Numerical solution of the general high-dimensional multi-term time-space-fractional diffusion equations

by   Xiaogang Zhu, et al.

In this article, an advanced differential quadrature (DQ) approach is proposed to obtain the numerical solutions of the two- and three-dimensional multi-term time-space-fractional diffusion equation (TSFDE) on general domains. The fractional derivatives in space are firstly discretized by deriving a new class of differential quadrature (DQ) formulas with radial basis functions (RBFs) being the trial functions. Then, the original problems are converted to a group of multi-term fractional ordinary differential equations (ODEs), which are further treated by a class of high-order difference schemes based on the weighted and shifted Lubich difference operators. The presented DQ method is high accurate and convergent on arbitrary domains. Finally, several numerical examples are carried out to illustrate its accuracy and effectiveness.



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1 Introduction

During the last ten years, fractional calculus came into the focus of interest in academic circles as an important branch of mathematics. Due to its non-locality and self-similarity, it has been proven to be very adequate to describe the dynamics phenomena with long term memory memory and hereditary effects like the anomalous transport in complex heterogeneous aquifer Adams and Gelhar (1992). The application of fractional calculus has been very extensive recently, including biology, chemistry, physics, economics, and many other disciplines.

The fractional partial differential equations (PDEs) are the results of mathematical modelling based on fractional calculus, which provides a new powerful tool for the study of mathematical physics or even for the whole scientific research. Nevertheless, there raises a challenge to solve these type of equations and few of fractional PDEs can be solved by analytic techniques. As an result, numerical methods are clearly a priority for development. Until now, the numerical algorithms for the time-fractional PDEs is on the way to maturity after recent years’ development

Lin and Xu (2007); Murio (2008); Jiang and Ma (2011); Yuste (2006); Ren et al. (2013); Zhang and Yan (2016); Hosseini and Ghaffari (2014); Huang et al. (2015), while these for high-dimensional space-fractional PDEs have to be further developed and advanced, especially for the ones defined on general domains. Regardless of the difficulties in constructing numerical algorithms for the space-fractional PDEs, many numerical methods have been invented to solve them, covering finite difference (FD) methods Meerschaert and Tadjeran (2004); Tian et al. (2015); Huang et al. (2014); Jia and Wang (2015), spectral methods Bhrawy and Baleanu (2013); Karaa et al. (2016), finite element (FE) methods Ervin and Roop (2006); Deng (2008); Zhang et al. (2010); Zhu et al. (2017), and finite volume methods Karaa et al. (2016); Liu et al. (2014). Liu et. al considered a space-fractional FitzHugh-Nagumo monodomain model by an implicit semi-alternative direction FD scheme on approximate irregular domains Liu et al. (2015a). Qiu et al. considered a nodal discontinuous Galerkin method and solved a L-shaped domain problem Qiu et al. (2015). Yang proposed a fully discrete FE method for the two-dimensional space-fractional diffusion equation on convex domains Yang et al. (2017). Bhrawy and Zaky developed a spectral tau method for the multi-term time-space fractional PDEs Bhrawy and Zaky (2015). In Dehghan and Abbaszadeh (2018), an efficient algorithm based on FD and FE methods was addressed for the two-dimensional multi-term time-space-fractional Bloch-Torrey equations. In Fan et al. (2018), a fully discrete FE method was given for the two-dimensional multi-term time–space fractional diffusion-wave equation. Although the numerical treatment of multi-term time-space-fractional PDEs is also an active area of research, few works can be found to solve these equations on general high-dimensional domains.

Meshless methods eliminate tedious mesh generation and reconstruction or only use easily generable meshes in a flexible manner, and can reduce computational cost, which serve as a promising alternative in dealing with structure destruction, high-dimensional crack propagation, and large deformation problems. Meshless methods have been achieved great progress, such as diffuse element method Nayroles et al. (1992), reproducing kernel particle method Han and Meng (2001), hp-cloud method, element-free Galerkin method Belytschko et al. (1994), meshless local Petrov-Galerkin method Atluri and Zhu (1998), boundary element-free method, RBF collocation methods (Kansa’s methods) Fedoseyev et al. (2002); Kansa (1990)

, point interpolation (PI) method

Liu and Gu (2001), DQ methods Du et al. (1994); Wu and Shu (2002), and so forth. They offer more advantages over mesh-dependent methods in treating the space-fractional PDEs but seldom works have bee reported. Liu et. al proposed a PI method for the space-fractional diffusion equation Liu et al. (2015b). Cheng et. al analysed an improved moving least-squares approximation to the solution of two-dimensional space-fractional wave equation Cheng et al. (2018). Pang et. al extended DQ method to solve the space-fractional diffusion equation and use the as Lagrange interpolating basis functions as trial functions to determine the weighted coefficients Pang et al. (2014).

