Showing papers on "Homotopy analysis method published in 2010"

Solving nonlinear fractional partial differential equations using the homotopy analysis method

Abstract: In this article, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations. On the basis of the homotopy analysis method, a scheme is developed to obtain the approximate solution of the fractional KdV, K(2,2), Burgers, BBM-Burgers, cubic Boussinesq, coupled KdV, and Boussinesq-like B(m,n) equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The homotopy analysis method for partial differential equations of integer-order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions of the studied models are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010

Homotopy analysis method for the Kawahara equation

Abstract: The homotopy analysis method (HAM) is used to find a family of travelling-wave solutions of the Kawahara equation. This approximate solution, which is obtained as a series of exponentials, has a reasonable residual error. 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. This method is reliable and manageable.

Homotopy based solutions of the Navier–Stokes equations for a porous channel with orthogonally moving walls

Abstract: This paper focuses on the theoretical treatment of the laminar, incompressible, and time-dependent flow of a viscous fluid in a porous channel with orthogonally moving walls Assuming uniform injection or suction at the porous walls, two cases are considered for which the opposing walls undergo either uniform or nonuniform motions For the first case, we follow Dauenhauer and Majdalani Phys Fluids 15, 1485 2003 by taking the wall expansion ratio to be time invariant and then proceed to reduce the Navier‐Stokes equations into a fourth order ordinary differential equation with four boundary conditions Using the homotopy analysis method HAM, an optimized analytical procedure is developed that enables us to obtain highly accurate series approximations for each of the multiple solutions associated with this problem By exploring wide ranges of the control parameters, our procedure allows us to identify dual or triple solutions that correspond to those reported by Zaturska et al Fluid Dyn Res 4, 151 1988 Specifically, two new profiles are captured that are complementary to the type I solutions explored by Dauenhauer and Majdalani In comparison to the type I motion, the so-called types II and III profiles involve steeper flow turning streamline curvatures and internal flow recirculation The second and more general case that we consider allows the wall expansion ratio to vary with time Under this assumption, the Navier‐ Stokes equations are transformed into an exact nonlinear partial differential equation that is solved analytically using the HAM procedure In the process, both algebraic and exponential models are considered to describe the evolution of t from an initial 0 to a final state 1 In either case, we find the time-dependent solutions to decay very rapidly to the extent of recovering the steady state behavior associated with the use of a constant wall expansion ratio We then conclude that the time-dependent variation of the wall expansion ratio plays a secondary role that may be justifiably ignored © 2010 American Institute of Physics doi:101063/13392770

Numerical study of the solution of the Burgers and coupled Burgers equations by a differential transformation method

Abstract: In this paper, the Differential Transformation Method (DTM) is employed to obtain the numerical/analytical solutions of the Burgers and coupled Burgers equations. We begin by showing how the differential transformation method applies to the linear and nonlinear parts of any PDE and give some examples to illustrate the sufficiency of the method for solving such nonlinear partial differential equations. We also compare it against three famous methods, namely the homotopy perturbation method, the homotopy analysis method and the variational iteration method. These results show that the technique introduced here is accurate and easy to apply.

The solution of the linear fractional partial differential equations using the homotopy analysis method

Abstract: In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the homotopy analysis method for partial differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

Unsteady flow with heat and mass transfer of a third grade fluid over a stretching surface in the presence of chemical reaction

Abstract: This paper describes the unsteady flow with heat and mass transfer characteristics in a third grade fluid bounded by a stretching sheet. The resulting problems are solved by means of homotopy analysis method (HAM). Convergence of derived series solutions is explicitly discussed. Graphical results for various interesting parameters are presented and analyzed.

Analytic approximate solutions for unsteady boundary-layer flow and heat transfer due to a stretching sheet by homotopy analysis method

Abstract: In this work, the homotopy analysis method is applied to study the unsteady boundary-layer flow and heat transfer due to a stretching sheet. 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 non-dimensional parameter on the heat transfer is discussed in detail. The validity of our solutions is verified by the numerical results.

The application of homotopy perturbation method for MHD flows of UCM fluids above porous stretching sheets

Abstract: In this study, by means of homotopy perturbation method (HPM) an approximate analytical solution of the magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell (UCM) fluid over a porous stretching sheet is obtained The main feature of the HPM is that it deforms a difficult problem into a set of problems which are easier to solve HPM produces analytical expressions for the solution of nonlinear differential equations The obtained analytic solution is in the form of an infinite power series In this work, the analytical solution obtained by using only two terms from HPM solution The results reveal that the proposed method is very effective and simple and can be applied to other nonlinear problems Also it is shown that this method coincides with homotopy analysis method (HAM) for the studied problem

Solution of the Jeffery-Hamel flow problem by optimal homotopy asymptotic method

Abstract: The present article addresses Jeffery-Hamel flow: fluid flow between two rigid plane walls, where the angle between them is [email protected] A new analytical method called the optimal homotopy asymptotic method (OHAM) is briefly introduced, and then employed to solve the governing equation. The validity of the homotopy asymptotic method is ascertained by comparing our results with numerical (Runge-Kutta method) results. The effects of the Reynolds number (Re) and the angle between the two walls ([email protected]) are highlighted in the proposed work. The results reveal that the proposed analytical method can achieve good results in predicting the solutions of such problems.

He's homotopy perturbation method for solving the space-and time-fractional telegraph equations

Abstract: In this study, homotopy perturbation method (HPM) is used to obtain analytic and approximate solutions of the space-and time-fractional telegraph equations. The space-and time-fractional derivatives are considered in the Caputo sense. The analytic solutions are calculated in the form of a series with easily computable terms. Some examples are given. The results reveal that HPM is very effective and convenient.