared with the known solutions.It is found that the approximate solution agrees very well with the known numerical solutions,showing the reliability and validity of the present work.     

Application of He's homotopy perturbation method to boundary layer flow and convection heat transfer over a flat plate

In this Letter, the problem of forced convection over a horizontal flat plate is presented and the homotopy perturbation method (HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing on the problem. It has been attempted to show the capabilities and wide-range applications of the homotopy perturbation method in comparison with the previous ones in solving heat transfer problems. The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.     

MHD flow of nanofluids over an exponentially stretching sheet in a porous medium with convective boundary conditions

This article concentrates on the steady magnetohydrodynamic(MHD) flow of viscous nanofluid. The flow is caused by a permeable exponentially stretching surface. An incompressible fluid fills the porous space. A comparative study is made for the nanoparticles namely Copper(Cu), Silver(Ag), Alumina(Al2O3) and Titanium Oxide(TiO2). Water is treated as a base fluid. Convective type boundary conditions are employed in modeling the heat transfer process. The non-linear partial differential equations governing the flow are reduced to an ordinary differential equation by similarity transformations. The obtained equations are then solved for the development of series solutions. Convergence of the obtained series solutions is explicitly discussed. The effects of different parameters on the velocity and temperature profiles are shown and analyzed through graphs.     

Application of homotopy perturbation method to the RLW and generalized modified Boussinesq equations

In this Letter, He's homotopy perturbation method (HPM) is implemented for finding the solitary-wave solutions of the regularized long-wave (RLW) and generalized modified Boussinesq (GMB) equations. We obtain numerical solutions of these equations for the initial conditions. We will show that the convergence of the HPM is faster than those obtained by the Adomian decomposition method (ADM). The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.     

Unsteady MHD flow and heat transfer near stagnation point over a stretching/shrinking sheet in porous medium filled with a nanofluid

In this article, the unsteady magnetohydrodynamic （MHD） stagnation point flow and heat transfer of a nanofluid over a stretching/shrinking sheet is investigated numerically. The similarity solution is used to reduce the governing system of partial differential equations to a set of nonlinear ordinary differential equations which are then solved numerically using the fourth-order Runge-Kutta method with shooting technique. The ambient fluid velocity, stretching/shrinking velocity of sheet, and the wall temperature are assumed to vary linearly with the distance from the stagnation point. To investigate the influence of various pertinent parameters, graphical results for the local Nusselt number, the skin friction coefficient, velocity profile, and temperature profile are presented for different values of the governing parameters for three types of nanoparticles, namely copper, alumina, and titania in the water-based fluid. It is found that the dual solution exists for the decelerating flow. Numerical results show that the extent of the dual solution domain increases with the increases of velocity ratio, magnetic parameter, and permeability parameter whereas it remains constant as the value of solid volume fraction of nanoparticles changes. Also, it is found that permeability parameter has a greater effect on the flow and heat transfer of a nanofluid than the magnetic parameter.     

Series solutions of stagnation slip flow and heat transfer by the homotopy analysis method

An analytical approximation for the similarity solutions of the two- and three-dimensional stagnation slip flow and heat transfer is obtained by using the homotopy analysis method. This method is a series expansion method, but it is different from the perturbation technique, because it is independent of small physical parameters at all. Instead, it is based on a continuous mapping in topology so that it is applicable for not only weakly but also strongly nonlinear flow phenomena. Convergent [m,m] homotopy Padé approximants are obtained and compared with the numerical results and the asymptotic approximations. It is found that the homotopy Padé approximants agree well with the numerical results. The effects of the slip length ℓ and the thermal slip constant β on the heat transfer characteristics are investigated and discussed. Supported by the National Natural Science Foundation of China (Grant No. 10872129)     

Thermophysical effects of carbon nanotubes on MHD flow over a stretching surface

This article is intended for investigating the effects of magnetohydrodynamics (MHD) and volume fraction of carbon nanotubes (CNTs) on the flow and heat transfer in two lateral directions over a stretching sheet. For this purpose, three types of base fluids specifically water, ethylene glycol and engine oil with single and multi-walled carbon nanotubes are used in the analysis. The convective boundary condition in the presence of CNTs is presented first time and not been explored so far. The transformed nonlinear differential equations are solved by the Runge–Kutta–Fehlberg method with a shooting technique. The dimensionless velocity and shear stress are obtained in both directions. The dimensionless heat transfer is determined on the surface. Three different models of thermal conductivity are comparable for both CNTs and it is found that the Xue [1] model gives the best approach to guess the superb thermal conductivity in comparison with the Maxwell [2] and Hamilton and Crosser [3] models. And finally, another finding suggests the engine oil provides the highest skin friction and heat transfer rates.     

