Numerical Investigation of the Fractional Diffusion Wave Equation with the Mittag–Leffler Function

: A spline is a sufficiently smooth piecewise curve. B-spline functions are powerful tools for obtaining computational outcomes. They have also been utilized in computer graphics and computer-aided design due to their flexibility, smoothness and accuracy. In this paper, a numerical procedure dependent on the cubic B-spline (CuBS) for the time fractional diffusion wave equation (TFDWE) is proposed. The standard finite difference (FD) approach is utilized to discretize the Atangana–Baleanu fractional derivative (ABFD), while the derivatives in space are approximated through the CuBS with a θ -weighted technique. The stability of the propounded algorithm is analyzed and proved to be unconditionally stable. The convergence analysis is also studied, and it is of the order O ( h 2 + ( ∆ t ) 2 ) . Numerical solutions attained by the CuBS scheme support the theoretical solutions. The B-spline technique gives us better results as compared to other numerical techniques.


Introduction
Fractional calculus (FC) is an area of mathematics that studies the integrals and derivatives of non-integer order as well as their applications.Leibniz [1] inquired about the derivative of order µ = 1  2 in a letter to L'Hôpital dated 30 September 1695.This day can be viewed as FC's birthday.A comprehensive history of FC was discussed in [2,3].However, it has attracted significant study interest in the last two decades due to its ability to solve a wide range of engineering and science problems: for example, bioengineering [4], heat and mass transfer [5], transport problems [6], damage and fatigue [7], groundwater flow problem [8,9], PID controllers [10], anomalous diffusion models [11], magnetic resonance imaging [12], hydrology cycle [13], economic processes [14] and coronavirus [15].In many cases, fractional problems characterized by fractional partial differential equations (FPDEs) frequently behave more efficiently than their conventional integer-order counterparts.In general, it is impossible to acquire the exact solution of the maximum of FPDEs.As a consequence, finding numerical methods to analyze these models is becoming increasingly important.
In this paper, the following TFDWE with damping and reaction terms for different numerical outcomes shall be studied: ∂ µ w(q, t) ∂t µ + ψ ∂w(q, t) ∂t + ϑw(q, t) − ∂ 2 w(q, t) ∂q 2 = p(q, t), µ ∈ (1, 2), q ∈ [a, b], t ∈ [t 0 , T], with initial conditions (ICs): w(q, t 0 ) = α 1 (q), w t (q, t 0 ) = α 2 (q) (2) and boundary conditions (BCs): where ψ and ϑ are coefficients of the damping and reaction terms, respectively, p(q, t) is the source term, and w t (q, t 0 ) represents the derivative of the function w(q, t) with respect to time t at t = t 0 .The ABFD ∂ µ ∂t µ w(q, t) is described as: where R(µ) is the normalization function and fulfills R(0) = R(1) = 1.E µ,ς (q) is the Mittag-Leffler (ML) function satisfying E µ,1 (q) = E µ (q), which is given as: Several papers have recently discussed the ABFD's applicability.For example, Bas and Ozarslan [16] investigated population growth, logistics equations, Newton's law of cooling and blood alcohol models with ABFD.Gómez-Aguilar et al. [17] employed the ABFD for electromagnetic waves.Ghanbari and Atangana [18] used ABFD in image processing.AIDS, Zika and macroeconomic models have been investigated by Gao et al. [19] with the help of ABFD.Ravichandran et al. [20] formulated a computer virus model involving ABFD.Hanif and Butt et al. [21] presented an optimal control method with ABFD for a dengue fever model.A coronavirus model using ABFD has been proposed by Goyal et al. [22].
