Purpose -The purpose of this paper is to consider the time-fractional diffusion-wave equation. The time-fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order 2 (0, 2]. The fractional derivatives are described in the Caputo sense. Design/methodology/approach -The two methods in applied mathematics can be used as alternative methods for obtaining an analytic and approximate solution for different types of differential equations. Findings -Four examples are presented to show the application of the present techniques. 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. Originality/value -In this paper, the variational iteration and homotopy perturbation methods are used to obtain a solution of a fractional diffusion equation.
IntroductionA fractional diffusion-wave equation is a linear integro partial differential equation obtained from the classical diffusion or wave equation by replacing the first-or secondorder time derivative term by a fractional derivative of order 2 (0, 2]. These equations arise in anomalous diffusion and sub-diffusion systems, the description of fractional random walk and the unification of diffusion and wave propagation phenomena. The nature of the diffusion is characterized by the temporal scaling of the mean-square displacement hr 2 (t)i % t . For standard diffusion ¼ 1, whereas in anomalous subdiffusion < 1, and in anomalous super-diffusion > 1. Both types of anomalous diffusion have been unified in continuous time random walk models with spatial and temporal memories, see e.
In this study, linear and nonlinear partial differential equations with the nonhomogeneous initial conditions are considered. We used Variational iteration method (VIM) and Homotopy perturbation method (HPM) for solving these equations. Both methods are used to obtain analytic solutions for different types of differential equations. Four examples are presented to show the application of the present techniques. 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.
In this paper, a new tool for the solution of nonlinear differential equations is presented. It is named rational homotopy perturbation method (RHPM). It delivers a high precision representation of the nonlinear differential equation using a few linear algebraic terms. In order to assess the benefits of this proposal, three nonlinear problems are solved and compared against other semi-analytic methods or numerical methods. Furthermore, in order to deal with BVP problems, we propose a modification of RHPM method. The obtained results show that RHPM is a powerful tool capable to generate highly accurate rational solutions.
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