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Research ArticleFully Discrete Local Discontinuous Galerkin Approximation forTime-Space Fractional Subdiffusion/Superdiffusion Equations

Meilan Qiu,1,2 LiquanMei,1 and Dewang Li2

1School of Mathematics and Statistics, Xian Jiaotong University, Xian 710049, China2Department of Mathematics, Huizhou University, Huizhou, Guangdong 516007, China

Correspondence should be addressed to Liquan Mei; lqmei@mail.xjtu.edu.cn and Dewang Li; ldwldw1976@126.com

Received 16 September 2016; Revised 24 December 2016; Accepted 19 January 2017; Published 16 March 2017

Academic Editor: Giampaolo Cristadoro

Copyright 2017 Meilan Qiu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We focus on developing the finite difference (i.e., backward Euler difference or second-order central difference)/local discontinuousGalerkin finite element mixed method to construct and analyze a kind of efficient, accurate, flexible, numerical schemesfor approximately solving time-space fractional subdiffusion/superdiffusion equations. Discretizing the time Caputo fractionalderivative by using the backward Euler difference for the derivative parameter (0 < < 1) or second-order central differencemethod for (1 < < 2), combined with local discontinuous Galerkinmethod to approximate the spatial derivative which is definedby a fractional Laplacian operator, two high-accuracy fully discrete local discontinuous Galerkin (LDG) schemes of the time-space fractional subdiffusion/superdiffusion equations are proposed, respectively. Through the mathematical induction method,we show the concrete analysis for the stability and the convergence under the 2 norm of the LDG schemes. Several numericalexperiments are presented to validate the proposed model and demonstrate the convergence rate of numerical schemes. Thenumerical experiment results show that the fully discrete local discontinuous Galerkin (LDG) methods are efficient and powerfulfor solving fractional partial differential equations.

1. Introduction

During the last few decades, fractional calculus emerges as anatural description for a broad range of nonclassical phenom-ena in the applied sciences and engineering. For example,anomalous diffusion is one of the most important conceptsinmodern physics [13] and is presented in extremely diverseengineering fields such as vibration and acoustic dissipationin soft matter [4], polymer dynamics [5], turbulence flow[6], glassy and porous media [7], charges transport in amor-phous semiconductor [8], chemistry and biochemistry [9],contaminant transport in groundwater flow [10], and chaoticdynamics of classical conservative systems [11].

To date, many solution techniques for fractional differ-ential equations have exploited the properties of the Fourierand Laplace transforms of the operators to confirm a classicalsolution. Typically, Gorenflo et al. in [12] used the methodof Laplace transform to obtain the scale-invariant solutionof the time fractional diffusion-wave equation according

to the Wright function. Starting from the Fourier-Laplacerepresentation, Mainardi et al. in [13] derived the explicitexpression of the Green function (i.e., the fundamentalsolution) for the Cauchy problem with respect to its scalingand similarity properties. Since fractional order differen-tial operators possess their own properties, (i) they arenonlocal operators and (ii) the adjoint operator itself offractional differential operator is a nonnegative operator.Therefore, most fractional differential equations do not haveexact solutions. Thus, some potent and accurate numericalmethods for these equations are developed. For example,a finite difference method was applied to construct thenumerical approximation for the time fractional diffusionequation and introduce an implicit difference scheme forthe space-time fractional diffusion equation was studied in[1417]. Ervin and Roop in [18] considered the stationaryfractional advection dispersion equation by using the stan-dard finite element method. Zhao et al. adopted the similarmethod to discuss the numerical solutions of the multiterm

HindawiAdvances in Mathematical PhysicsVolume 2017, Article ID 4961797, 20 pageshttps://doi.org/10.1155/2017/4961797

https://doi.org/10.1155/2017/4961797

2 Advances in Mathematical Physics

time-space Riesz fractional advection-diffusion equations intheir paper [19] and Deng in [20] presented the finite elementapproximation for the space-time fractional Fokker-Planckequation. Lin and Xu in [21] and Li and Xu in [22] proposeda spectral method for time or space fractional diffusionequations based on a weak formulation and the detailederror analysis was carried out. A new matrix approach tosolve fractional partial differential equations numericallyhas been developed by Podlubny in [23]. Li et al. in [24]used the fractional difference method in time and the finiteelement method in space, for the time-space fractional ordersubdiffusion and superdiffusion equations in which theypresented the error estimates for the full discrete schemes.And for the subdiffusion linear problem, they obtain theapproximation of order 2 + 2 in theoretical; for thesuperdiffusion linear problem, they got the approximation oforder 2 + 12. However, their numerical experimentscannot be consistent with the corresponding theoreticalresults very well, particularly for the superdiffusion prob-lems.

