Local meshless differential quadrature collocation method for time-fractional PDEs

This paper is concerned with the numerical solution of time- fractional partial differential equations (PDEs) via local meshless differential quadrature collocation method (LMM) using radial basis functions (RBFs). For the sake of comparison, global version of the meshless method is also considered. The meshless methods do not need mesh and approximate solution on scattered and uniform nodes in the domain. The local and global meshless procedures are used for spatial discretization. Caputo derivative is used in the temporal direction for both the values of \begin{document}$ \alpha \in (0,1) $\end{document} and \begin{document}$ \alpha\in(1,2) $\end{document} . To circumvent spurious oscillation casued by convection, an upwind technique is coupled with the LMM. Numerical analysis is given to asses accuracy of the proposed meshless method for one- and two-dimensional problems on rectangular and non-rectangular domains.


1.
Introduction. Fractional calculus has attracted significant interests in the field of science and engineering in the last few years. The elementary knowledge of fractional calculus can be found in [7]. Fractional differential equations contain derivatives of any real or complex order, being considered as general form of differential equations. Applications of such models appear in numerous real world problems in physics and applied mathematics [21,17].
Variety of meshless methods have been applied for solving different kind of PDE models arising in almost all disciplines of engineering. Meshless character is one of the most important reason for the rising demand of such type of methods. These methods eliminate the need to construct complicated meshes and are easily extendible to multi dimensional PDEs. Other reasons can be their implementation on complex geometries in uniform and non-uniform nodal settings [22,27,25,1,2].
In the case of shape parameter dependent global meshless method (GMM), selection of optimum value of the shape parameter value c and dense ill-conditioned matrix are counted as a major deficiencies. To ward off the effects of these deficiencies of the GMM, researchers suggested a counterpart local meshless method (LMM) [23,26,29]. In the recent literature, local meshless methods have been used for stable behaviour and better accuracy of complex PDE models (see [25,26,3,4]).
Most of the numerical methods failed to capture flow dynamics in convectiondominated flows. Like other numerical methods, several strategies have been suggested to circumvent the numerical instabilities of the LMM [9,15,19,22,26,3]. Following the idea of [22,26,3], the proposed LMM is combined with a technique based on local supported domain, called an upwind technique, in case of convection dominated PDE models. This technique has the ability to avoid mild spurious oscillatory solution. Two types of local supported domains are used i.e. central and upwind as shown in  Current research work is devoted to use, the local meshless method for numerical solution of 1D time-fractional Burgers' model equation [11,13], 2D time-fractional diffusion model equation [28], 2D time-fractional Burgers' model equation [5]. The space derivatives are approximated by the local and global meshless procedure using the multiquadric and the Gaussian radial basis functions whereas time-fractional derivative is approximated by using Caputo definition. In case of convectiondominated time-fractional models, the LMM is coupled with a stencil based upwind technique. Both rectangular and non-rectangular geometries are considered in numerical experiments.
2. Time-fractional models. Consider the 1D nonlinear time-fractional Burgers' equation (TFBE) [11,13] which is given as with the following initial and boundary conditions where Re is known as Reynolds number.
The 2D linear time-fractional diffusion equation (TFDE) [28] is with initial and boundary conditions The 2D nonlinear time-fractional Burgers' equation (TFBE) [5] is with initial and boundary conditions where Re is known as Reynolds number.
3. Formulation of the LMM. The LMM [26] is extended to the time-fractional Burgers' and diffusion models. The derivatives of V (x, t) are approximated at the centers x i by function values at a set of nodes in the neighborhood of . . , N n . For 1D case, when n = 1 then x = x and for 2D case, when n = 2 then x = (y, z). Now in 1D case, we have To find the corresponding coefficient λ k , radial basis function ψ( x − x l ) can be substituted in equation (4) as follows where in case of multiquadric (MQ) and Gaussian (GA) RBFs respectively. Matrix form of equation (5) is where for each k = i1, i2, . . . , in i . In matrix notation, equation (6) can be written as where From equation (7), we obtain From equation (4) and equation (8), we get For 2D case, the derivatives of V (y, z, t) with respect to y and z are approximated in the similar way as follows To find the corresponding coefficients γ 3.1. Discretization of space derivatives. Now we apply the LMM for spatial discretization to the model equations (1)-(3) along with the prescribed initial and boundary conditions. The semi-discretized can be written as where the sparse coefficient matrix (C) of order N n × N n (n = 1 in 1D case, n = 2 in 2D case). The matrix C can be calculated once V and its space derivatives are discretized by the local meshless method. The vectors g and h denote the initial and boundary conditions of the problem respectively. Orders of g and h are N n × 1 where n = 1, 2.

