Nonpolynomial Spline Interpolation for Solving Fractional Subdiffusion Equations

e nonpolynomial spline interpolation is proposed to distinguish numerical analysis from the senes boundary conditions, accurance error estimations. e idea used in this article is readily applicable to obtain numerical solution of nonpolynomial spline interpolation. ese analyze the methods that are suitable for the numerical solution of subdiusion equation. e method has been shown to be stable by using von Neumann technique. e accuracy and eciency of the scheme are checked by several examples to obtain numerical tests.


Introduction
In recent years, the derivation of solutions to fractional di erential equations has become a hot topic in many areas of applied science and engineering.
ere are still not enough extremely precise numerical methods, although there have been many works on numerical methods for fractional di erential equations. It is also very important to conduct research on fractional-order di erential equations in unconstrained elds. A popular approach to solving these problems is to use arti cial boundary conditions (ABCs) [1].
ere are several applications of spline methods to the numerical solution of di erential equations and, in particular, fractional di erential equations [2][3][4][5][6]. e derivation of solutions to fractional di erential equations has become a rst-rate topic in many areas of applied science and engineering. Since fractional derivatives provide more accurate models than integer derivatives for most real-world problems, fractional di erential equations are used to formulate a considerable number of applied problems. Applications are diverse and include viscoelastic and viscoelastic ows, control theory, transport problems, tumor evolution, random walks, continuum mechanics, and turbulence [7][8][9][10][11][12][13]. Finite di erences, Fourier method, orthogonal spline collocation, implicit schemes with alternating direction, and compact nite di erence methods have been successfully used in [14][15][16][17][18][19][20][21][22] to solve fractional subdi usion equations. e author of [23] has presented an implicit numerical method for the fractional di usion equation, where the fractional derivative is discretized by spline and the Crank-Nicolson discretization is used for the time variable. e authors of [24] have developed a novel nonpolynomial spline method for solving second-order hyperbolic equations, and their results are numerically more accurate than some nite di erence methods, see [25][26][27][28]. e authors of [29] presented a quadratic nonpolynomial spline approach to approximate the solution of a system of second-order boundary value problems associated with a one-sided obstacle, and compared to collocation, nite di erence, and certain common polynomial spline approaches, we used contact problems and produced approximations that were more accurate.

Basic Definition
Definition 1. Let u(x) be a function defined on (a, b), then the Riemann-Liouville fractional derivative is of the following form [12]: Definition 2. Let u(x) be a function defined on (a, b), then the Caputo fractional derivative is of the following form [12]:

Lemma 1.
Let Q is an estimation of the smoothness function at class C 2 , then the error estimation obtained as Proof. e above relations are due to the expansion of can be obtained as follows: Also, we have the same way.

Mathematical Problems in Engineering
And this is the end of the proof.
□ e local truncation error for the first-and second-order spline method is obtained as follows: To obtain unique solution for the nonlinear system equation (10), we need four more equations. We define the following identities: by using Taylor's expansion, we obtain the unknown coefficients in equation (16) where from equation (1), we can write M j i in the form Now, we need to use the definition of discrete approximation of fractional derivative in time of any appropriate function z2, 10 as where a � 1/Γ(2 − α)τ α and R j,q � (j − q) 1− α − (j − q − 1) 1− α . en, from equations (18) and (19), we can express M j i in the following form: By substituting equations (20) in (10), we have following systems: for i � 3, 4, . . . , n − 3, if j � 1, and Substituting equations (20) in (16), we get Mathematical Problems in Engineering 5 In the above systems, for each j � 1, 2, . . . , k, clearly, we have n linear equations in terms of n unknowns u j 1 , u j 2 , . . . , u j n .

Mathematical Problems in Engineering
From the denoted values of x and y, the equation (30) can be simplified as en, we obtain Now, from this equation, it is obtained that |ξ j | ≥ |ξ 0 | for all j � 2, 3, . . . , k (see [3]). Hence, the nonpolynomial spline method is unconditionally stable.

Numerical Results
In this section, the wealth of the presented method nonpolynomial spline method will be tested on three different examples, which are widely used in the literature. All problems were solved with best error estimations, and their results were compared with analytical or spline solutions     given in the literature. In the following section, detailed information about this comparison will be presented.

Example 1. Consider problem equations (1)-(4) with
e exact solution of the problem is as follows [7]: is problem is solved using the methods described in Section 2 for T � 1, 2, 3. e results are reported in Tables 1  and 2.
It can be checked that the exact solution is u(x, t) � t 2 sin(2πx) [10].
is problem is solved using the methods described in Section 2 for T � 1/4, 1/2, 1.
e results are reported in Table 3.

Conclusion
In this paper, we constructed the application of the nonpolynomial spline method to obtain the numerical point solution of fractional partial differential equations. Numerical results showed that these techniques achieve accuracy, in numerical results illustrative graphs, the Figure 1 the comparison of the present results with the result of [10] shows the efficiency of the suggested technique. Also, the stability and error estimates of the methods are investigated and shown some figures of the results. Moreover, illustrations for the stability region of the schemes are derived.

Data Availability
No data were used to support this study.

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