Abstract
Non-local fractional derivatives are generally more effective in mimicking real-world phenomena and offer more precise representations of physical entities, such as the oscillation of earthquakes and the behavior of polymers. This study aims to solve the 2D fractional-order diffusion-wave equation using the Riemann–Liouville time-fractional derivative. The fractional-order diffusion-wave equation is solved using the modified implicit approach based on the Riemann–Liouville integral sense. The theoretical analysis is investigated for the suggested scheme, such as stability, consistency, and convergence, by using Fourier series analysis. The scheme is shown to be unconditionally stable, and the approximate solution is consistent and convergent to the exact result. A numerical example is provided to demonstrate that the technique is more workable and feasible.
1 Introduction
Fractional calculus is the non-integer or fractional-order differentiation and integration that has received significant attention over the last two decades for its application in describing real-world problems. Non-linear fractional-order differential equations (NL-FDEs) play a major role in applied sciences and social sciences, such as material science, signal processing, control theory, finance, and food supplements [1–3]. Recently, many advanced and reliable approaches have been discussed by the researchers. For example, Shen et al. [4] discussed the analytical and numerical solutions for a 2D multi-term time-fractional DWE. They used the approach of the variable of separation to drive analytical solutions, as well as the properties of Mittag–Leffler functions, and showed that stability and convergence analyses were studied. Ruzhansky et al. [5] considered the multi-term DWE and used the Caputo derivative, a non-local initial problem, and the Mittag–Leffler function in this study. Fan et al. [6] investigated the inverse problem to recognize the initial value for the space–time FDWE. They utilized the Landweber iterative regularization method for the time–space FDWE and calculated the error between validity and stability. The approximate solutions for 1D and 2D multi-term time-fractional sub-diffusion and DWEs with smooth and non-smooth solutions have been discussed by Rashidinia et al. [7]. They applied the Legendre collocation method and the Caputo derivative for solving the proposed equation. Moreover, they found the convergence analysis and concluded that this method has the benefit of using limited Legendre polynomials to obtain accurate and acceptable results. Feng et al. [8] researched a novel approach based on the finite element method (FEM) 2D diffusion type non-integer order equation. The utilized scheme is approximated for three different types of equations. The matrix form of the scheme is generated using the FEM, and the formulation of the equation is determined. The theoretical analysis, such as stability and convergence, is established. The suggested technique is versatile and resilient and may be used to solve various multi-term time-fractional diffusion problems directly. The Caputo derivative is in the temporal direction of the 2D time-dependent FDWE, as studied by Yang et al [9]. They used the finite difference method (FDM) to discretize the fractional-order derivative and provide a meshless approximation in spatial directions using moving least squares (MLS), which may be used to solve more complicated problem domains. The study also investigated the time-related convergence and stability properties of the semi-discretized scheme. In another study, Yang et al. [9] considered a 2D non-linear time-fractional DWE solved using a Crank–Nicolson Legendre spectral technique. The time-stepping is performed using the Crank–Nicolson difference method, while the spatial discretization is performed using the Legendre spectral approach. Lyu et al. [10] suggested a fast and linearized FDM for solving the non-linear multi-term FDWE. The suggested technique is based on a weighted approach, fast discretization, and multi-term -type discretization. The obtained truncation error in the suggested weighted discretization to the multi-term Caputo derivative rigorously proved unconditional convergence by demonstrating certain crucial aspects of the refined coefficients of full discretization. Salehi [11] investigated multi-term FDWS in 2D and solved it using a meshless collocation approach. The shape functions for spatial approximation are constructed using the moving least squares reproducing kernel particle approximation. They also created a semi-discrete technique by discretizing Caputo’s time derivatives using a finite difference approximation. The obtained difference schemes demonstrated unconditional stability and convergence. Ghafoor et al. [12] developed a technique based on the approximations of finite difference and Haar wavelets. The approach is used to numerically solve -D and -D time fractional PDEs. Zhuang and Liua [13] worked on a finite domain and the 2D temporal fractional diffusion equation (2D-TFDE). The 2D-TFDE is modeled using an implicit difference approximation. The mathematical induction approach is used to assess the stability and convergence analysis. Heydari