%0 Journal Article
%T Solving the Fractional Rosenau-Hyman Equation via Variational Iteration Method and Homotopy Perturbation Method
%A R. Yulita Molliq
%A M. S. M. Noorani
%J International Journal of Differential Equations
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/472030
%X In this study, fractional Rosenau-Hynam equations is considered. We implement relatively new analytical techniques, the variational iteration method and the homotopy perturbation method, for solving this equation. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for fractional Rosenau-Hynam equations. In these schemes, the solution takes the form of a convergent series with easily computable components. The present methods perform extremely well in terms of efficiency and simplicity. 1. Introduction Recent advances of fractional differential equations are stimulated by new examples of applications in fluid mechanics, viscoelasticity, mathematical biology, electrochemistry, and physics. For example, the nonlinear oscillation of earthquake can be modeled with fractional derivatives [1], and the fluid-dynamic traffic model with fractional derivatives [2] can eliminate the deficiency arising from the assumption of continuum traffic flow. Based on experimental data fractional partial differential equations for seepage flow in porous media are suggested in [3], and differential equations with fractional order have recently proved to be valuable tools to the modeling of many physical phenomena [4]. Fractional partial differential equations also have studied and successfully solved such as the space-time fractional diffusion-wave equation [5¨C7], the fractional advection-dispersion equation [8, 9], the fractional KdV equation [10], and the linear inhomogeneous fractional partial differential equations [11]. Most nonlinear differential equations are usually arising from mathematical modeling of many physical systems. In most cases, it is very difficult to achieve analytic solutions of these equations. Perturbation techniques are widely used in science and engineering to handle nonlinear problems and do great contribution to help us understand many nonlinear phenomena. However, perturbation techniques are based on the existence of small/large parameter. Therefore, these techniques are not valid for strongly nonlinear problems. The homotopy perturbation method (HPM) is the new approach for finding the approximate analytical solution of linear and nonlinear problems. The method was first proposed by He [12, 13] and was successfully applied to solve nonlinear wave equation by He [14¨C16]. The convergence of Homotopy perturbation series to the exact solution is considered in [17]. Similarly, applying the variational iteration method,
%U http://www.hindawi.com/journals/ijde/2012/472030/