Homotopy analysis method for MHD viscoelastic fluid flow and heat transfer in a channel with a stretching wall

In this paper, we analyze the flow and heat transfer characteristics of a magnetohydrodynamic (MHD) viscoelastic fluid in a parallel plate channel with a stretching wall. Homotopy analysis method (HAM) is used to obtain analytical solutions of the governing nonlinear differential equations. The analytical solutions are obtained in the form of infinite series and the convergence of the series solution is discussed explicitly. The obtained results are presented through graphs for several sets of values of the parameters, and the salient features of the solutions are analyzed. A comparison of our HAM results (for a special case of the study) with the available results in the literature (obtained by other methods) shows that our results are accurate for a wide range of parameters. Further, we point that our analysis finds application to the study of the haemodynamic flow of blood in the cardiovascular system subject to external magnetic field.     

Analytical approximations to nonlinear vibration of an electrostatically actuated microbeam

This paper employs the homotopy analysis method (HAM) to derive analytical approximate solutions for the nonlinear problem with high-order nonlinearity. Such a problem corresponds to the large-amplitude vibration of electrostatically actuated microbeams. The HAM is also optimized to accelerate the convergence of approximate solutions. To verify the accuracy of the present approach, illustrative examples are provided and compared with other analytical and exact solutions.     

Study on asymptotic analytical solutions using HAM for strongly nonlinear vibrations of a restrained cantilever beam with an intermediate lumped mass

Presented herein is to establish the asymptotic analytical solutions for the fifth-order Duffing type temporal problem having strongly inertial and static nonlinearities. Such a problem corresponds to the strongly nonlinear vibrations of an elastically restrained beam with a lumped mass. Taking into consideration of the inextensibility condition and using an assumed single mode Lagrangian method, the single-degree-of-freedom ordinary differential equation can be derived from the governing equations of the beam model. Various parameters of the nonlinear unimodal temporal equation stand for different vibration modes of inextensible cantilever beam. By imposing the homotopy analysis method (HAM), we establish the asymptotic analytical approximations for solving the fifth-order nonlinear unimodal temporal problem. Within this research framework, both the frequencies and periodic solutions of the nonlinear unimodal temporal equation can be explicitly and analytically formulated. For verification, numerical comparisons are conducted between the results obtained by the homotopy analysis and numerical integration methods. Illustrative examples are selected to demonstrate the accuracy and correctness of this approach. Besides, the optimal HAM approach is introduced to accelerate the convergence of solutions.     

Homotopy Analysis Method for the heat transfer of a non-Newtonian fluid flow in an axisymmetric channel with a porous wall

In this article, a powerful analytical method, called the Homotopy Analysis Method (HAM) is introduced to obtain the exact solutions of heat transfer equation of a non-Newtonian fluid flow in an axisymmetric channel with a porous wall for turbine cooling applications. The HAM is employed to obtain the expressions for velocity and temperature fields. Tables are presented for various parameters on the velocity and temperature fields. These results are compared with the solutions which are obtained by Numerical Methods (NM). Also the convergence of the obtained HAM solution is discussed explicitly. These comparisons show that this analytical method is strongly powerful to solve nonlinear problems arising in heat transfer.     

The homotopy analysis method for explicit analytical solutions of Jaulent–Miodek equations

In this work, the homotopy analysis method (HAM) is applied to obtain the explicit analytical solutions for system of the Jaulent–Miodek equations. The validity of the method is verified by comparing the approximation series solutions with the exact solutions. Unlike perturbation methods, the HAM does not depend on any small physical parameters at all. Thus, it is valid for both weakly and strongly nonlinear problems. Besides, different from all other analytic techniques, the HAM provides us a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter ?. Briefly speaking, this work verifies the validity and the potential of the HAM for the study of nonlinear systems. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Mixed convection in the stagnation-point flow of a Maxwell fluid towards a vertical stretching surface

