Solution of Multi-Term Time-Fractional PDE Models Arising in Mathematical Biology and Physics by Local Meshless Method

: Fractional differential equations depict nature sufﬁciently in light of the symmetry properties which describe biological and physical processes. This article is concerned with the numerical treatment of three-term time fractional-order multi-dimensional diffusion equations by using an efﬁcient local meshless method. The space derivative of the models is discretized by the proposed meshless procedure based on the multiquadric radial basis function though the time-fractional part is discretized by Liouville–Caputo fractional derivative. The numerical results are obtained for one-, two- and three-dimensional cases on rectangular and non-rectangular computational domains which verify the validity, efﬁciency and accuracy of the method.


Introduction
Over the most recent couple of decades, fractional-order differential equations have been effectively utilized for modeling a wide range of processes and systems in the applied sciences and engineering. The basic information on fractional calculus can be found in [1][2][3]. The extensive applications in science and engineering are portrayed by fractional partial differential equation (PDEs) [4][5][6][7][8]. It is observed that the multi-term time-fractional PDEs are suggested to improve the modelling accuracy in depicting the anomalous diffusion process, modeling different sorts of viscoelastic damping, precisely catching power-law frequency dependence and simulating flow of a fractional Maxwell fluid [9].
Meshless methods have been got more attention for handling different sort of PDE models arise in various fields of science and technology. Particularly the meshless method based on radial basis functions (RBFs) are one of the most well-known sorts among these techniques. Unlike the traditional methods of finite difference and finite element, the meshless techniques do not need mesh in computational domains and can be applied efficiently to multi-dimensional PDE models with complex domains [23]. As per these realities, meshless strategies are known to be truly adaptable and valuable and are broadly utilized to numerous useful problems [24]. Despite the fact that the standard global meshless collocation strategy depends on the globally supported RBFs is known to be a proficient computational procedure. In this procedure the RBFs are used to obtain the coefficients, so that the derivatives of a function f (x) can be written as a linear combination of the functional values at the predetermined nodes, but the technique leads to an ill-conditioned and dense system of algebraic equations due to shape parameters sensitivity and a large number of collocation points. Tragically, the computational cost of execution and ill-conditioning of the procedure will increment drastically by increasing the number of collocation points. To avoid these limitations, the researchers recommended local meshless techniques [25,26]. The beauty of the local meshless technique is utilizing just neighbouring collocation points which results in a sparse matrix system and ward off the main deficiency of ill-conditioning. This sparse system of equations can effectively be solved [27][28][29].
According to [30], Symmetry is a fundamental property of nature and its phenomena. Therefore, the fractional-order diffusion equations are able to adequately describe physical and biological processes and have symmetry properties, which follow from some fundamental laws. The current work is dedicated to utilize the local meshless method (LMM) for the numerical investigation of three-term time fractional-order diffusion model equations up to three space dimensions. The space derivatives of the model equations are calculated by the proposed local meshless algorithm utilizing multiquadric (MQ) radial basis functions (RBFs) while time-fractional part is approximated by using Liouville-Caputo definition. The suggested algorithm is tested on non-rectangular domains as well in two-and three-dimensional case in numerical examinations. Consider the unsteady three-term time-fractional diffusion PDE where the initial and boundary conditions are as follows U(y, 0) = U 0 , C(y, t) = g(y, t), y ∈ ∂Ω, where a 4 is the diffusion coefficient and for one-dimensional (1D) case y = x, for two-dimensional (2D) case y = (x, y) and for three-dimensional (3D) case y = (x, y, z).

Proposed Methodology
The LMM [26,31] is utilized for the solution of time-fractional convection-diffusion models. The derivatives of U(y, t) at the centers y h are approximated by the neighborhood of y h , {y h1 , y h2 , y h3 , ..., y hn h } ⊂ {y 1 , y 2 , . . . , y N n }, n h N n , where h = 1, 2, . . . , N n . The local meshless procedure for 1D case is as follows where ψ( Matrix form of Equation (4) can be written as From Equation (5), we obtain Equations (3) and (6) implies The derivatives of U(x, y, t) in term of x and y are calculated for 2D case by utilizing the above procedure as follows To find coefficients γ The above technique can be rehashed for the three-dimensional case.

A θ-Weighted Technique
A θ-weighted technique is adopted to approximate the model equation in the form (1) with now using Equation (15) we get similarly for q = 0 After applying the LMM (discussed in Section 2), Equations (18) and (19) lead to where L is known to be differential operator and L is the corresponding weights matrix, additionally I is an identity matrix. For θ = 1, Equations (20) and (21) reduce to an implicit method.

Numerical Results and Discussions
This section is concerned with the numerical results of the one-, two-and three-dimensional three-term time-fraction diffusion model equations utilizing the suggested efficient local meshless method (LMM). The implicit time discretization scheme is coupled with the LMM based on multiquadric radial basis function with shape parameter c = 1 in all numerical simulation. Every single numerical trial are performed utilizing local supported domain n i = 3, n i = 5 and n i = 7 in one-, two-and three-dimensional case respectively, additionally the source functions in each case can be computed easily in the accordance of the corresponding exact solution. The accuracy is measure through different error norms which are defined as follows where U, U and ∆h denote the exact solutions, approximate solution and space step size respectively.

Test Problem
Consider Equation (1) as 1D three-term diffusion equation with a 1 = a 2 = a 3 = a 4 = 1, having exact solution [34,35] Initial and boundary conditions can found in accordance to the exact solution.
The simulation results for Test Problem in Section 3.1 are calculated utilizing the suggested LMM and Tabulated in Table 1 using various values of time-step size τ and nodal points N. It can be revealed from Table 1 that the accuracy increments to some extent with the increase in N and decrease in τ.  Figure 2. Figure 3 represents the comparison of the proposed LMM and the method given in [35], and we can see for this figure that the LMM method produced better results.

Test Problem
Consider the Equation (1) as 2D three-term diffusion equation with a 1 = a 2 = a 3 = a 4 = 1, having exact solution [35] is given as The simulation results for Test Problem in Section 3.2 are calculated in terms of absolute error, Max(ε) and RMS norms, are visualized in Figures 4-8 using the suggested LMM. In Figures 4 and 5, we have compared the numerical results, in term of Max(ε) and RMS error norms, of the suggested LMM and the method given in [35]. It can be seen that in this test problem the LMM gives accurate results when contrasted with the technique in [35]. Figure 6 shows the exact, numerical and absolute error for Test Problem in Section 3.2 which is the evidence of the proposed method for better accuracy. The LMM is tested on non-rectangular domains as shown in Figures 7 and 8. It is obvious from these figures that the LMM gives great numerical results irrespective of the domains.
The simulation results for Test Problem in Section 3.3 are appeared in Figure 9 for various fractional-order α and N. It tends to be see from the figure that the accuracy increments with increment in nodal points N to some extent. To testify the performance of the LMM on non-rectangular domains in three-dimensional case, the results are shown in Figures 10 and 11.

Conclusions
The current research is concerned with an efficient computational algorithm, named local meshless method, utilizing radial basis functions to approximate three-term time fractional-order diffusion PDE models in one-, two-and three-dimensions. The Liouville-Caputo definition is utilized in order to compute the time derivative the algorithm is constructed for 0 < γ ≤ β ≤ α < 1. To test the accuracy of the suggested LMM both rectangular and non-rectangular domains are considered in the test problems. The simulation results are evidence that the suggested LMM is a flexible interpolation method. In light of the current work, we can say that the proposed technique is powerful and effective to approximate the solution of multi-term time-fractional PDEs, so it can be also applied to a wide range of complex problems that occur in natural sciences and engineering.