Hang Xu - Academia.edu (original) (raw)

Papers by Hang Xu

$Research paper thumbnail of Series solutions of non-linear Riccati differential equations with fractional order$

Chaos Solitons & Fractals, 2009

In this paper, based on the homotopy analysis method (HAM), a new analytic technique is proposed ... more In this paper, based on the homotopy analysis method (HAM), a new analytic technique is proposed to solve nonlinear Riccati differential equation with fractional order. 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 h. Besides, it is proved that well-known Adomian's decomposition method is a special case of the homotopy analysis method when h = À1. This work illustrates the validity and great potential of the homotopy analysis method for the non-linear differential equations with fractional order. The basic ideas of this approach can be widely employed to solve other strongly non-linear problems in fractional calculus.

$Research paper thumbnail of Analysis of nonlinear fractional partial differential equations with the homotopy analysis method$

Communications in Nonlinear Science and Numerical Simulation, 2009

In this article, the homotopy analysis method is applied to solve nonlinear fractional partial di... more 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.

$Research paper thumbnail of Analysis of a time fractional wave-like equation with the homotopy analysis method$

Physics Letters A, 2008

The time fractional wave-like differential equation with a variable coefficient is studied analyt... more The time fractional wave-like differential equation with a variable coefficient is studied analytically. By using a simple transformation, the governing equation is reduced to two fractional ordinary differential equations. Then the homotopy analysis method is employed to derive the solutions of these equations. The accurate series solutions are obtained. Especially, whenh f =h g = −1, these solutions are exactly the same as those results given by the Adomian decomposition method. The present work shows the validity and great potential of the homotopy analysis method for solving nonlinear fractional differential equations. The basic idea described in this Letter is expected to be further employed to solve other similar nonlinear problems in fractional calculus.

Journal of Non-newtonian Fluid Mechanics, 2005

In this paper, the unsteady magnetohydrodynamic viscous flows of non-Newtonian fluids caused by a... more In this paper, the unsteady magnetohydrodynamic viscous flows of non-Newtonian fluids caused by an impulsively stretching plate are studied by means of an analytic technique, namely the homotopy analysis method. We give the analytic series solutions which are accurate and uniformly valid for all dimensionless time in the whole spatial region 0 ≤ η < ∞. To the best of authors' knowledge, such kind of analytic solutions have been never reported. Besides, the effects of the integral power-law index (n = 1, 2, 3) of the non-Newtonian fluids and the magnetic parameter M = 0, 1, 2 on the flows are investigated.

Chaos Solitons & Fractals, 2007

In this paper, an analytic technique, namely the homotopy analysis method, is employed to solve t... more In this paper, an analytic technique, namely the homotopy analysis method, is employed to solve the Fisher equation, which describes a family of travelling waves with a front. The explicit series solution for all possible wave speeds 0 < c < +1 is given. Such kind of explicit series solution has never been reported, to the best of authorÕs knowledge. Our series solution indicates that the solution contains an oscillation part when 0 < c < 2. The proposed analytic approach is general, and can be applied to solve other similar nonlinear travelling wave problems.