A reliable treatment of the homotopy analysis method for viscous flow over a non-linearly stretching sheet in presence of a chemical reaction and under influence of a magnetic field

The similarity solution for the steady two-dimensional flow of an incompressible viscous and electrically conducting fluid over a non-linearly semi-infinite stretching sheet in the presence of a chemical reaction and under the influence of a magnetic field gives a system of non-linear ordinary differential equations. These non-linear differential equations are analytically solved by applying a newly developed method, namely the Homotopy Analysis Method (HAM). The analytic solutions of the system of non-linear differential equations are constructed in the series form. The convergence of the obtained series solutions is carefully analyzed. Graphical results are presented to investigate the influence of the Schmidt number, magnetic parameter and chemical reaction parameter on the velocity and concentration fields. It is noted that the behavior of the HAM solution for concentration profiles is in good agreement with the numerical solution given in reference [A. Raptis, C. Perdikis, Int. J. Nonlinear Mech. 41, 527 (2006)].      

Flow and heat transfer of a nanofluid over a hyperbolically stretching sheet

This article explores the boundary layer flow and heat transfer of a viscous nanofluid bounded by a hyperbolically stretching sheet. Effects of Brownian and thermophoretic diffusions on heat transfer and concentration of nanoparticles are given due attention. The resulting nonlinear problems are computed for analytic and numerical solutions. The effects of Brownian motion and thermophoretic property are found to increase the temperature of the medium and reduce the heat transfer rate. The thermophoretic property thus enriches the concentration while the Brownian motion reduces the concentration of the nanoparticles in the fluid. Opposite effects of these properties are observed on the Sherwood number.     

Conformable variational iteration method,conformable fractional reduced differential transform method and conformable homotopy analysis method for non-linear fractional partial differential equations

In this paper, we introduce conformable variational iteration method (C-VIM), conformable fractional reduced differential transform method (CFRDTM) and conformable homotopy analysis method (C-HAM). Between these methods, the C-VIM is introduced for the first time for fractional partial differential equations (FPDEs). These methods are new versions of well-known VIM, RDTM and HAM. In addition, above-mentioned techniques are based on new defined conformable fractional derivative to solve linear and non-linear conformable FPDEs. Firstly, we present some basic definitions and general algorithm for proposal methods to solve linear and non-linear FPDEs. Secondly, to understand better, the presented new methods are supported by some examples. Finally, the obtained results are illustrated by the aid of graphics and the tables. The applications show that these new techniques C-VIM, CFRDTM and C-HAM are extremely reliable and highly accurate and it provides a significant improvement in solving linear and non-linear FPDEs.     

Higher accuracy analytical approximations to a nonlinear oscillator with discontinuity by He's homotopy perturbation method

He's homotopy perturbation method is used to calculate higher-order approximate periodic solutions of a nonlinear oscillator with discontinuity for which the elastic force term is proportional to sgn(x). We find He's homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. Only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate period of less than 1.56% for all values of oscillation amplitude, while this relative error is 0.30% for the second iteration and as low as 0.057% when the third-order approximation is considered. Comparison of the result obtained using this method with those obtained by different harmonic balance methods reveals that He's homotopy perturbation method is very effective and convenient.     

Solving fractional diffusion and wave equations by modified homotopy perturbation method

This Letter applies the modified He's homotopy perturbation method (HPM) suggested by Momani and Odibat to obtaining solutions of linear and nonlinear fractional diffusion and wave equations. The fractional derivative is described in the Caputo sense. Some illustrative examples are given, revealing the effectiveness and convenience of the method.     

Numerical simulation of the regularized long wave equation by He's homotopy perturbation method

In this Letter, we present the homotopy perturbation method (shortly HPM) for obtaining the numerical solution of the RLW equation. We obtain the exact and numerical solutions of the Regularized Long Wave (RLW) equation for certain initial condition. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Comparison of the results with those of other methods have led us to significant consequences. The numerical solutions are compared with the known analytical solutions.     