et al. [14] investigated the approximate solution to the variable-order (VO) space–time fractional non-linear diffusion-wave problem. The established approach combines the collocation and tau methods with the Chebyshev cardinal functions and their operational matrix of VO-FDs. A resilient and reliable numerical approach for solving a 2D VO non-linear FDWE on arbitrary domains was suggested by Shekari et al. [15] using the MLS meshless procedure for the space domain and the FDS for the time domain . The method was devised in such a way that it is independent of the uniformity of the domain in consideration and the solution of the algebraic system of equations. The proposed method was tested using a variety of space domains, including quadrilateral, rounded, triangular, and polar domains, as well as different types of non-linearity. Kumar et al. [16]provided the local collocation method that depends on radial basis functions to investigate the solution of time-fractional non-linear DWEs. They demonstrated the numerical schemes that are unconditionally stable and converge in semi-discrete. Ding [17] researched the creation of a high-order numerical approach for the 2D time-space FDWEs. A new approach with order is derived based on the fourth-order fractional-compact difference operator, where the temporal step size\tau and and are the spatial step sizes, respectively. The energy approach is used to analyze the algorithm’s stability and convergence, and a numerical experiment is conducted to confirm the numerical algorithm’s viability. Li et al. [18] analyzed a 2D non-linear FDWE in both time and space. The spatial component is discretized using the Galerkin FEM, while the temporal part is discretized using the new ADI method,” which also proved stability and convergence. They presented a 2D FDWE with the fractional derivative of order . Li et. al. [19] considered the ADI analysis based on the Crank–Nicolson method and the Galerkin FEM, both of which are analyzed. The ADI scheme is unconditionally stable, and norm convergence is rigorously illustrated. Datsko et al. [20] proposed that the TFDE with mass absorption in a sphere is observed under the harmonic impact on the surface of a sphere. The TFD of “Caputo” is implemented. The Mittag–Leffler function is also used to express the Laplace transform with respect to time and the finite sin-Fourier transform with respect to the spatial coordinates. Ren and Sun [21] discussed the fourth-order compact algorithm for solving the TFDWE with Neumann boundary conditions. For the time-fractional derivative, discretization is used, whereas the compact difference methodology is used for spatial discretization. The compact difference scheme is unconditionally stable, and global convergence is carefully shown. In addition, the Crank–Nicolson scheme with second-order spatial precision is described and perhaps an error estimate as well. The 2D case is solved using the compact ADI difference algorithm with Robin boundary conditions. Yang et al. [22] considered the initial value problem with mixed initial conditions of TFDWEs. They used the truncated regularization method and investigated conditional stability and error approximations, as well as some acceptable numerical examples, to establish the method’s validity. The convergence estimates in this publication are not saturated when compared with other authors’ works, and the non-homogeneous factors are correlated. Ali and Abdullah [23] discussed the FDWEs of arbitrary order.” They explain efficiency and fruitfulness through numerical test examples. There are many related studies available in the literature discussing different types of fractional-order models. For instance, Nawaz et al. [24] considered two numerical approaches, found the solution for two fractional-order models, and compared their solution with classical solutions. The fractional-order operator is in the Caputo sense, and we obtained the results numerically and graphically. They claimed that the methods are quickly convergent and yield encouraging results. Farid et al. [25] examined the Laplace transform with an iterative method for the space–time fractional-order models. They compared the obtained results with other existing schemes in the literature, which are more feasible and effective. Sayevand and Jafari [26] introduced the fractional-order KdV model in a fractal domain and applied the transformation to convert the fractional order into an ordinary-order derivative. They analyzed the theoretical analyses and presented many numerical examples to confirm the accuracy of the suggested approach. Li et al. [27] studied the significant properties of the Caputo fractional derivative in the real line and further developed it in the complex plane, which is used in signal processing. Guariglia [28] derived the functional equation for the fractional-order derivative Hurwitz function and proved the relation between the fractional-order zeta function and Bernoulli numbers. In another study, Guariglia [29] presented the functional equation with the Grunwald–Letnikov fractional derivative and discussed the link with the distribution of prime numbers. The other comprehensive literature can be studied in [30–37].
According to the previously described literature, fractional calculus is still a relatively new field that requires more accurate numerical approaches to examine the more practicable FDEs. The goal of this research is to develop a more accurate and reliable numerical technique for the FDWE. An attempt to discretize the R-L integral operator numerically and implement it in the Riemann–Liouville fractional-order derivative to approximate the FDWE has not yet been made. This approach reduces computational complexity and increases accuracy. Analysis, including aspects such as stability, consistency, and convergence, was also investigated based on the Fourier method and Taylor’s series expansion.
The remainder of the paper is arranged as follows: the associated preliminaries are described in Section 2. Section 3 explains the proposed method. The stability analysis is discussed in Section 4. Consistency and convergence are addressed in Section 5. Section 6 presents the numerical results, and the conclusion is provided in Section 7.
Here, we consider the FDWE as follows [23]:where the conditions arewhere represents the non-homogenous term and denotes the R-L fractional-order derivative operator of order lying between 0 and 1.
2 Preliminaries
The R-L fractional differential operator is the most important extension of the classical differential operator. The R-L fractional-order derivative is defined as follows [38]:
The R-L integral operator can be defined as follows:where represents the R-L fractional-order integral operator of order lying between 0 and 1. Equation 5 can also be written as
We use the Jumarie property in [25] as
By discretization of the aforementioned equation, we obtain the following equations:
Applying the Jumarie property ,where is the time step and can be defined as and , where .
Lemma 1The -order R-L fractional integral of in is defined as
Lemma 2The coefficient satisfies the following properties [39]:
3 The proposed implicit difference scheme
In this section, the implicit difference scheme (IDS) for the FDWE is constructed using Lemma 1 for the fractional-order part, and the space-derivative is reduced to the central difference approximation. The step for space is and , and the step for time is , where 1 , respectively. We substitute Eq. 2 in Eq. 1 at mesh point as follows:
To eliminate the second-order time derivatives, we applied the backward difference approximation, then integrated from to , and also, used the trapezoidal rule for the forcing term. Finally, we obtained
We substituted Lemma 1, and after simplification, we obtained the approximated scheme for the FDWE as follows:
We know that . Therefore,where
4 Stability
The von Neumann method is used to determine the stability of the proposed scheme, and the approach in [40]is followed. Let represent the exact solution for Eq. 12, and we obtain
The error is defined as , and satisfies Eq. 15 as follows:
We assume that the growth factor takes the form of a single Fourier mode aswhere and and are the mode number and step sizes, respectively. Eq. 17 can be the solution of the aforementioned error in Eq. 16.
Dividing both sides by and then replacing , we obtainwhere
Proposition 1Suppose , where , is the solution of Eq. 19; then, we need to prove thatProof. Let us consider in Eq. 19 to prove the proposition using the induction method as follows:We get by substituting Eqs 13, 17 into Eq. 21. The provided equation becomes easier after simplification.We obtain the following relation:Suppose holds true for .Using Eqs 19–, 23 and Lemma 2, we haveFrom Lemma 2, the value is , so it is clear thatHere, and , so it is written as . It reveals that the proposed method is unconditionally stable.
5 Consistency
Here, following the approach in [41], we assume that is the closed-form solution, W is the estimated solution, and the function is the approximated scheme for the proposed equation at the mesh point to discover the consistency analysis. The local truncation error at was subsequently indicated by
The local truncation error for the suggested scheme is .
Using the Taylor series, we obtain
After the cancelation of the same terms with opposite signs, we obtain
Replacing the values of and in the aforementioned equation, we obtain
From the aforementioned equation, it is implied that if the time and space steps approach zero, we obtain the following simplified form:
The obtained order of approximation is as follows:
The IDS of the FDWE is consistent if , and ; subsequently, the local truncation error approaches zero.
According to Lax equivalence theorem, if the method is consistent and stable, then it is convergent [42]. Hence, it is proven that the proposed scheme is convergent.
6 Numerical results
This section considers time fractional-order DWE examples to determine the exactness and viability of the technique. The numerical example is coded in Maple 15, and the maximum error is as follows:
The norms error is
Example 1. In Eqs 1–3, we consider the source term and the closed solution .
