A collocation method based on cubic trigonometric B-splines for the numerical simulation of the time-fractional diffusion equation

Fractional differential equations sufficiently depict the nature in view of the symmetry properties, which portray physical and biological models. In this paper, we present a proficient collocation method based on cubic trigonometric B-Splines (CuTBSs) for time-fractional diffusion equations (TFDEs). The methodology involves discretization of the Caputo time-fractional derivatives using the typical finite difference scheme with space derivatives approximated using CuTBSs. A stability analysis is performed to establish that the errors do not magnify. A convergence analysis is also performed The numerical solution is obtained as a piecewise sufficiently smooth continuous curve, so that the solution can be approximated at any point in the given domain. Numerical tests are efficiently performed to ensure the correctness and viability of the scheme, and the results contrast with those of some current numerical procedures. The comparison uncovers that the proposed scheme is very precise and successful.


Introduction
The time-fractional diffusion equation (TFDE) is given as C a D γ t u(s, t) - subject to the following initial condition (IC) and boundary conditions (BCs): u(a, t) = ψ 1 (t), u(b, t) = ψ 2 (t), t ≥ 0, where the diffusion exponent is denoted by γ , and C a D γ t u(s, t) is the Caputo fractional derivative (CFD) of order γ given by [1] Note that γ = 0 and γ = 1 correspond to the classical Helmholtz and standard diffusion equations, respectively.
The fractional calculus [1][2][3] has gained keenness in numerous fields such as chemistry, plasma physics, material science, biology, fluid mechanics, and so on. The fractional-order differential and integral equations are reliable tools to describe physical models of interest more exactly than their integer-order counterparts. A variety of applications of fractional calculus and TFDEs can be found in [3][4][5][6][7][8][9][10]. The numerical and approximate solutions play an important role in exploring applications of fractional partial differential equations. It is emphasized in many research papers that the fractional derivatives and integrals are more efficient tools for modeling the hereditary and memory effects of different processes and materials, in contrast with integer-order models, in which such effects are ignored.
Numerous analytical schemes are available for TFDEs [1,[11][12][13]. Numerical techniques are developed continuously because exact solutions are available in very few cases. Numerous numerical procedures for solving TFDEs have been developed recently. Esmaeili and Garrappa [14] obtained numerical solutions of TFDEs by a pseudospectral scheme.
Zhuang and Liu [16] obtained implicit difference approximations for TFDEs. Karatay et al. [17] used the Crank-Nicholson approach to construct a scheme for TFDEs. A weighted average and explicit finite difference schemes were developed in [18,19] for TFDEs. Murio [20] presented an unconditionally stable implicit scheme for TFDEs on a finite slab. Tasbozan et al. [21] introduced a numerical scheme using B-spline basis functions for space fractional subdiffusion equations. Huang et al. [22] presented a fully discrete discontinuous Galerkin method for TFDEs. Chen et al. [23] used the Fourier method to find approximate solutions of the fractional diffusion equation describing subdiffusion. Gao and Sun [24] presented a compact finite difference scheme for fractional subdiffusion equations using a compact finite difference scheme.
In this paper, we present a cubic trigonometric B-spline collocation method to obtain numerical solutions of TFDEs. The main motivation behind using B-splines is that the solutions are obtained in the form of piecewise continuous sufficiently smooth functions, enabling us to approximate the solution at any desired location in the domain. The stability and convergence analysis are also discussed to establish that the scheme does not propagate errors. Numerical tests are performed to affirm the feasibility and applicability of the method. The results are compared with those presented in [22,23].
The rest of the paper is organized as follows. In Sect. 2, we derive a numerical procedure. The stability and convergence analysis of the scheme are presented in Sects. 3 and 4, respectively. In Sect. 5, we show a contrast of our numerical results with those of [22,23].
Section 6 contains the outcomes of this study.

Space discretization
Let the solution domain be [a, b] × [0, T]. For given positive integers M and N , let τ = T N be the temporal and h = b-a M the spatial step sizes, respectively. The space interval [a, b] is uniformly partitioned as a = s 0 < s 1 < · · · < s M = b, where s i = a + ih, i = 0, . . . , M. In this partition, the CuTBS function TB 4 i (x) [25] is defined as where where c i (t) are unknowns, which are to be determined using the collocation method by utilizing the initial and boundary conditions. Using (5) and (6), the values of U(x, t) and its necessary derivatives at the nodal points are determined in terms of the parameters c i as follows: 3 sin 2 ( h 2 ) -2 sin 2 ( h 4 ) -sin 2 ( 5h 4 ) , 5 = -3 (4 cos(h) + 2)(tan 2 ( h 2 )) .

Stability analysis
Here we test scheme (11) for the stability analysis. The Duhamels principle [39] states that for an inhomogeneous case, the stability estimates are the same as those of the corresponding homogeneous case. So we present the stability analysis only for the case f = 0.
Let ω n i andω n i be the growth factor and its approximation, respectively, of a Fourier mode.
Proof By Proposition 1 and relation (17) we have which establishes the unconditional stability.