Thermal radiation and slip effects on MHD stagnation point flow of nanofluid over a stretching sheet

Present model is devoted for the stagnation point flow of nanofluid with magneto-hydrodynamics (MHD) and thermal radiation effects passed over a stretching sheet. Moreover, we have considered the combined effects of velocity and thermal slip. Condition of zero normal flux of nanoparticles at the wall for the stretched flow phenomena is yet to be explored in the literature. Convinced partial differential equations of the model are transformed into the system of coupled nonlinear differential equations and then solved numerically. Graphical results are plotted for velocity, temperature and nanoparticle concentration for various values of emerging parameters. Variation of stream lines, skin friction coefficient, local Nusselt and Sherwood number are displayed along with the effective parameters. Final conclusion has been drawn on the basis of both numerical and graphs results.     

Coupled effects of nano-size,stretching, and slip boundary conditions on nonlinear vibrations of nano-tube conveying fluid by the homotopy analysis method

In this paper, natural frequency and nonlinear response of carbon nano-tube (CNT) conveying fluid based on the coupling of nonlocal theory and von Karman's stretching have been obtained. The homotopy analysis method (HAM) has been used for solving nonlinear differential equation of system and convergence region of approach presented. Effects of mid-plane stretching, nonlocal parameter and their coupling in the model have been investigated. It has been concluded that stretching effect is significant only for higher-amplitude initial excitations and lower beam aspect ratios. Moreover, by including the slip boundary condition, the effect of nano-size flow has been revealed in the nonlinear vibration model. We have concluded that small-size effects of nano-tube and nano-flow have impressed critical velocity of fluid significantly specially for gas fluid. Analytical results obtained from HAM solution show satisfactory agreement with numerical solutions such as Runge–Kutta. Having an analytical approach, we have been able to investigate the unbounded growth of amplitude of vibrations for flow velocities near the critical value. Moreover, by employing the second-order approximation of Galerkin's method, the estimated natural frequency of the first mode is verified. The obtained results would indicate that the effects of higher mode on the first natural frequency are negligible for the doubly-clamped CNT.     

New homotopy analysis transform method for solving the discontinued problems arising in nanotechnology

We present a new reliable analytical study for solving the discontinued problems arising in nanotechnology.Such problems are presented as nonlinear differential–difference equations.The proposed method is based on the Laplace transform with the homotopy analysis method(HAM).This method is a powerful tool for solving a large amount of problems.This technique provides a series of functions which may converge to the exact solution of the problem.A good agreement between the obtained solution and some well-known results is obtained.     

On the solution of magneto-hydrodynamics three-dimensional flow due to a stretching sheet in a porous medium using the successive linearization method

In this paper, we propose a new application of the successive linearization method for solving a system of nonlinear third order equations that govern the steady of laminar three dimensional flow for an incompressible, viscous fluid past a stretching sheet. A comparison is made with results from existing methods reported in the literature to test the validity, accuracy and convergence of the method. It was found that the method is easy to implement, yields accurate results, and performs better than some numerical methods. Also, the effects of the physical parameters the fluid velocity are depicted graphically and are analyzed in detail.     

Dual solutions in boundary layer flow of Maxwell fluid over a porous shrinking sheet

An analysis is carried out for dual solutions of the boundary layer flow of Maxwell fluid over a permeable shrinking sheet. In the investigation, a constant wall mass transfer is considered. With the help of similarity transformations, the governing partial differential equations（PDEs） are converted into a nonlinear self-similar ordinary differential equation（ODE）. For the numerical solution of transformed self-similar ODE, the shooting method is applied. The study reveals that the steady flow of Maxwell fluid is possible with a smaller amount of imposed mass suction compared with the viscous fluid flow. Dual solutions for the velocity distribution are obtained. Also, the increase of Deborah number reduces the boundary layer thickness for both solutions.     

Soliton solutions of the two-dimensional KdV-Burgers equation by homotopy perturbation method

In this Letter, the He's homotopy perturbation method (HPM) to finding the soliton solutions of the two-dimensional Korteweg–de Vries Burgers' equation (tdKdVB) for the initial conditions was applied. Numerical solutions of the equation were obtained. The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. The results reveal that the HPM is very effective and simple.