Discretization of time derivatives. The time derivative
∂t α is discretized by using Caputo derivative. In this paper two cases 0 < α < 1 and 1 < α < 2 are considered. The Caputo fractional derivative for α ∈ (0, 1) is Consider N + 1 equally spaced time levels t 0 , t 1 , . . . , t N in the interval [0, t], such that t n = nτ , n = 0, 1, 2, . . . , N , τ being the time step and finite difference scheme is used to approximate the first-order derivative involved in the time fractional term The term ∂V (x,ξ k ) ∂ξ is approximated as follows . . , n, we can write the above equation in more precise form as The Caputo fractional derivative for α ∈ (1, 2) is defined by Like the previous case, the approximate fame work can be constructed as can be approximated as The simplified form of the fractional derivative becomes Letting a 0 = τ −α Γ(3−α) and b k = (k + 1) 2−α − k 2−α , k = 0, 1, . . . , n, one can write the above equation in a more precise form as where V is exact solution and V is the approximate solution.
Test Problem 1. Consider the 1D TFBE (1) as a first test problem with initial and boundary conditions taken form the exact solution In Table 1, numerical results in the form of maximum absolute error L ∞ norm are obtained by the LMM for Test Problem 1 using MQ RBF and the value of shape parameter in this case is taken c = 1, N = 60, time step size τ = 0.0002, α = 1, σ = 0.4, µ = 0.3, Re = 10 and λ = 0.8. Table 1 shows that the numerical results of the LMM are more accurate than the method reported in [8].
Test Problem 2. Consider 1D TFBE (1) with initial and boundary conditions given as V (x, 0) = sin(πx), V (a, t) = V (b, t) = 0. The exact solution of this problem is not known.         Test Problem 3. Consider 2D TFDE (2) with initial condition where the source function is Table 2 shows the numerical results obtained by the LMM using Crank-Nicolson (CN) for different nodal points N , τ = 0.001, t = 1, shape parameter value c = 1 and local support domain n i = 15. The table shows that good accuracy has been achieved for α = 1.5 and α = 1.8. Numerical results of the proposed LMM using Crank-Nicolson are compared in Table 3 for N = 81, n i = 11 and different values of τ . It can be observed from the table that the LMM is more accurate than the method reported in [28]. In Table 4 we have compared numerical result obtained by the LMM (Explicit, Implicit and Crank-Nicolson) for different values of α using regular and Chebyshev nodal points. The results are obtained in terms of Ave.L abs error norms, N = 20, n i = 11 and t = 1. The table shows that the results of explicit method using regular nodes is comparatively better than the other methods for small time step.

The exact solution is
The numerical results of the 2D TFDE (2) on non-rectangular domains are shown in Figures 6-10          The exact solution to this problem is unknown.

5.
Conclusion. In the present work, a local meshless method based on radial basis functions is applied to one-and two-dimensional time-dependent fractional PDE models. The time derivative is defined and simplified in Caputo sense and the scheme is constructed for 0 < α < 1 and 1 < α < 2. To check accuracy and efficacy of the proposed scheme different test problems have been considered on both rectangular and non-rectangular domains. Results of the local meshless method are compared with exact/approximate solutions reported in the existing literature. The stable results (in the case of high Reynolds number) of the LMM combined with upwind technique strongly supported the advantage of the LMM over other conventional methods. The LMM has been found to be a flexible interpolation method with respect to accuracy and well-conditioned coefficient matrix.