Numerical Solution of Fractional Order Anomalous Subdiffusion Problems Using Radial Kernels and Transform

By coupling of radial kernels and localized Laplace transform, a numerical scheme for the approximation of time fractional anomalous subdiffusion problems is presented. The fractional order operators are well suited to handle by Laplace transform and radial kernels are also built for high dimensions. The numerical computations of inverse Laplace transform are carried out by contour integration technique. The computation can be done in parallel and no time sensitivity is involved in approximating the time fractional operator as contrary to finite differences. The proposed numerical scheme is stable and accurate.


Introduction
In the last decades, many researchers have studied the fractional calculus [1][2][3]. Differential equations of fractional order have many applications in the field of science and engineering [4][5][6][7]. Analytical solution of many fractional differential equations is not possible or very hard to find, so we need a new numerical technique to find its approximate solution. Various phenomena in viscoelastic materials, economics, chemistry, finance, control theory, hydrology, physics, cosmology, solid mechanics, bioengineering, statistical mechanics, and control theory can be mathematically modeled from fractional calculus [8][9][10][11][12][13][14][15][16][17]. In literature, various numerical approaches are available for modeling anomalous diffusive behavior such as Carlo simulations [18]. An introduction of diffusion equations can be found in [19][20][21].
Recently, RBF-based methods were used in solving fractional partial differential equations (FPDEs) [22][23][24]. ese methods have been employed in approximation of partial differential equations with complex domains. An implicit meshless technique based on the radial basis functions for the numerical simulation of the anomalous subdiffusion equation can be found in [25]. e convergence and stability of these mesh-free methods can be found in [26,27]. ese globally defined RBF methods cause illcondition system matrices [28]. To overcome the problem of ill-conditioning, local RBF techniques were used in [29][30][31]. Unlike global RBF methods, the RBF method in local setting uses center points in each subdomain area of influence, surrounding each spatial point due to which there is reduction in the computational cost.
Recently, Laplace transform is combined with RBF method in [32,33]. In [34][35][36][37], the authors use Laplace transform as tool in spectral method and other mesh-based methods such as finite element methods and finite difference method. To avoid the issues of computational efficiency and instability of the system matrix, we introduce a new technique Laplace transform-based local RBF method in solving the time fractional modified anomalous subdiffusion equations in irregular domain.

Preliminaries
Here, we introduce some fundamental definitions related to fractional calculus [39,40].
Definition 1. Let n − 1 < α < n ∈ Z + and α > 0, then the Caputo derivative of fractional order is defined as Definition 2. Let w(t), t ≥ 0, be a given function, then its Laplace transform is defined by provided this integral converges.

Lemma 1.
If w(t) ∈ C p [0, ∞), with α ∈ (n − 1, n) ∈ Z + , then the Laplace transform of the fractional order Caputo derivative is given by Theorem 1. the Bromwich inversion theorem [41]). Let w(t) have a continuous derivative and let |w(t)| < Ke ct , where K and c are positive constants. Define then

Time Discretization.
Here, we apply Laplace transform to models (1)-(3) which gives In more compact form, we have e transformed problems (10) and (11) will be solved for the solution w(x, z) using local RBF method. e solution w(x, t) of the given models (1)-(3) will be found by using numerical inversion.

Local Radial Basis Functions Method.
Here, the linear operators B and L are discretized by using local RBF [42,43]. Consider the centers where Ω is the bounded domain. For each point x i , i � 1, 2, 3, . . . , N, we can find a subdomain Ω j such that n < N. e unknown function w(x, t) can be approximated with RBF in each local subdomain Ω i , i � 1, 2, . . . , N, by the following equation: are the unknown coefficients, and r ij � ‖x i − x j ‖ is the norm between nodes x i and x j , ϕ(r), r ≥ 0 is a radial kernel (multiquadric radial basis function), and Ω j ⊂ Ω is a local domain for around each x i , containing n neighboring nodes around the node x i . So, we have N small size linear systems each of order n × n given by which can be denoted by where Now, applying the operator L to (12) gives e vector form of (15) is given by where G i is given by 2 Journal of Mathematics From equation (14), the unknown coefficients λ i are given by and by inserting the values of λ i in (16), we have where Hence, the discretized form is given by where matrix H is called the sparse differentiation matrix of order N × N.

