Soliton solutions for the Fitzhugh–Nagumo equation with the homotopy analysis method

An analytic technique, the homotopy analysis method (HAM), is applied to obtain the soliton solution of the Fitzhugh–Nagumo equation. The homotopy analysis method (HAM) is one of the most effective method to obtain the exact solution and provides us with a new way to obtain series solutions of such problems. HAM contains the auxiliary parameter ?, which provides us with a simple way to adjust and control the convergence region of series solution.     

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.     

Analytic approximate solutions for heat transfer of a micropolar fluid through a porous medium with radiation

This paper aims to present complete analytic solution to heat transfer of a micropolar fluid through a porous medium with radiation. Homotopy analysis method (HAM) has been used to get accurate and complete analytic solution. The analytic solutions of the system of nonlinear ordinary differential equations are constructed in the series form. The convergence of the obtained series solutions is carefully analyzed. The velocity and temperature profiles are shown and the influence of coupling constant, permeability parameter and the radiation parameter on the heat transfer is discussed in detail. The validity of our solutions is verified by the numerical results (fourth-order Runge–Kutta method and shooting method).     

Numerical solution of the generalized Zakharov equation by homotopy analysis method

In this paper, an analytic technique, namely the homotopy analysis method (HAM) is applied to obtain approximations to the analytic solution of the generalized Zakharov equation. The HAM contains the auxiliary parameter , which provides us with a simple way to adjust and control the convergence region of the solution series.     

Homotopy analysis method for generalized Benjamin–Bona–lMahony equation

An analytic technique, the homotopy analysis method (HAM), is applied to solve the generalized Benjamin–Bona–Mahony (BBM) equation. An explicit series solution is given, different from traditional analytic techniques, our approach is independent of knowing some parameters. This analytic method provides us with a new way to obtain series solutions of such problems. The homotopy analysis method contains the auxiliary parameter ħ, which provides us with a simple way to adjust and control the convergence region of series solution.     

Approximate analytical solutions of Schnakenberg systems by homotopy analysis method

In this paper, the homotopy analysis method (HAM) has been employed to obtain analytical solution of a two reaction–diffusion systems of fractional order (fractional Schnakenberg systems) which has been modeling morphogen systems in developmental biology. Different from all other analytic methods, HAM provides us with a simple way to adjust and control the convergence region of solution series by choosing proper values for auxiliary parameter h. The fractional derivative is described in the Caputo sense. The reason of using fractional order differential equations (FOD) is that FOD are naturally related to systems with memory which exists in most biological systems. Also they are closely related to fractals which are abundant in biological systems. The results derived of the fractional system are of a more general nature. Respectively, solutions of FOD spread at a faster rate than the classical differential equations, and may exhibit asymmetry. However, the fundamental solutions of these equations still exhibit useful scaling properties that make them attractive for applications.     

An approximation of the analytic solution of some nonlinear heat transfer in fin and 3D diffusion equations using HAM

In this article, the approximate solution of nonlinear heat diffusion and heat transfer equation are developed via homotopy analysis method (HAM). This method is a strong and easy‐to‐use analytic tool for investigating nonlinear problems, which does not need small parameters. HAM contains the auxiliary parameter ?, which provides us with a simple way to adjust and control the convergence region of solution series. By suitable choice of the auxiliary parameter ?, we can obtain reasonable solutions for large modulus. In this study, we compare HAM results, with those of homotopy perturbation method and the exact solutions. The first differential equation to be solved is a straight fin with a temperature‐dependent thermal conductivity and the second one is the two‐ and three‐dimensional unsteady diffusion problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Analytical solution of magnetohydrodynamic stagnation-point flow of a power-law fluid towards a stretching surface

A new kind of analytic technique, namely the homotopy analysis method (HAM), is employed to give an explicit analytical solution of the steady two-dimensional stagnation-point flow of an electrically conducting power-law fluid over a stretching surface when the surface is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. A uniform transverse magnetic field is applied normal to the surface. An explicit analytical solution is given by recursive formulae for the first-order power-law (Newtonian) fluid when the ratio of free stream velocity and stretching velocity is not equal to unity. For second and real order power-law fluids, an analytical approach is proposed for magnetic field parameter in a quite large range. All of our analytical results agree well with numerical results. The results obtained by HAM suggest that the solution of the problem under consideration converges.     

Simultaneous effects of partial slip and thermal-diffusion and diffusion-thermo on steady MHD convective flow due to a rotating disk

The purpose of present research is to derive analytical expressions for the solution of steady MHD convective and slip flow due to a rotating disk. Viscous dissipation and Ohmic heating are taken into account. The nonlinear partial differential equations for MHD laminar flow of the homogeneous fluid are reduced to a system of five coupled ordinary differential equations by using similarity transformation. The derived solution is expressed in series of exponentially-decaying functions using homotopy analysis method (HAM). The convergence of the obtained series solutions is examined. Finally some figures are sketched to show the accuracy of the applied method and assessment of various slip parameter γ, magnetic field parameter M, Eckert Ec, Schmidt Sc and Soret Sr numbers on the profiles of the dimensionless velocity, temperature and concentration distributions. Validity of the obtained results are verified by the numerical results.     

Heat transfer analysis of unsteady boundary layer flow by homotopy analysis method

This paper aims to present complete analytic solution to the unsteady heat transfer flow of an incompressible viscous fluid over a permeable plane wall. The flow is started due to an impulsively stretching porous plate. Homotopy analysis method (HAM) has been used to get accurate and complete analytic solution. The solution is uniformly valid for all time τ ∈ [0, ∞) throughout the spatial domain η ∈ [0, ∞). The accuracy of the present results is shown by giving a comparison between the present results and the results already present in the literature. This comparison proves the validity and accuracy of our present results. Finally, the effects of different parameters on temperature distribution are discussed through graphs.     

