An alternating direction implicit Galerkin finite element method for the distributed-order time-fractional mobile–immobile equation in two dimensions

Abstract

In this paper, we shall present the alternating direction implicit (ADI) Galerkin finite element method (FEM) for solving the distributed-order time-fractional mobile–immobile equation in two dimensions. In the time direction, the backward Euler method is used to deal with the temporal first-order derivative, and the weighted and shifted Grünwald formula is employed to discretize the distributed-order time-fractional derivative. Galerkin FEM is used for discretization of the spatial direction, and then an ADI Galerkin finite element scheme is constructed. The stability and $L^2$-norm convergence are proved. Several numerical experiments are provided to verify our theoretical analysis.

Introduction

In recent years, fractional order problems become increasingly popular, which promotes the development of the fractional partial differential equations (FPDEs) to some extent. Compared with integer-order partial differential equations, FPDEs provide a simpler and more accurate description of complex mechanical and physical processes that are historically dependent and spatially nonlocal, which has aroused the interest of a large number of scholars. For solving FPDEs, various numerical methods were proposed, which include the finite difference methods [1], [2], [3], [4], [5], [6], [7], the finite element method [8], the finite volume method [9], the weak Galerkin method [10], the two-grid methods [11], [12], the spline collocation methods [13], [14], [15], [16], etc. The numerical researches on the single-order time-fractional differential equations were considered by the papers above. However, for some special cases, such as the case of logarithmic increasing mean square displacement, a single-order diffusion model is not enough to describe it, which leads to the development of a distributed-order diffusion equation [17]. This kind of diffusion is called ultra-slow diffusion or strong anomaly [18], and it usually appears in polymer physics, quenching random force fields, iterative mapping models, etc. [19]. Actually, Luchko [17] pointed out that the multi-term fractional-order diffusion equation can be interpreted as a special case of distributed-order fractional diffusion equation with a linear combination of the Dirac $\delta$-functions as the weight function.

In the past few years, the distributed-order differential equations were studied by many researchers. In the field of mathematical theory, via using the separation of variables and Laplace transform, Naber looked for the solution of distributed-order fractional sub-diffusion equations [20]. Kochubei considered and analyzed the derivatives and integrals of distributed order [18]. Atanackovic et al. considered the Cauchy problem of a time distributed-order diffusion-wave equation [21]. Meerschaert et al. discussed the explicit strong solutions and stochastic analogues [19]. In the field of numerical investigation, many researchers have designed various methods for solving the distributed-order fractional differential equations. In the integration interval $[\alpha ,\beta ]$, Katsikadelis [22] considered a new numerical method for solving distributed-order fractional differential equations of the general form. Then Morgado and Rebelo [23] developed an implicit scheme for solving a distributed-order time-fractional reaction–diffusion equation with a nonlinear source term. For solving the multi-term distributed-order time-fractional equations, Gao and Sun established the temporal second-order difference schemes based on the interpolation approximation [24]. The stability and convergence were proved. In [25], Du et al. formulated and analyzed a high order difference schemes with smooth solutions in one and two space dimensions. For the distributed-order time fractional diffusion equation with the non-smooth initial data, Jin et al. [26] proposed a rigorous numerical analysis of two fully-discrete schemes, and the theoretical analysis was presented. On a semi-infinite spatial domain, the spectral scheme and pseudospectral scheme were studied by Chen et al. in [27]. After that, Abbaszadeh and Dehghan [28] presented an improved meshless method with the error estimate, and Yang et al. developed an orthogonal spline collocation (OSC) scheme [29].

In this article, we consider an ADI Galerkin finite element scheme for the distributed-order time-fractional mobile–immobile equation in two dimensions

$$\begin{array}{c}{u}_t(x,y,t)+{\mathbb{D}}_t^{\omega }u(x,y,t)-\Delta u(x,y,t)=f(x,y,t),(x,y)\in \Omega ,t\in (0,T],\hfill \end{array}$$

the initial condition and the boundary conditions are

$$u(x,y,0)=0,(x,y)\in \Omega ,$$$$\begin{array}{c}u(x,y,t)=0,(x,y)\in \partial \Omega ,t\in (0,T],\hfill \end{array}$$

respectively, where the domain $\Omega =\mathcal{I}\times \mathcal{I}$, $\mathcal{I}=(0,L)$ with the boundary $\partial \Omega$, the notation

$$\begin{array}{c}{\mathbb{D}}_t^{\omega }u(x,y,t)={\int }_0^1\omega \left(\alpha \right)D_t^{\alpha }u(x,y,t)d\alpha ,\hfill \end{array}$$

the Caputo fractional derivative [30] is defined by

$$D_t^{\alpha }u(x,y,t)=\left\{\begin{array}{c}\frac{1}{\Gamma (1-\alpha )}{\int }_0^t{(t-s)}^{-\alpha }\frac{\partial u}{\partial s}(x,y,s)ds,0\le \alpha <1,\hfill \\ \frac{\partial u}{\partial t}(x,y,t),\alpha =1,\hfill \end{array}\right.$$

the weight function $\omega \left(\alpha \right)\ge 0$, ${\int }_0^t\omega \left(\alpha \right)d\alpha =c_0>0$, and the source term $f(x,y,t)$ is given smooth function. Without loss of generality, the zero initial condition (1.2) is supposed. Besides, if non-zero initial value problem is encountered, we can take into account the analogous problem for $v(x,y,t)=u(x,y,t)-u(x,y,0)$.

As can be seen, the advantage of the ADI methods is that they reduce the computational cost by dividing a multi-dimensional problem into sets of independent one-dimensional problems. For the study of fractional-order models, a lot of work has been done by ADI methods, e.g., ADI finite difference methods were used for the single-order time-fractional equations [1], [31] and the distributed-order time-fractional equations [32], [33], ADI OSC methods were utilized for single-order time-fractional equations [13], [14], [34], [35], ADI finite element methods were applied to the researches of single-order time/space fractional equations [36], [37], [38], etc. However, the temporal distributed-order ADI finite element method has not been studied yet. Below, we shall consider and analyze that.

The main purpose of this paper is to construct an ADI Galerkin finite element scheme for the distributed-order time-fractional mobile–immobile equation in two dimensions. For the temporal discretization, the backward Euler scheme is utilized to discretize the time derivative, and the weighted and shifted Grünwald difference approximation is used to deal with the time-fractional distributed-order derivative. For the spatial discretization, the finite element method is applied to the approximation of spatial derivative. Meanwhile, the fully-discrete ADI Galerkin finite element scheme is obtained by utilizing an ADI method. The stability and convergence in $L^2$ norm are derived. The results demonstrate that our scheme is stable and convergent with the convergence of order 1 for time, order 2 for space and order 2 for distributed order.

The outline of this article is organized as follows. In Section 2, some preliminaries are presented. In Section 3, we devote to the derivation of the ADI Galerkin finite element scheme. The stability and convergence of our scheme are proved in Section 4. Numerical experiments are given to validate the theoretical analysis in Section 5. The article ends with a brief summary section.