Abstract
The objective of this paper is to present an efficient numerical technique for solving time fractional modified anomalous subdiffusion equation. Anomalous diffusion equation has its role in various branches of biological sciences. B-spline is a piecewise function to draw curves and surfaces, which maintain its degree of smoothness at the connecting points. B-spline provides an active process of approximation to the limit curve. In current attempt, B-spline curve is used to approximate the solution curve of time fractional modified anomalous subdiffusion equation. The process is kept simple involving collocation procedure to the data points. The time fractional derivative is approximated with the discretized form of the Riemann-Liouville derivative. The process results in the form of system of algebraic equations, which is solved using a variant of Thomas algorithm. In order to ensure the convergence of the procedure, a valid method named Von Neumann stability analysis is attempted. The graphical and tabular display of results for the illustrated examples is presented, which stamped the efficiency of the proposed algorithm.
1. Introduction
The current study deals with the investigation of modified fractional anomalous subdiffusion equation using hybrid B-spline-based collocation method. The general form of the equation iswith BCand ICwhere is the Riemann-Liouville version of the fractional partial derivatives of order (), are the fractional parameters with for are fractional parameters appears in the equation. Here, , , , and are smooth functions, and 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 [2–6]; 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 aswherein whichand . Modified fractional anomalous subdiffusion equation have not been solved by hybrid B-spline collocation method yet. So, we solved modified fractional equation using hybrid B-spline collocation method. The proposed method is evaluated by Von Neumann stability analysis for convergence, which stamped that iterative method is stable without any condition over the domain. Numerical examples have been tested to highlight the performance of the numerical method. Numerical results are presented through tables and graphs. It shows that result very much agree with closed form of solution.
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.
2. Description of Proposed Method
Hybrid B-spline collocation method is utilized to resolve the modified anomalous subdiffusion equation. The approximate solution to the analytical solution is considered as
Here, time-dependent unknowns are denoted by , and is a hybrid cubic B-spline basis function of third order and given as
The value of plays a very significant role in the hybrid cubic basis function. The hybrid nature of the proposed method occurred in , and at boundaries of this hybrid parameter, it produces trigonometric and polynomial B-spline. The nodal values are placed in Table 1.
Here, , and are only three nonzero basis functions that are included over subinterval owing to the local support property of B-spline basis function. Hence, the approximate solution and its derivatives with respect to at , by using (8), are
2.1. Numerical Formulation
In order to apply the hybrid B-spline collocation method, we take (1) at 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 getwhere
3. Initial State
The numerical procedure of the iterative process can be initiated using initial vector . Initial conditions have been utilized to attain the initial vector by using the following procedure:where are unknown parameters. We need the initial approximation to satisfy the following conditions:
This produce a square matrix of order of the formwhere and . The penta-diagonal system can be solved by the Thomas algorithm.
4. 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. The Von Neumann stability analysis provides us growth of the error in terms of Fourier series. For this, we usedin equation, where is a rational number and is a imaginary symbol. Utilizing (13), we get
Simplifying the equation in the presence of (18), we have
After simplification, we get
Thus, .
Hence, our described method is stable unconditionally.
5. 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 problemin the domain and for the interval , with the boundary conditionsand the ICwhereThe exact solution is
Table 2 describes the numerical errors at distinct rational values of , and it is evident that 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 is shown for , , and . Table 4 shows the numerical display of solution for different values of at final time and . Table 5 shows the numerical display of solutions at different values of when , , and . Figure 1(a) shows the graphical error when varies at final time but remains same. Figure 1(b) shows the graphical representation of errors at different terminal time when and 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 is presented in Figure 2(b).
Example 2. Consider the problemin the domain and for the time
with the BCand the ICwhereThe exact solution is
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In Table 6, the absolute error at different values of is displayed, and it shows that gives better result. Table 7 shows the numerical solution for different values of with final time and . In Table 8, the numerical display of solutions for different values of when , , and is presented. Figure 3(a) represents absolute error Example 2 of various values of with fixed . Figure 3(b) describes the graphical representation of errors at various values of terminal time when and remains same. Figure 4(a) represents the comparison of numerical and approximate solution in entire domain; on the other hand, the comparison at different values of is presented in Figure 4(b).
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6. Conclusion
A B-spline collocation method is utilized to find the numerical approximation to the solution curve of fractional form of anomalous subdiffusion equation in modified form. First, formulate the formula for the modified fractional anomalous subdiffusion equation using discretized form of Riemann-Liouville Derivative for the time fractional derivative. Then, we show the stability of B-spline method using the Fourier version of stability method, i.e., Von Neumann stability analysis. The iterative method is applied on numerical examples to show the applicability of described method. The graphs and tables demonstrate the numerical results. It shows that results are very much in agreement with exact solution. The technique can be further extended to higher dimension and nonlinear fractional PDEs in future.
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.
Acknowledgments
This work is supported by the Higher Education Commission of Pakistan under the NRPU project grant number 9306/Punjab/NRPU/R&D/HEC/2017 and by the Taif University Researchers Supporting Project number (TURSP-2020/77), Taif University, Taif, Saudi Arabia. The research was supported by the National Natural Science Foundation of China (Grant nos. 11971142, 11871202, 61673169, 11701176, 11626101, and 11601485).