Abstract
Aiming at the initial boundary value problem of variable-order time-fractional wave equations in one-dimensional space, a numerical method using second-order central difference in space and H2N2 approximation in time is proposed. A finite difference scheme with second-order accuracy in space and order accuracy in time is obtained. The stability and convergence of the scheme are further discussed by using the discrete energy analysis method. A numerical example shows the effectiveness of the results.
1. Introduction
In recent years, due to the non-locality of fractional calculus, more and more problems in physical science, electromagnetism, electrochemistry, diffusion and general transport theory can be described by the fractional calculus approach, among which the Riemann-Liouville fractional derivative and the Caputo fractional derivative are the most widely used [1–4]. At the same time, more and more researchers found that a variety of important dynamical problems exhibit fractional-order behavior that may vary with time, space, or other conditions. This phenomenon indicates that variable-order fractional calculus is a natural choice to provide an effective mathematical framework for the description of complex problems.
In 2020, Shen et al. proposed a new numerical approximation method—the H2N2 approximation [5] for the numerical differential formula of the Caputo fractional derivative of and applied it for the constant-order time-fractional wave equations in the following multidimensional space where are given sufficiently smooth functions, is the boundary of , . When satisfy consistency conditions . It was proved that the proposed scheme has the accuracy of order of in time and 2 in space, and it is clear that its theoretical analysis is similar to the L1 method applied in solving the constant-order time-fractional slow diffusion equations.
Motivated by the above literature [6–9], in this work, we consider the numerical solution of the following variable-order time-fractional wave equations in one-dimensional space where is the variable-order Caputo fractional derivative, are given suffificiently smooth functions and satisfy . Suppose its solution function .
The rest of this paper is organized as follows. In the next section, some necessary notations are introduced. In Section 3, the H2N2-based finite difference scheme for the variable-order time-fractional wave equations is derived. In Section 4, the stability and convergence of the difference scheme are studied. In Section 5, a numerical result is listed to verify the theoretical prediction and the effectiveness of the difference scheme. Finally, a brief conclusion is provided.
2. Preliminary Knowledge and Relevant Lemmas
Definition 1 (see [10]). Suppose the function is defined on the interval , then the variable-order Caputo fractional derivative is defined as
Next, mesh the solution intervals and , take integers and , denote , and are called space step and time step, respectively. Denote . Define the following grid function spaces
For grid function defined on , introduce the following notations
For any grid functions , denote the following notations
For any function defined on the interval , using the data to make the quadratic Hermite interpolation polynomial of Taking the twice derivative arrives at
For any function defined on the interval , using three points to make the quadratic Newton interpolation polynomial of Taking the second-order derivative yields
On the basis of the above interpolation polynomial, we next discuss the high-precision approximation formula of the variable-order Caputo fractional derivative.
Here, we denote . Suppose , then at the half-grid point , we have
Here
where .
Then, it can be calculated that
Denote
we have
Here,, the proof process is similar to Theorem 2.1 in Reference [5].
Lemma 2. For any , according to defined by (14)–(15), we have
Proof. According to the formula (14)–(15), we have When , it can be obtained by calculation From equations (20) and (21), we have Therefore, it can be obtained When , we have From the above formula And when , we have Therefore, it can be seen that To sum up, Lemma 2 is proved.
Lemma 3 (see [11]). If the function , there is
Lemma 4. For any positive integer and any , when we have where .
Proof. On the basis of [12], it can be seen from the condition When satisfies the following condition namely Then, we have
Remark 5. Consider the function
We have
If and is an non-increasing function on , then , consequently
(30) is valid.
If is an non-increasing function on the interval and is a constant on the interval , (30) is also valid.
Lemma 6 (see [11]). For any , there is
Lemma 7 (see [11]). For any grid function , there is
Lemma 8 (see [13]). Suppose are two non-negative sequences, does not decrease with , if where is an non-negative constant, when , then
3. Establishment of the Difference Scheme
Denote , consider (2) at the point , we have
Applying (13) to approximate the temporal fractional derivative and central difference quotient (29) to approximate the spatial derivative, we can obtain
There exists a positive constant such that
Noticing the initial and boundary value conditions (3) and (4), we have
Omitting the small term in the equation and replacing the grid function by its numerical approximation , we construct the difference scheme for solving the problems (2)–(4) as follows
4. Stability and Convergence of the Difference Scheme
Theorem 9. Suppose is the solution of the following difference scheme where is a given perturbation term, when , it holds that and are given in (56) and (64), respectively.
Proof. Taking an inner product (50) with and summing from 1 to , we have Noticing that where Applying Lemma 4, we have Then, we have By Lemma 2, noticing that Then We use the Cauchy inequality for the inner product , the above equation can be simplified Multiplying by , then we have Note that is decreasing on the interval . Since , . Then Let Then It is easy to know does not decrease with . According to Lemma 8, when , we have Theorem 9 is proved. We can say that the difference scheme is stable.
Theorem 10. Assume and are solutions of problems (2)–(4) and difference scheme (47)–(49), respectively. Denote Then, there exists a positive constant , when , , such that where and .
Proof. Subtracting (44) and (46) from (47)–(49), we obtain the system of error equations
Applying Theorem 9 and (45), it yields
where .
Applying Lemma 7, it yields
The proof is ended.
5. Numerical Example
In order to verify the accuracy of the finite difference scheme, several different types of variable-order index are used to solve the variable-order fractional wave equations (2)–(4) in 1D case. The scheme is implemented in MATLAB (R2019a).
Here, we take , . The source term of equation (2) the initial value the boundary value the exact solution is given by
Define the error of the numerical solution the temporal convergence order the spatial convergence order
Denote , for different , , . The time step is varied from to , where , 1024, 2048, 4096. Table 1 shows the errors and temporal convergence orders of the difference scheme (47)–(49). It can be seen from Table 1 that the difference scheme (47)–(49) has a precision of approximately order in time. The computational results are in good agreement with theoretical results.
Take a fixed and sufficiently small time step , for different , , , verify space step from /5 to /40, where , . Table 2 shows the errors and spatial convergence orders of the difference scheme (47)–(49). It can be seen from Table 2 that the difference scheme (47)–(49) in the space has an accuracy of approximately 2 order, which is consistent with the theoretical results.
6. Conclusions
In this paper, we consider a numerical approximation method for the variable-order Caputo fractional derivative—H2N2 approximation, and give the corresponding calculation formula. Secondly, we use this formula to solve the one-dimensional variable-order time-fractional wave equations and discuss the stability and convergence of the equations by the discrete energy analysis method. Finally, a numerical example verifies the effectiveness of the scheme.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The work is supported by the National Natural Science Foundation of China (11761074), the Natural Science Foundation of Jilin Province (2020122336JC), and the Program for Young and Middle-aged Leading Talents in Scientific and Technological Innovation of Jilin Province (20200301053RQ).