Numerical Inversion Technique
In this section, the numerical inversion of Laplace transform for approximating the given models (1)-(3) is as follows: where Ψ is the suitable path joining ξ − ι∞ to ξ + ι∞. is Bromwich integral is numerically solved by using the following hyperbolic contour [37]: Integral in (22) gives Next applying trapezoidal rule for approximation of (24), we have where k is the step size.

Application of the Method
In this section, the proposed numerical scheme is applied to multidimensional problems. We solved four test problems and used various domain points N ∈ Ω, stencils points n ∈ Ω j , and quadrature points M. ree error formulas, the error estimate, L est � e (− cM/log(M)) , L ∞ , and L 2 norms are used. e radial kernel used in our computations is . e shape parameter ϵ is optimized by the uncertainty rule related to RBFs. Problem 1. Consider models (1)-(3) to the following form [38]: with the following boundary and initial conditions: respectively, where the actual solution is given by In our numerical scheme, we used the hyperbolic contour (23). e optimal parameter values are taken as ,  Table 1 with various values of fractional order α and β and nodal points N. For comparatively smaller values of fractional order α and β, better results in terms of L ∞ and L 2 error norms are obtained. In the upper part of Table 1, condition number increases, as we increase nodal points N. Error versus various quadrature points M at N � 21, n � 9, and t � 1 and various values of α and β are shown in Figure 1. e error estimate L est for c � 1 is well matched with L ∞ and L 2 error norms, as shown in Figure 1. Hence, our proposed method is stable and accurate.

Problem 2. Consider models (1)-(3) corresponding to the form [38]
zw(x, t) zt � 1 2 initial and boundary conditions given by e actual solution is e same domain and same parameter values as used in Problem 1 are incorporated. e numerical results are shown in Table 2 with the same as well as with various values of fractional order α and β and nodal points N. For comparatively identical values of fractional order α and β, better results in terms of L ∞ and L 2 error norms are obtained. In the upper part of Table 2, condition number of the system matrix is fixed for 11 ≤ N ≤ 71. Error versus various quadrature points M at N � 41, n � 9, and t � 1 and various values of α and β are depicted in Figure 2. e error estimate L est for c � 0.7 is well agreed with L ∞ and L 2 error norms, as shown in Figure 1. e results obtained by our proposed numerical scheme are comparatively identical with the results in Table 2 [38].
Problem 3. Next, we consider models (1)-(3) corresponding to the form [44] zw(x, y, t) zt where f(x, y, t) � 2t sin(2πx)sin(2πy) initial and boundary conditions given by e exact solution is w(x, y, t) � t 2 sin(2πx)sin(2πy).  Table 3, for various nodal points N and stencils points n � 11, 15 and with various values of α and β, the L ∞ error norm is well matched with L 2 error norm. e condition number is increasing steadily as we decrease both the values of α and β at the same time.

Conclusion
In this work, a numerical scheme is constructed which is based on Laplace transform and radial basis functions in the local setting. e proposed numerical scheme efficiently approximated time fractional anomalous subdiffusion equation.
e supremacy of this method particularly for fractional order equations is its nonsensitive nature in time as contrary to finite difference approximation for fractional order operators. Since the fractional order derivative is of integral convolution type and suited to handle by Laplace transform, the spatial operators in multidimensions can be approximated by RBF in the local setting which generates small size differentiation matrices in local subdomains and these are assembled as a single sparse matrix in the global domain. So, large amount of data can be manipulated very easily and accurately.
Data Availability e data supporting the results are available within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.