A New Variant of B-Spline for the Solution of Modified Fractional Anomalous Subdiffusion Equation

Department of Mathematics, The Govt. Sadiq College Women University, Bahawalpur, Pakistan School of Mathematical Sciences, Jiangsu Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing 210023, China Department of Mathematics, Huzhou University, Huzhou 313000, China Department of Mathematics, Ghazi University, D G Khan, Pakistan Department of Computer Engineering, Biruni University, Istanbul, Turkey Department of Mathematics, Science Faculty, Firat University, 23119 Elazig, Turkey Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan Department of Mechanical Engineering, College of Engineering, Taif University, P.O.Box 11099, Taif 21944, Saudi Arabia


Introduction
The current study deals with the investigation of modified fractional anomalous subdiffusion equation using hybrid Bspline-based collocation method. The general form of the equation is with BC ψ ξ 0 , τ ð Þ= p 0 τ ð Þ, and IC where D 1−α i τ is the Riemann-Liouville version of the fractional partial derivatives of order (1 − α i ), α i ′s are the fractional parameters with 1 ≥ α i > 0 for i = 1, 2, ⋯, n are fractional parameters appears in the equation. Here, g 0 ðτÞ, g 1 ðτÞ, h 0 ðξÞ, and h 1 ðξÞ are smooth functions, and ξ, τ are space and time variables.
Fractional PDEs have implementation in numerous fields of engineering and science. Some of them are placed in the great book of Podlubny [1] and in some recent work can be found in [2][3][4][5][6]; their applications can be found in the field of electromagnetic and mechanical engineering.
Liu et al. [7] proposed a finite difference technique for the modified anomalous subdiffusion equation. Liu et al. [8] presented two versions of finite element approximation with semi-and full discretizations in a finite domain and compact difference scheme by Wang et al. [9]. An unconditionally stable high-order scheme was proposed by Mohebbi et al. [10]. The proposed scheme has a nonlinear source term. In another paper by Mohebbi et al. [11], the solution of 2D modified anomalous subdiffusion equation was proposed, and this method was based on radial basis functions.
Dag et al. [12] presented a numerical solution of Burger's equation using B-spline collocation method. Zahra and Elkholy [13] use cubic splines to represent a numerical solution of fractional differential equations. Nonclassical diffusion problems by Ismail et al. [14], advection-diffusion problems by Nazir et al. [15], and fractional subdiffusion equation by Zhu et al. [16] are solved by using B-spline collocation methods. In recent times, Hashmi et al. [17,18] has solved Hunter Saxton equation and space fractional PDE by cubic trigonometric and hybrid B-spline method. The numerical solution of time-space fractional PDEs using B-spline wavelet method is presented by Kargar and Saeedi [19].
In present research, numerical formulation of the Riemann-Liouville derivative for anomalous subdiffusion equation described by Dehghan et al. [20] is used as time fractional derivative which is given as where in which The paper is organized as follows: the description of proposed method along with methodology is presented in Section 2. In Section 3, initial state is calculated. Section 4 presents the convergence analysis of the iterative method.
The accuracy of the iterative scheme is shown in Section 5, and Section 6 contains the conclusion.

Description of Proposed Method
Hybrid B-spline collocation method is utilized to resolve the modified anomalous subdiffusion equation. The approximate solution ψðξ, τÞ to the analytical solution uðξ, τÞ is considered as Here, time-dependent unknowns are denoted by ς j ðτÞ, and H 3,j ðξÞ is a hybrid cubic B-spline basis function of third order and given as The value of γ plays a very significant role in the hybrid cubic basis function. The hybrid nature of the proposed method occurred in 0 < γ < 1, and at boundaries of this hybrid parameter, it produces trigonometric and polynomial B-spline. The nodal values are placed in Table 1.   Here, H 3,j−1 ðξ j Þ, H 3,j ðξ j Þ, and H 3,j+1 ðξ j Þ are only three nonzero basis functions that are included over subinterval ½ξ j , ξ j+1 owing to the local support property of B-spline basis function. Hence, the approximate solution and its derivatives with respect to ξ at ðξ j , τ n Þ, by using (8), are 2.1. Numerical Formulation. In order to apply the hybrid B-spline collocation method, we take (1) at ðξ j , t n Þ and can be written in a form as Now, using the above discretized form of forward difference scheme and Riemann-Liouville derivative as a time fractional derivative will have the form Simplifying the equation, we get where w p+1 − w p = ðd p Þ α i , κ = ððdτÞ α 1 Þ/ðΓð1 + α 1 ÞÞ and λ = ððdτÞ α 2 Þ/ðΓð1 + α 2 ÞÞ:

Initial State
The numerical procedure of the iterative process can be initiated using initial vector T . Initial conditions have been utilized to attain the initial vector by using the following procedure: where ς j ð0Þ ′ s are unknown parameters. We need the initial approximation ψ N ðξ, 0Þ to satisfy the following conditions: This produce a square matrix of order ðN + 3Þ of the form and B = ½q 0 ′ ðξ 0 Þ, q 0 ðξ 0 Þ, ⋯, q 0 ðξ N Þ, q 0 ′ ðξ N Þ T . The pentadiagonal system can be solved by the Thomas algorithm.

Stability Analysis
The stability of numerical technique can be examined by performing Von Neumann stability analysis. The stability of numerical schemes is closely related with numerical errors.

Journal of Function Spaces
The Von Neumann stability analysis provides us growth of the error in terms of Fourier series. For this, we used in equation, where θ is a rational number and i is a imaginary symbol. Utilizing (13), we get     Journal of Function Spaces Simplifying the equation in the presence of (18), we have After simplification, we get Thus, jη n+1 j ≤ jη n j. Hence, our described method is stable unconditionally.

Numerical Experiments
In current section, numerical examples are computed to prove feasibility and accuracy of the scheme, and the results are displayed through graphs and tables.
Example 1. Consider the following problem in the domain 0 ≤ ξ ≤ 1 and for the interval 0 < τ ≤ 1, with the boundary conditions and the IC where

Journal of Function Spaces
The exact solution is u ξ, τ ð Þ= e ξ τ 1+α 1 +α 2 À Á : ð26Þ Table 2 describes the numerical errors at distinct rational values of γ, and it is evident that γ = 0 is an appropriate solution than other case. So, trigonometric spline is the obvious choice to proceed further. In Table 3, the numerical display of the solutions at different values of τ f is shown for α 1 = 0:2, α 2 = 0:8, and N = 50. Table 4 shows the numerical display of solution for different values of α 1 at final time τ f = 0:01 and α 2 = 0:3. Table 5 shows the numerical display of solutions at different values of N when α 1 = 0:6, α 2 = 0:9, and τ f = 0:001. Figure 1(a) shows the graphical error when α 1 varies at final time τ f = 0:05 but α 2 remains same. Figure 1(b) shows the graphical representation of errors at different terminal time τ f when α 1 and α 2 remains same. Figure 2(a) represents the comparison of numerical and approximate solution in the entire domain; on the other hand, the comparison at different values of τ f is presented in Figure 2(b).

Conclusion
A B-spline collocation method is utilized to find the numerical approximation to the solution curve of fractional form of

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors have no conflict of interests regarding the publication of this paper.