Convergence analysis
Here we give convergence estimates for the discrete-time problem (10). As in the case of stability analysis, we present the convergence analysis for the homogeneous problem only.
be the exact solution of (1), and let {u n } N-1 n=0 be the discretetime solution of (10). Then where e n+1 = u(s, t n+1 )u n+1 , and D is a constant.
Proof As before, we give a proof for f = 0 only. Note that the exact solution u also satisfies the semidiscrete scheme (10), so that we have u s, t n+1 = n-1 l=0 (κ lκ l+1 )u s, t n-l + κ n u s, t 0 + α 0 u s, t n+1 ss (23) and Subtracting (24) from (23), we obtain e n+1 = n-1 l=0 (κ lκ l+1 )e n-l + κ n e 0 + α 0 e n+1 ss + r n+1 where we have used the fact that e 0 = 0. Now taking the inner product of both sides of (25) with e n+1 and using x, where we have used the relations u ss , u =u s , u s and x, x = x 2 . Applying the Cauchy-Schwarz inequality x, y ≤ x y in (26), we obtain Dividing (27) throughout by e n+1 , we obtain where D n = max 0≤l≤n-1 e n-l and D = max 0≤n≤N D n . We have also used the relation (1κ n ) < 1.

Numerical results and discussions
In this section, we present the results of the numerical tests for the TFDE (1) with initial (2) and boundary conditions (3). We use the following error norms to measure the accuracy of the method: where U exact is the exact solutions, and (U N ) j is the approximate one. The order of convergence is given by where Error(M) and Error(2M) are the L ∞ norms at M and 2M, respectively.
Example 1 Consider the TFDE (1) [22] with initial condition u(s, 0) = sin s and boundary conditions u(0, t) = u(π, t) = 0. This problem has the exact solution u( is the ML function. The corresponding source term is f = 0. We apply the proposed algorithm (11) to the problem. The approximate solutions when τ = 0.01, h = π 20 , and γ = 0.5 at t = 0.5 and t = 1 are given by  respectively. Figure 1 displays the behavior of the numerical and exact solutions at different times. The graphs are in excellent affirmation. In Fig. 2 the absolute errors are presented in 2D and 3D at t = 0.5. Figure 3 demonstrates an excellent 3D contrast between the exact and numerical solutions at time step t = 1. In Table 1, a comparison of the error norms with those obtained in [22] is tabulated. Our methodology gives better precision for bigger τ over that obtained in [22]. The order of convergence is tabulated for the L ∞ norm in Table 2.    C a D 0.9 t u(s, t) - with zero initial and boundary conditions. This problem has the exact solution u(s, t) = t 2 sin(2πs).
We solve (29) by using the proposed scheme (11). The approximate solutions when τ = 0.01 and h = 1 20 at t = 0.5 and t = 1 are given by  respectively. We get the numerical results by utilizing the proposed scheme. A close comparison between the exact and numerical solutions at different times is shown in Fig. 4. In Fig. 5 the 2D and 3D error profiles are displayed at t = 0.5. Figure 6 deals with 3D comparison between the exact and approximate solutions. Table 3 reports a comparison of the error norms with those obtained in [22]. Although we have chosen a larger time step than that of [22], we still obtained a better accuracy. Table 4 records the convergence orders for the L ∞ norm.   Example 3 Consider the TFDE describing subdiffusion [23] ∂u(s, t) with initial condition u(s, 0) = 0, 0 ≤ s ≤ 1, and boundary conditions This problem has the exact solution u(s, t) = e s t 1+γ .
Following [24], equation (30) can be equivalently written as with same initial and boundary conditions. We apply scheme (11) to (30). The approximate solutions when τ = 0.01, h = 1 20 , and γ = 0.5 when t = 0.5 and t = 1 are given by   Figure 7 analyzes the graphs of the exact and approximate solutions when τ = 0.01 h = 1 80 , and γ = 0.5. Figure 8 depicts the 2D and 3D error profiles, which exhibit exactness of the method. Figure 9 shows exceptionally close comparison of 3D graphs of approximate and exact solutions using τ = 0.01, h = 1 60 , and γ = 0.5. In Tables 5-8 the maximum errors contrast with those presented in [23] for various values of τ and h to show that the present scheme is increasingly precise and gives better precision.

Conclusions
In this study, we developed a cubic trigonometric B-spline collocation method for numerical approximation of time-fractional diffusion equations. The time discretization is done using the typical finite difference method, whereas the derivatives in space are approximated by utilizing the trigonometric B-splines. The approximate solution is obtained as a piecewise continuous function, so that the solution can be approximated at any desired location in the domain of interest. We also presented a stability and convergence analysis of the scheme to affirm that the errors do not propagate. The obtained numerical results contrast with those of some current numerical procedures. We infer that the present scheme is more precise and provides better accuracy.