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 nonlocality and selfsimilarity, 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 timefractional 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 highdimensional spacefractional 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 spacefractional 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 spacefractional FitzHughNagumo monodomain model by an implicit semialternative direction FD scheme on approximate irregular domains Liu et al. (2015a). Qiu et al. considered a nodal discontinuous Galerkin method and solved a Lshaped domain problem Qiu et al. (2015). Yang et.al proposed a fully discrete FE method for the twodimensional spacefractional diffusion equation on convex domains Yang et al. (2017). Bhrawy and Zaky developed a spectral tau method for the multiterm timespace fractional PDEs Bhrawy and Zaky (2015). In Dehghan and Abbaszadeh (2018), an efficient algorithm based on FD and FE methods was addressed for the twodimensional multiterm timespacefractional BlochTorrey equations. In Fan et al. (2018), a fully discrete FE method was given for the twodimensional multiterm time–space fractional diffusionwave equation. Although the numerical treatment of multiterm timespacefractional PDEs is also an active area of research, few works can be found to solve these equations on general highdimensional 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, highdimensional 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), hpcloud method, elementfree Galerkin method Belytschko et al. (1994), meshless local PetrovGalerkin method Atluri and Zhu (1998), boundary elementfree 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 meshdependent methods in treating the spacefractional PDEs but seldom works have bee reported. Liu et. al proposed a PI method for the spacefractional diffusion equation Liu et al. (2015b). Cheng et. al analysed an improved moving leastsquares approximation to the solution of twodimensional spacefractional wave equation Cheng et al. (2018). Pang et. al extended DQ method to solve the spacefractional 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 meshfree, and the adaptability to highdimensional problems. Regarding this point, in this article, we showcase an efficient RBFsbased DQ method for the following multiterm TSFDE on general domains:

twodimensional multiterm TSFDE:
(1.1) where , , , with being its boundary, are the diffusion coefficients with .

threedimensional multiterm TSFDE:
(1.2) 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 leftside derivatives, while those tagged with ”–” are the rightside 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 multiterm fractional derivative operator:
(1.3)  
(1.4) 
where
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 RBFsbased DQ formulas, we construct a fully discrete DQ method for the multiterm 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 RiemannLiouville fractional integrals of order are defined by
and if , and .
Defition 2.2.
The left and right RiemannLiouville 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 frequentlyused fractional derivatives are equivalent with exactness to an additive factor:
(2.5)  
(2.6) 
Moreover, we easily realize the following properties:
(2.7)  
(2.8)  
(2.9) 
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 rightside 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:
(2.10) 
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
(2.11) 
Putting Eqs. (2.10)(2.11)in matrix form leads to a matrix system:
(2.12) 
where the matrix elements for are and for [B] are .
The RBFs provides an effective tool for the interpolation of scattered data in multidimensional 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, dimensionindependent and isotropic properties. In particular, in the application of solving highdimensional 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 preprocessing 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) 
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 rightside of Eq. 2.10 can be removed for Reciprocal Multiquadrics, Inverse Quadratics and Gaussians while it is indispensable to Multiquadrics to maintain the wellposedness 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):
(3.13) 
where , , are weighted coefficients. By realizing the essence of DQ formula, we propose the following DQ approximations to the lefthand fractional derivaitves:
(3.14) 
where , , , , are the related weighted coefficients, which satisfy
(3.15) 
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 rightside 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 rightside 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
(3.16) 
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
(3.17) 
where
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
(3.18) 
through the linear property of fractional derivatives. In addition, it is sure that the above convergent result is exactly true for the rightside fractional derivatives with regard to the others variables.
4 The description of RBFsbased DQ method
4.1 Discretization of timefractional derivative
In this subsection, we introduce highorder schemes to discretize the fractional derivative in temporal direction. To this end, we define a time lattice , , , , and review the relationship between Caputo and RiemannLiouville derivatives:
(4.19) 
where , . Using the property (2.9), we obtain a class of highorder schemes for Caputo derivaitve by applying the weighted and shifted Lubich difference operators to discretize RiemannLiouville derivatives on the rightside of Eq. (4.19), which reads
(4.20) 
where are the discrete coefficients Chen and Deng (2014). For instance, if , we have , , and if , we have
(4.21) 
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(4.22) 
Theorem 4.1.
Assume that , with are smooth enough with regard to , the we have
(4.23) 
where
Proof.
Applying the operator to the multiterm fractional derivative operator and set , we finally obtain
(4.24) 
4.2 Fully discrete RBFsbased DQ scheme
In this subsection, we develop a fully discrete DQ scheme for the multiterm TSFDE, which utilizes the operator to treat the Caputo derivative in time and RBFsbased 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:
(4.25) 
where , and .
Let be a set of nodes in . Replacing the spacefractional derivatives by RBFsbased DQ formulas, we obtain the following multiterm fractional ODEs:
(4.26) 
where . Furthermore, using the operator to discretize arrives at
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