DQ method is a kind of meshless methods which uses the weighted sum of functional values at sampling points along ordinate directions to discretize derivatives Bellman et al. (1975). Subsequently, this kind of method enjoys the advantages as high accuracy, truly mesh-free, and the adaptability to high-dimensional problems. Regarding this point, in this article, we showcase an efficient RBFs-based DQ method for the following multi-term TSFDE on general domains:

  • two-dimensional multi-term TSFDE:


    where , , , with being its boundary, are the diffusion coefficients with .

  • three-dimensional multi-term TSFDE:


    where , , , with being its boundary, are the diffusion coefficients with .

The fractional derivatives in Eqs. (1.1)-(1.2) are defined in Caputo sense. More precisely, the ones tagged with ”+” are the left-side derivatives, while those tagged with ”–” are the right-side derivatives. For example, assuming , the left and right boundaries of , in which,

then , are defined as follows:

with the Gamma function .

The fractional derivatives in other coordinate directions can be defined in the same fashion. denotes the multi-term fractional derivative operator:



The outline is as follows. In Section 2, some preliminaries on fractional calculus and RBFs are introduced. In Section 3 the DQ approximation of fractional derivative is proposed based on RBFs. In Section 4, using the RBFs-based DQ formulas, we construct a fully discrete DQ method for the multi-term TSFDE on general domains and its algorithm is further studied in Section 5. In Section 6, some numerical tests are presented to confirm its convergence. In the last section, a brief summary is drawn.

2 Preliminaries

We recall some basic preliminaries on fractional calculus and RBFs required for further discussions.

2.1 Fractional calculus

Defition 2.1.

The left and right Riemann-Liouville fractional integrals of order are defined by

and if , and .

Defition 2.2.

The left and right Riemann-Liouville fractional derivatives of order are defined by

where , , and if , and .

Defition 2.3.

The left and right Caputo fractional derivatives of order are defined by

where , , and if , and .

The two frequently-used fractional derivatives are equivalent with exactness to an additive factor:


Moreover, we easily realize the following properties:


where is a constant and with being the ceiling function, which outputs the smallest integer greater than or equal to . Another point need to be noticed is that these properties are also true for the right-side fractional calculus. For more details, we refer the readers to Kilbas et al. (2006) for overview.

2.2 Radial basis functions

Letting , and be Euclidean norm, the approximation of can be written as a weighted sum of RBFs in the form:


where , , , are unknown weights, are the basis functions of the polynomial space of degree at most , and are RBFs, which defined by . To ensure that the interpolant properly behave at infinity, the above equations are augmented by


Putting Eqs. (2.10)-(2.11)in matrix form leads to a matrix system:


where the matrix elements for are and for [B] are .

The RBFs provides an effective tool for the interpolation of scattered data in multi-dimensional domains. Since they only take the space distance as independent variable, as compared to the traditional basis function defined along coordinates, RBFs have the advantages of concise form, dimension-independent and isotropic properties. In particular, in the application of solving high-dimensional PDEs, the RBFs interpolation shows great flexibility. The numerical methods constructed by RBFs not only have the simplicity in the implementation, but also avoid the tedious mesh generation, and thereby reducing the pre-processing process of algorithms. If , then the commonly used RBFs are listed in Table 1.

Name RBF
Multiquadric (MQ)
Reciprocal Multiquadric (RMQ)
Inverse Quadratic (IQ)
Polyharmonic Spline (PS)
Gaussian (GA)
Table 1: Some commonly used RBFs.
Defition 2.4.

A function is completely monotonic if and only if , for , and .

