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 [14]. 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 [69], 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