Moreover, high order accuracy numerical methods forsolving fractional differential equations are still very limited.Discontinuous Galerkin method is famous for extreme flex-ibility and high-accuracy properties [25]. By applying thismethod, differential equations can be solved locally and accu-rately while we only need to assemble and solve a large linearsystem. Deng and Hesthaven in [26] studied the convergencerate analysis of local discontinuous Galerkin method (LDG)for solving spatial fractional diffusion equation, which wasthe first attempt to solve fractional derivative equations byusing the LDG method; at the same time, the authors inthis paper speculated that the LDG method has significantpotential in solving fractional calculus problems. After thatlots of researchers use the LDG method to analyze the timediscretization of fractional differential equations [27, 28].Recently, Xu and Hesthaven [29] obtained a consistent andhigh-accuracy numerical scheme for fractional convection-diffusion equationswith a space fractional Laplacian operatorof order (1 < < 2) through exploiting LDGmethod.Theyintroduced a new technique by rewriting the fractional Lap-lacian operator as a composite of first-order derivativesand a fractional integral; then they converted the frac-tional convection-diffusion equation into a system of loworder equations. Optimal order of convergence (+1)for the space fractional diffusion problem and suboptimalorder of convergence (+1/2) for the general spatial frac-tional convection-diffusion problem are established, respec-tively.

In this paper, motivated by the research work of [24],we intend to construct and analyze the numerical schemesfor solving the time-space fractional subdiffusion equation asfollows: (, ) = ( ()/2) (, ) , , (0, ] , (, 0) = 0 () , ,

(1)

and the time-space fractional superdiffusion equation asfollows: (, ) = ( ()/2) (, ) , , (0, ] , (, 0) = () , , (, 0) = () , ,

(2)

where = (, ), > 0, 1 < < 2 denotes the ordernumber of space fractional derivatives, 0 < < 1 describesthe fractional subdiffusion equation (1), 1 < < 2 describesthe fractional superdiffusion equation (2), is a real positiveparameter, and 0, (), () 10 () are the given smoothfunctions.

For the sake of simplicity, we just consider the periodicboundary conditions in this paper. Notice that this assump-tion is not essential; our method can be directly extended tothe problems with aperiodic boundary condition.

The time fractional derivative in (1) uses the Caputofractional partial derivative of order , defined as [30, 31] (, )

= {{{{{1 (1 ) 0 (, ) ( ) , 0 < < 1, (, ) , = 1,

(3)

and the time fractional Caputo derivative in (2) defined as[24] (, )

= {{{{{{{1 (2 ) 0 ( )1 2 (, )2 , 1 < < 2,2 (, )2 , = 2,

(4)

where () is the Gamma function.The space fractional terms in (1) and (2) are defined

through the fractional Laplacian operator (( )/2)(, ),1 < < 2, which is also known as the Riesz fractionalderivative (/||)(, )under the assumption that(, ),(, ), . . . , (1)(, ) vanishes at the end point of an infinitedomain (i.e., = ) [32].

The spatial fractional differential operator is also animportant tool to describe the anomalous diffusion phe-nomenon; when 1 < < 2, it represents a Levy -stable flight[33]; when 2, it models a Brownian diffusion process.

The rest of this paper is organized as follows. In thenext section, we review the appropriate functional spacesand some important definitions and properties of fractionalderivatives. By using the technique which is proposed in[29], that is, through converting the space fractional operatorof order into a composite of first-order derivative and

Advances in Mathematical Physics 3

a fractional integral of order 2 , the fully discrete LDGschemes for the time-space fractional subdiffusion equationand the time-space fractional superdiffusion equation areconstructed, respectively, in Section 3. Sections 4 and 5provide the detailed stability and 2 error analysis for thetwo fully discrete schemes, respectively. Several examples arepresented to test our theoretical results in Section 6. Section 7contains the concluding remarks.