Analytical and numerical results for the Swift-Hohenberg equation

The problem of the Swift-Hohenberg equation is considered in this paper. Using homotopy analysis method (HAM) the series solution is developed and its convergence is discussed and documented here for the first time. In particular, we focus on the roles of the eigenvalue parameter α and the length parameter l on the large time behaviour of the solution. For a given time t, we obtain analytical expressions for eigenvalue parameter α and length l which show how different values of these parameters may lead to qualitatively different large time profiles. Also, the results are presented graphically. The results obtained reveal many interesting behaviors that warrant further study of the equations related to non-Newtonian fluid phenomena, especially the shear-thinning phenomena. Shear thinning reduces the wall shear stress.     

On comparison of series and numerical solutions for second Painlevé equation

This attempt presents the series solution of second Painlevé equation by homotopy analysis method (HAM). Comparison of HAM solution is provided with that of the Adomian decomposition method (ADM), homotopy perturbation method (HPM), analytic continuation method, and Legendre Tau method. It is revealed that there is very good agreement between the analytic continuation and HAM solutions when compared with ADM, HPM, and Legendre Tau solutions. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Some issues on HPM and HAM methods: A convergence scheme

The homotopy method for the solution of nonlinear equations is revisited in the present study. An analytic method is proposed for determining the valid region of convergence of control parameter of the homotopy series, as an alternative to the classical way of adjusting the region through graphical analysis. Illustrative examples are presented to exhibit a vivid comparison between the homotopy perturbation method (HPM) and the homotopy analysis method (HAM). For special choices of the initial guesses it is shown that the convergence-control parameter does not cover the HPM. In such cases, blindly using the HPM yields a non convergence series to the sought solution. In addition to this, HPM is shown not always to generate a continuous family of solutions in terms of the homotopy parameter. By the convergence-control parameter this can however be prevented to occur in the HAM.     

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     

Series solutions for unsteady laminar MHD flow near forward stagnation point of an impulsively rotating and translating sphere in presence of buoyancy forces

The similarity solution for the unsteady laminar incompressible boundary layer flow of a viscous electrically conducting fluid in stagnation point region of an impulsively rotating and translating sphere with a magnetic field and a buoyancy force gives a system of non-linear partial 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 magnetic parameter, buoyancy parameter and rotation parameter on the surface shear stresses and surface heat transfer. It is noted that the behavior of the HAM solution for the surface shear stresses and surface heat transfer is in good agreement with the numerical solution given in reference [H. S. Takhar, A. J. Chamkha, G. Nath, Unsteady laminar MHD flow and heat transfer in the stagnation region of an impulsively spinning and translating sphere in the presence of buoyancy forces, Heat Mass Transfer 37 (2001) 397].     

A new method for homoclinic solutions of ordinary differential equations

Consideration is given to the homoclinic solutions of ordinary differential equations. We first review the Melnikov analysis to obtain Melnikov function, when the perturbation parameter is zero and when the differential equation has a hyperbolic equilibrium. Since Melnikov analysis fails, using Homotopy Analysis Method (HAM, see [Liao SJ. Beyond perturbation: introduction to the homotopy analysis method. Boca Raton: Chapman & Hall/CRC Press; 2003; Liao SJ. An explicit, totally analytic approximation of Blasius’ viscous flow problems. Int J Non-Linear Mech 1999;34(4):759–78; Liao SJ. On the homotopy analysis method for nonlinear problems. Appl Math Comput 2004;147(2):499–513] and others [Abbasbandy S. The application of the homotopy analysis method to nonlinear equations arising in heat transfer. Phys Lett A 2006;360:109–13; Hayat T, Sajid M. On analytic solution for thin film flow of a forth grade fluid down a vertical cylinder. Phys Lett A, in press; Sajid M, Hayat T, Asghar S. Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt. Nonlinear Dyn, in press]), we obtain homoclinic solution for a differential equation with zero perturbation parameter and with hyperbolic equilibrium. Then we show that the Melnikov type function can be obtained as a special case of this homotopy analysis method. Finally, homoclinic solutions are obtained (for nontrivial examples) analytically by HAM, and are presented through graphs.     

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.     

Three-dimensional rotating flow induced by a shrinking sheet for suction

Series solution of magnetohydrodynamic (MHD) and rotating flow over a porous shrinking sheet is obtained by a homotopy analysis method (HAM). The viscous fluid is electrically conducting in the presence of a uniform applied magnetic field and the induced magnetic field is neglected for small magnetic Reynolds number. Similarity solutions of coupled non-linear ordinary differential equations resulting from the momentum equation are obtained. Convergence of the obtained solutions is ensured by the proper choice of auxiliary parameter. Graphs are sketched and discussed for various emerging parameters on the velocity field. The variations of the wall shear stress f″(0) and ?g′(0) are also tabulated and analyzed.     

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.     

A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations

In this paper, based on the homotopy analysis method (HAM), a powerful algorithm is developed for the solution of nonlinear ordinary differential equations of fractional order. The proposed algorithm presents the procedure of constructing the set of base functions and gives the high-order deformation equation in a simple form. Different from all other analytic methods, it provides us with a simple way to adjust and control the convergence region of solution series by introducing an auxiliary parameter ?$?$. The analysis is accompanied by numerical examples. The algorithm described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus.