A Fast Compact Finite Difference Method for Fractional Cattaneo Equation Based on Caputo–Fabrizio Derivative

)e Cattaneo equations with Caputo–Fabrizio fractional derivative are investigated. A compact finite difference scheme of Crank–Nicolson type is presented and analyzed, which is proved to have temporal accuracy of second order and spatial accuracy of fourth order. Since this derivative is defined with an integral over the whole passed time, conventional direct solvers generally take computational complexity of O(MN2) and require memory of O(MN), with M and N the number of space steps and time steps, respectively. We develop a fast evaluation procedure for the Caputo–Fabrizio fractional derivative, by which the computational cost is reduced to O(MN) operations; meanwhile, only O(M) memory is required. In the end, several numerical experiments are carried out to verify the theoretical results and show the applicability of the fast compact difference procedure.


Introduction
Fractional diffusion equations have become a strong and forceful tool to describe the phenomenon of anomalous diffusion, and more research works have been obtained in the last decades [1][2][3][4][5][6]. However, since the fractional derivative is nonlocal and has weak singularity, it is impossible to solve fractional diffusion equations analytically in most cases. Instead, seeking numerical solutions is becoming an indispensable tool for research work about fractional equations.
Different from the traditional derivative of the integer order, the fractional derivative depends on the total information in the correlative region, and this is the socalled nonlocal properties. Just because of this, it consumes computational time extremely to solve fractional equations. We hope to develop effective numerical schemes, which not only have better stability and higher accuracy but also require less storage memory and save computational cost.
About stability and convergence analysis of the numerical schemes for fractional equations, the readers can refers to [7,8] for spatial fractional order equation, [9][10][11][12][13][14][15][16][17][18] for temporal fractional diffusion equations, and [19][20][21][22] for space-time-fractional equations. About the complexity, i.e., storage requirement and computation cost of an algorithm, researchers devote themselves to reduce storage requirement and computational time by analyzing the particular structure of coefficient matrices arising from the discretization system or reutilizing the intermediate data skillfully. We call these algorithms fast methods, including fast finite difference methods [23][24][25][26][27][28], fast finite element methods [29], and fast collocation methods [30,31]. A fast method for Caputo fractional derivatives is proposed [32,33]. Lu et al. [34] presented a fast method of approximate inversion for triangular Toeplitz tridiagonal block matrix, which is successfully applied to the fractional diffusion equations. Comparatively, there is less research work about the fast method for temporal fractional derivative than that for spatial fractional operators.
where 1 < α < 2; Ω � (a, b) for one-dimensional case, and Ω � (a, b) × (c, d) for two-dimensional case; f(x, t) is the source term; ϕ(x) and ψ(x) are the prescribed functions for initial conditions; and z α u/zt α is a new Caputo fractional derivative without singular kernel, which is defined in the next section. Our purpose is to establish a fast finite difference scheme of high order for this equation. We will extract the recursive relation between the (k + 1) time step and the k time step of the finite difference solution. e computational work is significantly reduced from O(MN 2 ) to O(MN), and the memory requirement from O(MN) to O(M), where M and N are the total numbers of points for space steps and time steps, respectively. For improving the accuracy, a compact finite difference scheme is established. eoretical analysis shows that the fast compact difference scheme has spatial accuracy of fourth order and temporal accuracy of second order. Several numerical experiments are implemented, which verify the effectiveness, applicability, and convergence rate of the proposed scheme.
is paper is organized as follows: some definitions and notations are prepared in Section 2. e compact finite difference scheme is described and then the stability and convergence rates are rigorously analyzed for the scheme in Section 3. e compact finite difference scheme is extended to the case of two space dimensions in Section 4. Fast evaluation and efficient storage are established skillfully in Section 5. Some numerical experiments are carried out in Section 6. In the end, we summarize the major contribution of this paper in Section 7.

Some Notations and Definitions
We provide some definitions which will be used in the following analysis.
First, let us recall the usual Caputo fractional derivative of order α with respect to time variable t, which is given by By replacing the kernel function (t − s) − α with the exponential function exp(− α(t − s/1 − α)) and 1/(Γ(1 − α)) with M(α)/1 − α, Caputo and Fabrizio [35] proposed the following definition of fractional time derivative.

Remark 2.
An open discussion is ongoing about the mathematical construction of the CF operator. Ortigueira and Tenreiro Machado [37] indicated that the CF fractional derivative is neither a fractional operator nor a derivative operator, the authors of [38,39] showed that this operator cannot describe dynamic memory, and Giusti [40] indicated that this operator can be expressed as an infinite linear combination of Riemann-Liouville integrals with integer powers. As responses to these criticisms, Atangana and Gómez-Aguilar [41] pointed out the need to account for a fractional calculus approach without an imposed index law and with nonsingular kernels. Furthermore, Hristov [42] indicated that the CF operator is not applicable for explaining the physical examples in [37,40]; instead, he suggested that the CF operator can be used for the analysis of materials that do not follow a power-law behavior. e authors of [43] believe that models with CF operators produce a better representation of physical behaviors than do integer-order models, providing a way to model the intermediate (between elliptic and parabolic or between parabolic and hyperbolic) behaviors.
To obtain the accuracy of the fourth order in spatial directions, the following lemma is necessary.

Compact Finite Difference Scheme for One-Dimensional Fractional Cattaneo Equation
In order to construct the finite difference schemes, the interval [a, b] is divided into subintervals with where h � (b − a)/M and Δt � T/N are the spatial grid size and temporal step size, respectively.
. e values of the function u at the grid points are denoted as u k j � u(x j , t k ), and the approximate solution at the point ( . We also introduce the following notations for any mesh function v ∈ V h : and define the average operator It is easy to see that where I is the identical operator. We also denote the discrete inner products and norms are defined as By summation by parts, it is easy to see that For the average operator A, define Additionally, let V Δt � v|v � (v 0 , v 1 , · · · , v N ) be the space of grid function defined on Ω Δt . For any function v ∈ V Δt , a difference operator is introduced as follows: 3.1. Compact Finite Difference Scheme. We will consider the time-fractional Cattaneo equation equipped with the Caputo-Fabrizio derivative. Vivas-Cruz et al. [43] gave the theoretical analysis of a model of fluid flow in a reservoir with the Caputo-Fabrizio operator. ey proved that this model cannot be used to describe nonlocal processes since it can be represented as an equivalent differential equation with a finite number of integer-order derivatives. e finite difference methods usually lead to stencils through the whole history passed by the solution which consume too much computational work. In this paper, we will establish a high-order finite difference scheme and propose a procedure to reduce the computational cost. In [43], the authors proposed a recurrence formula of discretized CF operator and obtained an algorithm which can be considered a stencil with a one-step expression without the need of integrals over the whole history. It seems that the procedure in our paper and the algorithm in [43] are different in approach but equally satisfactory in result.
For obtaining effective approximation with high order, we introduce the numerical discretization for the fractional By the initial and boundary value conditions, we have A compact finite difference scheme can be established by omitting the truncation term R k+ (1/2) i and replacing the exact solution u k i in equation (21) with numerical solution u k i :

Stability Analysis.
e following Lemma about M n is useful for the analysis of stability.
Lemma 2 (see [45]). For the definition of M n , M n > 0 and M n+1 < M n , ∀n ≤ k, are held.
Multiplying hδ t u k+1 i on both sides of equation (24) and summing up with respect to i from 1 to M − 1, the following equation is obtained: Observing equation (13), we have By the triangle inequality and Lemma 2, we obtain 4 Mathematical Problems in Engineering Let Summing up with respect to k from 0 to N − 1 leads to 2 1 , and then Theorem 1. For scheme (24), we have the following stable conclusion:

Optimal Error Estimates.
Combining equations (21) and (23) with (24), we get an error equation as follows: i on both sides of equation (33) and summing up with respect to i from 1 to M − 1, we get By the triangle inequality and Lemma 2, we obtain Combining equation (34) with (35), we have By the definition of Q in stability analysis, the inequality (36) can be rearranged as

Mathematical Problems in Engineering
Summing up with respect to k from 0 to N − 1, we get Observing that the initial error e 0 � 0 implies Q(e 0 ) � 0. en, we have

Theorem 2. Suppose that the exact solution of the fractional Cattaneo equation is smooth sufficiently, then there exists a positive constant C, independent of h, k, and Δt such that
where

Compact Finite Difference Scheme in Two Dimensions
In this section, the following fractional Cattaneo equation in two dimensions will be considered: ϕ(x, y) and ψ(x, y) are the given functions, and z α u/zt α is defined by the new Caputo fractional derivative without singular kernel. In order to construct the finite difference schemes, the and Δt � T/N are the spatial grid and temporal step sizes, respectively.
denotes the values of function u at the grid points, and u k i,j denotes the values of the numerical solution at the point (x i , y j , t k ).
For any mesh function v ∈ V h , we use the following notations: and define the average operator It is clear that We also denote A For any gird function u, v ∈ V 0 h , the discrete inner product and norms are defined as follows: For the average operator A x A y , define

Compact Finite Difference Scheme.
At the node (x i , y j , t k+(1/2) ), the differential equation is rewritten as For the approximation of the time-fractional derivative, we have the following approximation [45]: where the truncation error R k+ (1/2) i,j � O(Δt 2 ) and Furthermore, we also have zu zt Substituting (48) and (50)∼(52) into (47) leads to and there exists a constant C, depending on the function u and its derivatives such that By the initial and boundary conditions, we have Omitting the truncation error R k+ (1/2) i,j and replacing the true solution u k i,j with numerical solution u k i,j , a compact finite difference scheme can be obtained as follows: Mathematical Problems in Engineering

Stability Analysis
Definition 4 (see [46]). For any gird function u ∈ V 0 h , define the norm e lemmas below is useful in the subsequent analysis of stability.
Multiplying h x h y δ t U k+1 i,j on both sides of equation (56) and summing up w.r.t. i, j from 1 to (M x − 1) and from 1 to (M y − 1), respectively, the following equation is obtained: Observing Lemma 4, we have By the triangle inequality and Lemma 2, we obtain Combining equation (60) with (61)∼(63),we get

Mathematical Problems in Engineering
Let Summing up with respect to k from 0 to N − 1, we get Noting Theorem 3. For the compact finite difference scheme (56), the following stability inequality holds: Similar to the stability, the convergence can also be analyzed.

Theorem 4. Suppose that the exact solution of the fractional Cattaneo equation is sufficiently smooth, then there exists a positive constant C independent of h, k, and Δt such that
where e k i,j � u k i,j − u k i,j and h � max h x , h y .