In the present analysis, we study the steady mixed convection boundary layer flow of an incompressible Maxwell fluid near the two-dimensional stagnation-point flow over a vertical stretching surface. It is assumed that the stretching velocity and the surface temperature vary linearly with the distance from the stagnation-point. The governing nonlinear partial differential equations have been reduced to the coupled nonlinear ordinary differential equations by the similarity transformations. Analytical and numerical solutions of the derived system of equations are developed. The homotopy analysis method (HAM) and finite difference scheme are employed in constructing the analytical and numerical solutions, respectively. Comparison between the analytical and numerical solutions is given and found to be in excellent agreement. Both cases of assisting and opposing flows are considered. The influence of the various interesting parameters on the flow and heat transfer is analyzed and discussed through graphs in detail. The values of the local Nusselt number for different physical parameters are also tabulated. Comparison of the present results with known numerical results of viscous fluid is shown and a good agreement is observed.     

Explicit series solutions of some linear and nonlinear Schrodinger equations via the homotopy analysis method

In this paper, by means of the homotopy analysis method (HAM), the solutions of some Schrodinger equations are exactly obtained in the form of convergent Taylor series. The HAM contains the auxiliary parameter ?, that provides a convenient way of controlling the convergent region of series solutions. This analytical method is employed to solve linear and nonlinear examples to obtain the exact solutions. HAM is a powerful and easy-to-use analytic tool for nonlinear problems.     

Solution of the laminar viscous flow in a semi-porous channel in the presence of a uniform magnetic field by using the homotopy analysis method

In this paper, the problem of laminar viscous flow in a semi-porous channel in the presence of a transverse magnetic field is presented and the homotopy analysis method (HAM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing the problem. It has been attempted to show the capabilities and wide-range applications of the homotopy analysis method in comparison with the numerical method in solving this problem. The obtained solutions, in comparison with the numeric solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical method’s (NM) results that the HAM provides highly accurate solutions for nonlinear differential equations.     

An analytical approach for solving nonlinear boundary value problems in finite domains

Based on the homotopy analysis method (HAM), a general analytical approach for obtaining approximate series solutions to nonlinear two-point boundary value problems in finite domains is proposed. To demonstrate its effectiveness, this approach is applied to solve three nonlinear problems, and the analytical solutions obtained are more accurate than the numerical solutions obtained via the shooting method and the sinc-Galerkin method.     

Analytical approximations for a population growth model with fractional order

In this paper, we apply the homotopy analysis method (HAM) to solve the fractional Volterra’s model for population growth of a species in a closed system. This technique is extended to give solutions for nonlinear fractional integro–differential equations. The whole HAM solution procedure for nonlinear fractional differential equations is established. Further, the accurate analytical approximations are obtained for the first time, which are valid and convergent for all time t. This indicates the validity and great potential of the homotopy analysis method for solving nonlinear fractional integro–differential equations.     

Prediction of multiplicity of solutions of nonlinear boundary value problems: Novel application of homotopy analysis method

The purpose of the present paper is to introduce a method, probably for the first time, to predict the multiplicity of the solutions of nonlinear boundary value problems. This procedure can be easily applied on nonlinear ordinary differential equations with boundary conditions. This method, as will be seen, besides anticipating of multiplicity of the solutions of the nonlinear differential equations, calculates effectively the all branches of the solutions (on the condition that, there exist such solutions for the problem) analytically at the same time. In this manner, for practical use in science and engineering, this method might give new unfamiliar class of solutions which is of fundamental interest and furthermore, the proposed approach convinces to apply it on nonlinear equations by today’s powerful software programs so that it does not need tedious stages of evaluation and can be used without studying the whole theory. In fact, this technique has new point of view to well-known powerful analytical method for nonlinear differential equations namely homotopy analysis method (HAM). Everyone familiar to HAM knows that the convergence-controller parameter plays important role to guarantee the convergence of the solutions of nonlinear differential equations. It is shown that the convergence-controller parameter plays a fundamental role in the prediction of multiplicity of solutions and all branches of solutions are obtained simultaneously by one initial approximation guess, one auxiliary linear operator and one auxiliary function. The validity and reliability of the method is tested by its application to some nonlinear exactly solvable differential equations which is practical in science and engineering.     