In the literature, different methods have been put out to solve the TFDWE.Liu et al. [23] developed an FD discrete method for the TFDWE involving the Caputo fractional derivative (CFD).Huang et al. [24] constructed two FD schemes for the TFDWE.A meshless scheme for the TFDWE has been analyzed by Dehghan et al. [25].For the TFDWE, Ali et al. [26] proposed a compact implicit difference algorithm.Wei [27] investigated the TFDWE using an FD/local discontinuous Galerkin scheme.Soltani Sarvestani et al. [28] established a wavelet method for a multi-term TFDWE with CFD.Huang et al. [29] developed efficient schemes for solving a non-linear TFDWE involving CFD.A pseudo-spectral scheme that depends on a reproducing kernel has been proposed by Fardi et al. [30].Chen et al. [31] presented numerical and analytical solutions for a TFDWE with damping.A multistep method has been used by Yang et al. [32] for the computational solution to the TFDWE.Chen and Li [33] established an FD scheme for solving the TFDWE.A fictitious time integration algorithm has been constructed by Hashemi et al. [34] to solve the TFDWE with CFD.
B-spline functions have been employed by several researchers for solving FPDEs.Some work has been done by using these functions, and their detailed comparison with the proposed method is reported in Table 1.The novelty of the proposed work is to discretize the secondorder time fractional derivative in the sense of the Atangana-Baleanu fractional operator µ ∈ (1, 2) with a second-order implicit finite difference scheme, while in reference [35], the authors used the Atangana-Baleanu fractional operator µ ∈ (0, 1) for discretization of the first-order time fractional derivative with a first-order implicit finite difference scheme, and in reference [36], the Caputo fractional derivative µ ∈ (0, 1) with the Crank-Nicholson finite difference scheme has been used to discretize the first-order time fractional derivative.It can be easily seen that the work done in this paper is different from [35,36] because we have used a different problem with second-order fractional derivative discretization.This secondorder time fractional derivative discretization in the sense of the Atangana-Baleanu fractional operator has never been used, as far as we are aware, for the case of the second-order TFDWE.A cubic trigonometric B-spline (CTBS) collocation scheme for the TF diffusion equation (DE) has been propounded by Dhiman et al. [37].Shafiq et al. [38] presented a numerical scheme for the TF Burgers' equation (BE) involving ABFD.Majeed et al. [39] constructed an extended-CuBS-based technique for numerical simulation of the modified BE with CFD.For the linear TF Klein-Gordon equation (KGE), Khader and Abualnaja [40] proposed computational solutions by employing quadratic spline functions.A new CuBS scheme for numerical results for the Burgers' and Fisher's problems has been propounded by Majeed et al. [41].Numerical simulations of the TFDE using a piecewise CuBS algorithm have been demonstrated by Shafiq et al. [42].
The propounded work is motivated by recent advances in the analysis of numerical simulations for the TFDWE.In this study, we designed and utilized a novel B-spline-based technique to solve the TFDWE.The current proposed method makes use of the secondorder ABFD and the θ-weighted scheme.Implementation of the ABFD for the TFDWE is novel in B-spline techniques.In addition, examination of the scheme's convergence and stability is carried out.To the greatest extent of the authors' awareness, the offered TFDWE method is novel and has not previously been mentioned in the literature.
This paper is formatted as follows: In Section 2, the CuBS functions and Parseval's identity (PI) are appended.The presented methodology is formulated in Section 3. The current method's stability and convergence are included in Sections 4 and 5, respectively.The efficacy and validity of the proposed technique are explored in Section 6, and finally, in Section 7, the research work is concluded.

Preliminaries Definition
then PI is defined as [43]: where ĝ(n) = b a g(q)e 2πinq dq is the Fourier transform for all integers n.
In addition, C −1 , C 0 , • • • , C N+1 have been established.For w(q, t), the estimation W(q, t) in terms of the CuBS can be presumed as [42]: where unknown points φ s (t) are to be computed at every temporal stage.Equations ( 7) and (8) provide the following approximations at the nodal points:

Description of Numerical Scheme
The given problem's time derivative is discretized using the ABFD.Suppose 0 M .Now we discretize the time derivative at t = t m+1 as Using the forward difference approach, Equation (10) becomes Hence, Equation (11) gives where E y = E µ,2 − µ 2−µ (y∆t) µ and v y = (y + 1)E y+1 − yE y .Furthermore, the truncation error ϖ m+1 ∆t is given by where 0 is a constant.Employing a θ-weighted scheme and Equation ( 12), Equation ( 1) is transformed to Discretizing ( 14) for θ = 1, we achieve where , and p m+1 s = p(q s , t m+1 ).It is detected that the term w −1 will appear when m = y or m = 0. To make w −1 vanish, we employ the IC to attain Using the CuBS approximation and its required derivatives at the knot q s in Equation (15), then Substituting ( 9) into (17), we get The system (18) has N + 1 linear equations with N + 3 unknowns.To get a consistent system, two more equations are acquired by employing the BCs (3).Hence, a matrix system of algebraic equations with (N + 3) × (N + 3) dimension is achieved; it can be uniquely solved using any suitable algorithm.Before utilizing (18), the initial vector The matrix form of Equation ( 19) is represented as where Any numerical algorithm can be used to solve Equation ( 20) for φ 0 .Mathematica 12 is utilized to accomplish the numerical results.

The Stability of the Presented Scheme
If the error does not increase while the algorithm is running, it can be deemed that the computational method is stable [44].The Fourier technique is utilized to ensure the proposed scheme's stability.For this, presume that εm s and ϵ m s indicate the growth factors numerically and analytically, respectively.The error Φ m s can be written as Thus, from Equation ( 18), we obtain ) .( 22) Through ICs and BCs, we can write and The grid function is as follows: The Fourier mode of Φ m (q) can be presented as: where Implementing the ∥.∥ 2 norm, we have .
Proof.Using expression (28) and Proposition 1, we achieve Consequently, the proposed computational technique is stable unconditionally.

The Convergence of the Presented Scheme
The methodology described in [45] is utilized to examine the convergence of the proposed approach.First and foremost, the following theorem is stated as [46,47]: If solution curve w(q, t) is interpolated by unique spline W(q, t) at q s ∈ ε, then for every t ≥ 0 there exist positive constants χ s independent of h; for s = 0, 1, 2, we have Lemma 1.The CuBS set {C s (q)} N+1 s=−1 in (7) fulfills the following inequality as provided in [42]: Theorem 3. If p is a member of C 2 [0, 1], and, furthermore, the computational solution W(q, t) to the analytical solution w(q, t) for TFDWE (1)-( 3) exists, then when h is small enough and χ > 0 is independent of h.
l m s (t)C s (q) be the unique CuBS interpolant for W(q, t).The triangular inequality yields Using inequality (32) for s = 0 in Equation ( 35), we get The current approach has collocation conditions p(q s , t) = LW(q s , t) = Lw(q s , t), s = 0, 1, • • • , N. Consider L W(q s , t) = p(q s , t).
Consequently, L( W(q s , t) − W(q s , t)) is written at t = t m as The BCs can be provided as (32), we attain (ψ where s = 0, 1, • • • , N. Employing IC, e 0 = 0: Taking absolute values of ℘ 1 s and ℑ 1 s and for an adequately small h, we get We get the values e 1 −1 and e 1 N+1 through BCs: where χ 1 does not depend on h.Now, mathematical induction is utilized for the proof of this theorem.Suppose that e x s ≤ χ x h 2 is true for 1 ≤ x ≤ n and χ = max{χ x : x = 0, 1, • • • , n}; then from Equation (37), we get Again, taking absolute values of ℘ m+1 s and ℑ m+1 s , we achieve Like before, we obtain the values of e m+1 −1 and e m+1 N+1 from the BCs: Consequently, for all m, we get In particular, Thus, from Lemma 1 and inequality (38), we attain Using (39), the inequality (36) yields where χ = χ 0 h 2 + 5 3 χ.Theorem 4. The numerical scheme (18) for TFDWE is convergent with respect to ICs and BCs.
Therefore, the proposed approach is second-order convergent.

