Homotopy Analysis Method

A scheme is developed for the numerical study of the Korteweg-de Vries (KdV) and the Korteweg-de Vries Burgers (KdVB) equations with initial conditions by a homotopy approach. Numerical solutions obtained by homotopy analysis method are compared with exact solution. The comparison shows that the obtained solutions are in excellent agreement. Full text

This work presents HomotopyDC, a package that carries out the DC analysis of a circuit by using homotopy methods, i.e. methods that are able to find more than one DC solution. The capabilities of MAPLE have been used to their full extent in order to formulate the equilibrium equation of the circuit in a fully symbolic form, which is then used to formulate the homotopic equation. Not only are the DC solutions calculated, but also their stability is assessed. Several homotopy schemes have been implemented within the package

This paper investigates the Magnetohydrodynamic flow and heat transfer towards a stationary/moving plate with convective boundary conditions in presence of thermal radiation. The governing equations are simplified by similarity transformations which are then solved by homotopy analysis method (HAM). Effects of emerging parameters such as Prandtl number, local Grashof number, Magnetic parameter, Radiation parameter and the Biot number on the velocity field and temperature field are plotted and discussed. It is found that the temperature of the stationary flat plate is higher than the temperature of the moving flat plate. The obtained results are compared with other results obtain in previous works so that the high accuracy of the present result is apparent. Keywords MHD boundary layer flow · Inclined plate · Grashof number · Radiation · HAM

In this paper, the boundary-layer natural convection flow on a permeable vertical plate with thermal radiation and mass transfer is studied when the plate moves in its own plane. A uniform temperature with uniform species concentration at the plate is affected and the fluid is considered to be a gray, absorbing–emitting. A viscous flow model is presented using boundary-layer theory comprising the momentum, energy, and concentration equations, which is solved analytically by means of an excellent method called homotopy analysis method (HAM). First, a comparison between HAM results and those obtained by means of a higher-order numerical method, namely differential quadrature method (DQM), is done. Close agreement of two sets of results indicates the accuracy of the HAM. The velocity, temperature, and concentration distributions are displayed graphically, and a parametric study is performed in which the effect of various parameters on the skin friction, the local Nusselt number (Nn), and the local Sherwood number (Mu) are investigated.

This article studies the motion of temperature dependent plastic dynamic viscosity and thermal conductivity of steady
incompressible laminar free convective magnetohydrodynamic (MHD) Casson fluid flow over an exponentially stretching surface with suction and exponentially decaying internal heat generation. It is assumed that the natural convection is driven by buoyancy
and space dependent heat generation. The viscosity and thermal conductivity of Casson fluid is assumed to vary as a linear function
of temperature. By using suitable transformation, the governing partial differential equations corresponding to the momentum and energy equations are converted into non-linear coupled ordinary differential equations and solved by the Homotopy analysis method. A new kind of averaged residual error is adopted and used to find the optimal convergence control parameter. A parametric study is performed to illustrate the influence of Prandtl number, Casson parameter, temperature dependent viscosity,
temperature dependent thermal conductivity, Magnetic parameter and heat source parameter on the fluid velocity and temperature profiles within the boundary layer. The flow controlling parameters are found to have a profound effect on the resulting flow profiles.

In this paper a novel hybrid spectral-homotopy analysis technique developed by Motsa et al. (2009) and the homotopy analysis method (HAM) are compared through the solution of the nonlinear equation for the MHD Jeffery–Hamel problem. An analytical solution is obtained using the homotopy analysis method (HAM) and compared with the numerical results and those obtained using the new hybrid method. The results show that the spectral-homotopy analysis technique converges at least twice as fast as the standard homotopy analysis method.

In this study, the problem of unsteady squeezing flow between circular parallel plates is performed. The similarity transformation is applied to reduce a governing partial differential equation (PDE) to a nonlinear ordinary differential equation (ODE) in dimensionless form. A new method, called the homotopy analytical method (HAM), is applied to obtain approximate analytical solution of this nonlinear equation. The obtained solutions, in comparison with the numerical solutions (fourth-order Runge–Kutta scheme), admit a remarkable accuracy.

The limit cycle of the van der Pol oscillator, $\ddot{x}+ \epsilon (x^2-1) \dot{x} + x =0$, is studied in the plane $(x,\dot{x})$ by applying the homotopy analysis method. A recursive set of formulas that approximate the amplitude and form of this limit cycle for the whole range of the parameter $\epsilon$ is obtained. These formulas generate the amplitude with an error less than 0.1%. To our knowledge, this is the first time where an analytical approximation of the amplitude of the van der Pol limit cycle, with validity from the weakly up to the strongly nonlinear regime, is given.

In the present investigation we have analyzed the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface. The effects of thermal radiation are carried out for two cases of heat transfer analysis known as (1) Prescribed exponential order surface temperature (PEST) and (2) Prescribed exponential order heat flux (PEHF). The highly nonlinear coupled partial differential equations of