On convergence of homotopy analysis method and its modification for fractional modified KdV equations

In this paper, the homotopy analysis method (HAM) and its modification (MHAM) are applied to solve the nonlinear time- and space-fractional modified Korteweg-de Vries (fmKdV). The fractional derivatives are described by Caputo’s sense. Approximate and exact analytical solutions of the fmKdV are obtained. The MHAM in particular overcomes the computing difficulty encountered in HAM. Convergence theorems for both the homogeneous and non-homogeneous cases are given. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the approach.     

Application of homotopy analysis method for natural convection of Darcian fluid about a vertical full cone embedded in pours media prescribed surface heat flux

In this paper, a powerful analytical method, called homotopy analysis method (HAM) is used to obtain the analytical solution for a nonlinear ordinary deferential equation that often appear in boundary layers problems arising in heat and mass transfer which these kinds of the equations contain infinity boundary condition. The boundary layer approximations of fluid flow and heat transfer of vertical full cone embedded in porous media give us the similarity solution for full cone subjected to surface heat flux boundary conditions. Nonlinear ODE which is obtained by similarity solution has been solved through homotopy analysis method (HAM). The main objective is to propose alternative methods of solution, which do not require small parameters and avoid linearization and physically unrealistic assumptions. The obtained analytical solution in comparison with the numerical ones represents a remarkable accuracy. The results also indicate that HAM can provide us with a convenient way to control and adjust the convergence region.     

同伦分析法求解非线性多孔收缩表面上黏性磁流体的流动

研究在非线性多孔收缩表面上黏性磁流体(MHD)的流动.先用相似变换简化其控制方程,然后用同伦分析法(HAM)求解该简化问题.用图表的形式对问题的相关参数进行讨论,发现在有磁流体时,收缩解存在.同时得到,在不同参数下f″(0)的解是收敛的.     

Thermal analysis of a longitudinal trapezoidal fin with temperature-dependent thermal conductivity and heat transfer coefficient

A homotopy analysis method (HAM) is used to develop analytical solution for the thermal performance of a straight fin of trapezoidal profile when both the thermal conductivity and the heat transfer coefficient are temperature dependent. Results are presented for the temperature distribution, heat transfer rate, and fin efficiency for a range of values of parameters appearing in the mathematical model. Since the HAM algorithm contains a parameter that controls the convergence and accuracy of the solution, its results can be verified internally by calculating the residual error. The HAM results were also found to be accurate to at least three places of decimal compared with the direct numerical solution of the mathematical model generated using a fourth–fifth-order Runge–Kutta–Fehlberg method. The HAM solution appears in terms of algebraic expressions which are not only easy to compute but also give highly accurate results covering a wide range of values of the parameters rather than the small values dictated by the perturbation solution.     

Homotopy analysis method for solving the MHD flow over a non-linear stretching sheet

The problem of the boundary layer flow of an incompressible viscous fluid over a non-linear stretching sheet is considered. Homotopy analysis method (HAM) is applied in order to obtain analytical solution of the governing nonlinear differential equations. The obtained results are finally compared through the illustrative graphs with the exact solution and an approximate method. The compression shows that the HAM is very capable, easy-to-use and applicable technique for solving differential equations with strong nonlinearity. Moreover, choosing a suitable value of none–zero auxiliary parameter as well as considering enough iteration would even lead us to the exact solution so HAM can be widely used in engineering too.     