Numerical Results and Discussion
In this section, numerical results of some experiments utilizing the proposed approach are demonstrated.To investigate the efficacy of the proposed technique, we utilize error norms L ∞ and L 2 as In addition, the convergence order is measured as All of the examples are investigated by taking R(µ) = 1.Numerical calculations are carried out by utilizing Mathematica 12 on an Intel(R) Core(TM) i5-3437U CPU @ 1.90 GHz (2.40 GHz Turbo), 12.0 GB RAM, an SSD and a 64-bit operating system (Windows 10).

Example 1. Consider the TFDWE
with ICs w(q, 0) = 0, w t (q, 0) = − sin(πq) and BCs w(0, t) = w(1, t) = 0, The expression w(q, t) = (t 2 − t) sin(πq) is the exact solution.For Example 1, the absolute errors for distinct q values at t = 0.2 are displayed in Table 2.For various temporal stages, error norms L ∞ and L 2 are demonstrated in Table 3. Table 4 comprises the analysis of the convergence order and error norms.Tables 5-7 describe the error norms at several values of ∆t and h. Figure 1 reveals the relation between numerical outcomes and analytical solutions at different temporal directions.A 3D plot of the analytical and approximate solutions is demonstrated in Figure 2. Figure 3 expounds on the descriptions of the 2D and 3D errors.Example 2. Consider the TFDWE with ICs w(q, 0) = 0, w t (q, 0) = 0 (40) and BCs w(0, t) = w(1, t) = 0, where p(q, t) The expression w(q, t) = t 2 q(1 − q) is the exact solution.Table 8 contains the approximate results and the absolute errors for Example 2 at distinct spatial grid values.Table 9 describes the analysis of the convergence order and the error norms.For various time stages, the error norms L ∞ and L 2 are expressed in Table 10.Table 11 expounds the error norms at distinct values of ∆t. Figure 4 illustrates strong agreement between the approximative and analytical outcomes for different times.The 3D precision of the existing technique is shown by graphs of the analytic results and the numerical solutions in Figure 5.The graphs of the errors in 2D and 3D forms are displayed in Figure 6.The figures and tables depict that the proposed numerical results are compatible with the exact solutions.The piecewise CBS numerical solution for µ = 1.7,N = 20, ∆t = 0.001 and t = 1 for Example 2 is given in Equation (42).

Conclusions
This work dealt with the problem of determining numerical solutions for the time FPDE.A collocation approach using B-splines for the TFDWE was established for this purpose.This approach used the standard FDM to estimate the time fractional derivative, while the CuBS was utilized to discretize the space derivative.In order to evaluate the numerical solutions of the TFDWE combining damping and reaction components, an efficient numerical scheme was provided.Further, we employed the θ-weighted technique with ABFD.The suggested technique is stable unconditionally and exhibits second-order temporal and spatial convergence.Three numerical examples were investigated.The presented method is precise and computationally very efficient according to the numerical and graphical comparisons.In the future, we might think about using spline functions to solve higher-dimensional and variable-order FPDEs.

Table 3 .
Error norms for different time stages for ∆t = 0.01 and µ = 1.9 for Example 1.

Table 5 .
Error norm comparison for distinct values of h and ∆t for Example 1 for t = 0.2 and µ = 1.5.

Table 6 .
Error norm comparison for distinct values of h and ∆t for Example 1 for t = 0.4 and µ = 1.7.

Table 7 .
Error norms for distinct values of ∆t for Example 1 for µ = 1.5 and t = 1.

Table 9 .
Error norms for Example 2 at various h values when t = 1 and µ = 1.5.

Table 12 .
Absolute error for Example 3 for distinct choices of µ for ∆t = 0.01 and N = 400 at t = 1.

Table 15 .
Error norms for Example 3 for distinct values of h when t = 1 and µ = 1.7.

Table 17 .
Error norms for Example 3 for distinct values of ∆t for t = 1 and µ = 1.3.