Elsevier

Applied Numerical Mathematics

Volume 158, December 2020, Pages 152-163
Applied Numerical Mathematics

A difference scheme for the time-fractional diffusion equation on a metric star graph

https://doi.org/10.1016/j.apnum.2020.07.022Get rights and content

Abstract

In this paper, we propose an unconditionally stable numerical scheme based on finite difference for the approximation of time-fractional diffusion equation on a metric star graph. The fractional derivative is considered in Caputo sense and the so-called L1 method is used for the discrete approximation of Caputo fractional derivative. The convergence and stability of the difference scheme has been proved by means of energy method. Test examples are illustrated in order to verify the feasibility of the proposed scheme.

Introduction

We consider a graph G=(V,E) consisting of a finite set of vertices (nodes) V=(vi)i=0k and a finite set of edges E (such as heat conducting elements) connecting these nodes. The graph considered in this work is a metric graph [22]. Therefore, each edge ei, 1ik is parametrized by an interval (0,li) such that 0<li<. Hence, we define a coordinate system on each edge ei of the star graph (see Fig. 1) by taking v0 as the origin and x(0,li) as the coordinate. We consider the following IBVP for the time-fractional diffusion equation (TFDE) on a metric star graph G,D0,tαCyi(x,t)=2yi(x,t)x2+fi(x,t),x(0,li),t(0,T),0<α<1,1ik,yi(x,0)=yi0(x),x(0,li),1ik,y1(0,t)=y2(0,t)==yk(0,t),t(0,T),i=1kyi(0,t)x=0,yi(li,t)=0,t(0,T),1ik. Here, D0,tαC is the Caputo fractional derivative of order α with respect to t, conditions (1.3) and (1.4) are called transmission conditions, namely, continuity and Kirchoff conditions, respectively and (1.5) denotes the Dirichlet conditions at boundary nodes vi. This paper aims to develop a difference scheme for the numerical solution of IBVP (1.1)-(1.5) along with the stability and convergence analysis of the proposed scheme. We refer to [21] for the well-posedness and the regularity of the solution of system (1.1)-(1.5) and [18], [19] for the existence and uniqueness of solution for fractional boundary value problems on metric star graph.

The study of differential equation on metric graphs dates back to the 1980s with Lumer's work [15] on ramification spaces. Evolutionary problems (such as parabolic equations) on metric graphs were studied by von Below in [31]. Domain decomposition technique for the approximation of differential equations on metric graphs were discovered by G. Leugering. For instance, in [8], dynamic domain decomposition method is applied for the problems of network of strings (modeled by wave equation) and Timoshenko beams, while the domain decomposition for the optimal control problems for dynamic networks of elastic strings has been studied in [10], see also [7], [9], [11]. Recently, finite element method [2] is used for the numerical solution of differential equations (initial and boundary value problems) on metric graphs. The approximation of Burger's equation on metric star graph using spectral graph wavelet method was studied in [29]. For more problems on metric graphs and their numerical solution, we refer to [4], [27] and references therein.

On the other hand, there are various complex system in different fields such as physics [5], bioengineering [16], hydrology [17], stochastic processes [30], neural networks [20] etc, which cannot be modeled using ordinary differential equations, and thus, fractional differential equation models are established in these fields. A strong motivation for the study of fractional diffusion equations comes from the fact that they efficiently describe the phenomenon of anomalous diffusion [14]. Various difference schemes have been constructed for the approximation of TFDE on Euclidean (bounded) domains. For instance in [13], Lin and Xu proposed the difference scheme based on finite difference in time and spectral methods in space for the numerical solution of TFDE. They constructed L1 method (having accuracy dependent on the fractional order α) for the discrete approximation of Caputo derivative which is widely used till date for the numerical solution of differential equations with Caputo fractional derivative. In [34], Zhang et al. numerically solved the TFDE on non-uniform mesh, where the L1 method was used for the time discretizations and compact difference scheme for the space. Later in [1], Alikhnov proposed the new difference analog, namely, L21σ formula for the Caputo derivative and solved the TFDE using finite difference with second and fourth order accuracy in space, while the approximation of TFDE using finite element method in space and fractional linear multistep method in time was studied in [33]. We refer to [3], [12], [23], [25] and references therein for more results on the numerical solution of fractional diffusion equations.

To the best of authors' knowledge, there has not been any published work related to numerical solution of the time-fractional diffusion equation on metric graphs so far. In this paper, we focus on developing a difference scheme for the approximation of TFDE on metric star graph as well as the stability and convergence analysis for the proposed scheme.

The rest of the paper is organized as follows. In Sec. 2, we give basic definition of fractional calculus, define the function spaces for star graph G and investigate the analytic solution for TFDE. In Sec. 3, we propose the difference scheme for the approximation of TFDE on metric star graph and prove the stability and the convergence of the proposed scheme. In Sec. 4, two test examples are illustrated in order to verify the accuracy of the proposed method. Sec. 5 concludes the work done.

Section snippets

Preliminaries

In this section, we give basic definitions of fractional calculus and investigate the analytic solution of time-fractional diffusion equation on a star graph.

Definition 2.1

The Caputo fractional derivative of order α>0 for a function fCn(a,b), is defined as [6]Da,tαCf(t)=1Γ(nα)(at(tξ)nα1f(n)(ξ)dξ), where n1<αn, nN and Γ(.) is the Euler's gamma function. In particular, when 0<α<1, then Caputo derivative is given byDa,tαCf(t)=1Γ(1α)(at(tξ)αf(ξ)dξ).

Definition 2.2

The Mittag-Leffler function is defined as

Finite difference approximation for TFDE on metric star graph

In this section, we develop an unconditionally stable numerical scheme based on the finite difference for the approximation of fractional diffusion equation (1.1)-(1.5). In what follows, we assume that IBVP (1.1)-(1.5) has a sufficient regular solution.

Now, we discretize each edge ei, i.e. the interval (0,li) as xi,m=mΔxi, where Δxi denotes the spatial discretization step for the edge ei given by Δxi=li/M, m=0,1,,M, xi,0=0 and xi,M=li,1ik. For the time grid, we define tn=nΔt, n=0,1,,N, t0=0

Numerical results

In this section, we consider two test examples in order to verify the stability and convergence properties of the proposed scheme proved in the previous section. We define the error between exact solution and the approximated solution for the time-fractional diffusion equation (1.1)-(1.5) asEi(N,M)=max0mM|yi(xm,tN)yi,mN| and the rate of convergence as(rate)i=log2(Ei(N,M/2)Ei(N,M)),1ik. Before embarking on the examples, we first demonstrate how to solve the difference scheme (3.3)-(3.7).

Conclusion

In this paper, the stability and the convergence analysis of the difference scheme approximating the time fractional diffusion equation on a metric star graph is studied. The stability of the proposed scheme has been proved using energy method. The accuracy of the proposed scheme is illustrated using two test examples. Since any metric graph can be decomposed into star graph, the proposed scheme can be applied to any general metric graph.

Acknowledgements

The second author acknowledges the support provided by the Department of Science and Technology, India, under the grant number SERB/F/11946/2018-2019. The authors would also like to thank the referees' valuable suggestion for the improvement of the manuscript.

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