Theorem 2.1.

Cheng (2012) Let an univariate function be such that is completely monotonic, but not a constant. Suppose further that . Then the interpolation matrix of the basis function is positive definite.

The Eq. (2.12) is solved for , and the interpolation matrix should be positive definite so that it invertible. According to Theorem 4.1, Reciprocal Multiquadrics, Inverse Quadratics and Gaussians are positive definite while Multiquadrics are only conditionally positive definite. Therefore, the polynomial term on the right-side of Eq. 2.10 can be removed for Reciprocal Multiquadrics, Inverse Quadratics and Gaussians while it is indispensable to Multiquadrics to maintain the well-posedness of the resulting algebraic system.

3 The DQ approximation of fractional derivative

In this section, we derive a new class of DQ formulas for fractional derivatives by RBFs as the trial functions. The DQ formula employs the linear sum of function values at discrete points to approximate partial derivatives with regard to an independent variable. Supposing , , we have the following DQ formula Wu and Shu (2002):


where , , are weighted coefficients. By realizing the essence of DQ formula, we propose the following DQ approximations to the left-hand fractional derivaitves:


where , , , , are the related weighted coefficients, which satisfy


with being the trial functions. The weighted coefficients are determined by reforming Eqs. (3.15) in matrix system in advance. Likewise, we can extend the above formulas to the right-side fractional derivatives with regard to the variables , and .

The calculation of weighted coefficients comprise a major part of DQ methods. In what follows, we show how to use the aforementioned RBFs to determine these coefficients. It should be noted that for Reciprocal Multiquadrics, Inverse Quadratics and Gaussians, their interpolation matrices invertible. Therefore, ignoring the polynomial terms on the right-side of RBFs interpolation (2.10) and substitute into Eq. (3.15), we obtain the weighted coefficients by solving the resulting algebraic systems for each nodal point . As for Multiquadrics, we prefer to


To make the problem be well posed, one more equation is required. From Eq. (2.11), it follows that , which further yields

By careful observation, we can consider , , , as the trial functions. Referring to the above discussion and , we obtain the algebraic system as follows



For each node , the values of unknown weighted coefficients are consequently computed by .

We note that the DQ approximation (3.14) works because the fractional derivatives are linear operators. In other words, if ,  , and the weighted coefficients fulfill Eqs. (3.15), then for , we realize


through the linear property of fractional derivatives. In addition, it is sure that the above convergent result is exactly true for the right-side fractional derivatives with regard to the others variables.

4 The description of RBFs-based DQ method

4.1 Discretization of time-fractional derivative

In this subsection, we introduce high-order schemes to discretize the fractional derivative in temporal direction. To this end, we define a time lattice , , , , and review the relationship between Caputo and Riemann-Liouville derivatives:


where , . Using the property (2.9), we obtain a class of high-order schemes for Caputo derivaitve by applying the weighted and shifted Lubich difference operators to discretize Riemann-Liouville derivatives on the right-side of Eq. (4.19), which reads


where are the discrete coefficients Chen and Deng (2014). For instance, if , we have , , and if , we have


where , , and i denotes imaginary unit.

Lemma 4.1.

The coefficients satisfy the following properties

  • ,  ,  ,

  • .

Lemma 4.2.

Chen and Deng (2014) Assume that , with

and their Fourier transforms belong to

with regard to , then the Lubich difference operators satisfy

Theorem 4.1.

Assume that , with are smooth enough with regard to , the we have




The proof is trivial by following Lemma 4.2 and references Chen and Deng (2014); Deng et al. (2015). ∎

Applying the operator to the multi-term fractional derivative operator and set , we finally obtain


4.2 Fully discrete RBFs-based DQ scheme

In this subsection, we develop a fully discrete DQ scheme for the multi-term TSFDE, which utilizes the operator to treat the Caputo derivative in time and RBFs-based DQ approximations for the fractional derivatives in space. For the ease of expression, letting , , we put Eqs. (1.1)-(1.2) in the unified form:


where , and .

Let be a set of nodes in . Replacing the space-fractional derivatives by RBFs-based DQ formulas, we obtain the following multi-term fractional ODEs:


where . Furthermore, using the operator to discretize arrives at