Convective heat transfer in a conducting fluid over a permeable stretching surface with suction and internal heat generation/absorption

We consider a convective flow in a porous medium of an incompressible viscous conducting fluid impinging on a permeable stretching surface with suction, and internal heat generation/absorption. Using a similarity transformation the governing equations of the problem are reduced to a coupled third-order nonlinear ordinary differential equations. We first examine a number of special cases for which we may obtain exact solutions. We then obtain analytical solutions (by the Homotopy Analysis Method) and numerical solutions (by a boundary value problem solver), in order to further study the behavior of the nonlinear differential equations, for various values of the physical parameters. Our numerical solutions are shown to agree with the available results in the literature. We then employ the numerical results to bring out the effects of the suction parameter, heat source/sink parameter, stretching parameter, porosity parameter, the Prandtl number and the free convection parameter on the flow and heat transfer characteristics. In the absence of suction and free convection, our findings are in agreement with the corresponding numerical results of Attia [H.A. Attia, On the effectiveness of porosity on stagnation point flow towards a stretching surface with heat generation, Comput. Mater. Sci. 38 (2007) 741-745].     

Approximate limit cycles of coupled nonlinear oscillators with fractional derivatives

In this paper, the homotopy analysis method (HAM) is presented to establish the accurate approximate analytical solutions for multi-degree-of-freedom (MDOF) coupled nonlinear oscillators with fractional derivatives. Approximate limit cycles (LCs) of two systems of the coupled fractional van der Pol (VDP) oscillators and the fractional damped Duffing resonator driven by a fractional VDP oscillator are exampled for illustrating the validity and great potential of the HAM. The presented approach can provide approximate LCs very accurately and efficiently compared with some direct simulation results. This method can keep high accuracy and efficiency for both weakly and strongly nonlinear problems with any given fractional order. Furthermore, it is capable of tracking unstable LCs which cannot be generated by some time-marching numerical algorithm. Based on the obtained results, we analyze effect of different fractional orders, coupling coefficient, and nonlinear coefficient of the coupled equations on amplitudes and frequencies of the LCs.     

Analysis of nonlinear BVPs motivated by fluid film flow over a surface

We study a coupled nonlinear boundary value problem which has been shown to have applications to fluid flow and heat transfer in a fluid film over a stretching surface for set values of the model parameters (one of which determines the size of the problem domain). For arbitrary values of these parameters we are able to establish the existence and uniqueness of a class of monotone solutions. Perturbation solutions are then constructed and used to approximate certain invariants for the solutions. We then study a related boundary value problem formed by imposing an additional boundary condition on one of the governing equations (which results in an ill-posed problem), and we arrive at conditions allowing for solutions to this four-parameter problem to agree with the solutions to the three-parameter problem.     

On the utility of the homotopy analysis method for non-analytic and global solutions to nonlinear differential equations

In recent work on the area of approximation methods for the solution of nonlinear differential equations, it has been suggested that the so-called generalized Taylor series approach is equivalent to the homotopy analysis method (HAM). In the present paper, we demonstrate that such a view is only valid in very special cases, and in general, the HAM is far more robust. In particular, the equivalence is only valid when the solution is represented as a power series in the independent variable. As has been shown many times, alternative basis functions can greatly improve the error properties of homotopy solutions, and when the base functions are not polynomials or power functions, we no longer have that the generalized Taylor series approach is equivalent to the HAM. In particular, the HAM can be used to obtain solutions which are global (defined on the whole domain) rather than local (defined on some restriction of the domain). The HAM can also be used to obtain non-analytic solutions, which by their nature can not be expressed through the generalized Taylor series approach. We demonstrate these properties of the HAM by consideration of an example where the generalizes Taylor series must always have a finite radius of convergence (and hence limited applicability), while the homotopy solution is valid over the entire infinite domain. We then give a second example for which the exact solution is not analytic, and hence, it will not agree with the generalized Taylor series over the domain. Doing so, we show that the generalized Taylor series approach is not as robust as the HAM, and hence, the HAM is more general. Such results have important implications for how iterative solutions are calculated when approximating solutions to nonlinear differential equations.