Efficient Storage and Fast Evaluation of the Caputo-Fabrizio Fractional Derivative
Since time-fractional derivative operator is nonlocal, the traditional direct method for numerically solving the frac- Let (71)       Table 5: Considering h � 0.001, the discrete l ∞ error and convergence rates of u with different α for Example 1.
Δt  zu(x, t) zt u(x, 0) � sin(πx), zu zt t�0 � sin(πx), In Tables 1 and 2, we take Δt � h 2 and h � �� Δt √ to examine the discrete l ∞ -norm (l 2 -norm) errors and corresponding spatial and temporal convergence rates, respectively. We list the errors and convergence rates (order) of the proposed compact finite difference (CD) scheme, which is almost O(Δt 2 + h 4 ) for different α. Additionally, Table 3 shows the CPU time (CPU) consumed by direct compact (DCD) scheme and fast compact difference (FCD) scheme, respectively. It is obvious that the FCD scheme has a significantly reduced CPU time over the DCD scheme. For instance, when α � 1.5, we choose h � 0.1 and Δt � 1/50, 000 and observe that the FCD scheme consumes only 94 seconds, while the DCD scheme consumes 3692 seconds. We can find that the performance of the FCD scheme will be more conspicuous as the time step size Δt decreases.
In Figure 1, we set h � 0.1 and α � 1.75 and change the total number of time steps N to plot out the CPU time (in seconds) of the FCD scheme and DCD scheme. We can observe that the CPU time increases almost linearly with respect to N for the FCD scheme, while the DCD scheme scales like O(N 2 ). Tables 4 and 5 show the discrete l ∞ errors and convergence rates of the compact finite difference scheme for Example 1.
e space rates are almost O(h 4 ) for fixed Δt � 2 − 13 , and the time convergence rates are always O(Δt 2 ) for fixed h � 0.001. We can conclude that the numerical convergence rates of our scheme approach almost to O(Δt 2 + h 4 ).

Example 2.
e example is described by Note that the exact solution of the above problem is      We apply the fast compact difference scheme to discretize the equation. In Figure 2, we set c � 0.001, α � 1.5, and M � N � 100 and plot exact and numerical solutions at time T � 1 for Example 2 with different x 0 . For x 0 � 0.5, α � 1.5, and M � N � 100, we also plot exact and numerical solutions with the different c in Figure 3. In Figure 4, for h � 0.1 and α � 1.5, we vary the total number of time steps N to plot out the CPU time (in seconds) of the FCD scheme and DCD scheme. e numerical experiments verified our theoretical results. In Table 6, by equating Δt � h 2 and fixing x 0 � 0.5, we compute the discrete l ∞ error and convergence rates with different fractional derivative orders α and different c. It shows that the compact finite difference scheme has space accuracy of fourth order and temporal accuracy of second order. We set h � �� Δt √ and fix c � 0.01, and the discrete l 2 error and convergence rates with different α and x 0 are displayed in Table 7.

Example 3.
If the exact solution is given by u(x, t) � e t sin(πx)sin(πy), we have different f(x, y, t) for different α accordingly: In Figure 5, h � 0.1 and α � 1.75 are fixed, and the total number of time steps N vary to plot out the CPU time (in seconds) of the FCD procedure and DCD procedure, and it presents an approximately linear computation complexity for FCD procedure. We set Δt � h 2 in Table 8, and h � �� Δt √ in Table 9, the discrete l ∞ error, discrete l 2 error, and convergence rates with different derivative orders α are presented. e fourth-order space accuracy and secondorder temporal accuracy can be observed clearly.

Conclusion
In this paper, we develop and analyze a fast compact finite difference procedure for the Cattaneo equation equipped with time-fractional derivative without singular kernel. e timefractional derivative is of Caputo-Fabrizio type with the order of α(1 < α < 2). Compact difference discretization is applied to obtain a high-order approximation for spatial derivatives of integer order in the partial differential equation, and the Caputo-Fabrizio fractional derivative is discretized by means of Crank-Nicolson approximation. It has been proved that the proposed compact finite difference scheme has spatial accuracy of fourth order and temporal accuracy of second order. Since the fractional derivatives are history dependent and nonlocal, huge memory for storage and computational cost are required. is means extreme difficulty especially for a long-time simulation. Enlightened by the treatment for Caputo fractional derivative [32], we develop an effective fast evaluation procedure for the new Caputo-Fabrizio fractional derivative for the compact finite difference scheme. Several numerical experiments have been carried out to show the convergence orders and applicability of the scheme.
Inspired by the work [43], the topic about modelling and numerical solutions of porous media flow equipped with fractional derivatives is very interesting and challenging and will be our main research direction in the future.

Data Availability
All data generated or analyzed during this study are included in this article.

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