Example 2. We consider the two-dimensional fractional-order Rayleigh–Stokes problem, which is provided as follows [23]:where the forcing term is
Therefore, the closed-form solution is
7 Discussion
The 2D fractional-order DWE is solved by the modified implicit numerical scheme. The formulated scheme is established by the Riemann–Liouville fractional integral operator that is mentioned in Lemma 1. The discretized Riemann–Liouville fractional integral operator is used with the implicit scheme, which is very easy to implement and find in the theoretical analysis. The numerical results are provided in the form of tables for various values of space steps, time steps, and different fractional orders. As Table 1 is plotted for various values of step sizes and varying the values of fractional order, the error is reduced when increasing the number of step sizes for various values of fractional order. In Table 2, the value of fractional order is fixed and the step sizes of space and time are varied, which shows that the error is reducing. Table 3 confirmed the feasibility and agreement with theoretical analysis by the rate of convergence for the fixed value of of the proposed formulated scheme. The graphical representation in Figures 1, 2 also shows that the approximate solution has excellent performance compared to the exact solution. For more confirmation and to check the accuracy of the proposed scheme, we solved the 2D fractional-order Rayleigh–Stokes problem mentioned in example 2 and compared the numerical values with the high-order approximated scheme in [43] in Table 4, which shows better accuracy. Although the proposed scheme is not of high order, Figure 3 represents the graphical solution for example 2, which also shows that the obtained solution is more accurate and feasible.
TABLE 1
| 1\4 | 1\2 | 1.588 E-03 | 3.338 E-03 | 4.815 E-03 | 6.271 E-03 |
| 1\16 | 1\4 | 1.050 E-03 | 1.734 E-03 | 2.370 E-03 | 2.143 E-03 |
| 1\64 | 1\8 | 4.620 E-04 | 6.780 E-04 | 9.03 E-04 | 1.144 E-03 |
| 1\100 | 1\10 | 3.431 E-04 | 4.910 E-04 | 6.522 E-04 | 8.240 E-04 |
Numerical results of the proposed scheme for various values of fractional order space steps , and time step sizes at T = 1.0.
TABLE 2
| 20 | 1.927 E-03 | 9.539 E-04 | 7.670 E-04 | 7.014 E-04 |
| 40 | 1.746 E-03 | 1.606 E-03 | 5.798 E-04 | 4.302 E-04 |
| 60 | 1.650 E-03 | 6.685 E-04 | 5.453 E-04 | 3.295 E-04 |
Numerical results of the proposed scheme for various values of and a fixed value of
TABLE 3
| order | ||
|---|---|---|
| 9.2878 E-03 | - | |
| 1.7302 E-03 | 2.425 | |
| 5.3546 E-03 | - | |
| 9.8240 E-04 | 2.446 |
Error and order of convergence of the IDS for the example for different values of and a fixed value of
FIGURE 1
Graphical representation between the exact and approximated solutions at .
FIGURE 2
Graphical representation between the exact and approximated solutions at
TABLE 4
| For | ||||
|---|---|---|---|---|
| Test problem 1 [34] | Proposed scheme | Test problem 1 [34] | Proposed scheme | |
| 1/10 | 6.9182E-03 | 6.3169 E-03 | 5.5331 E-03 | 5.2677 E-03 |
| 1/20 | 3.5625E-03 | 3.0504 E-03 | 2.9268 E-03 | 2.4844 E-03 |
| 1/40 | 1.8210E-03 | 1.3302 E-03 | 1.5059 E-03 | 1.1844 E-03 |
| 1/80 | 9.3190E-04 | 7.7013 E-04 | 7.5969 E-04 | 6.8401 E-04 |
Comparison between the numerical results of the proposed scheme with the previous study for various values of fractional order , space steps , and time step sizes at T = 1.0.
FIGURE 3
Graphical representation between the exact and approximated solutions at
8 Conclusion
A practical and quick numerical approach was designed for the FDWE. The discretization of the Riemann–Liouville integral, as described in Lemma 1, serves as the basis for the approximation. Through employing mathematical induction and demonstrating consistency and convergence, we successfully showed the theoretical analysis of stability, consistency, and convergence. The numerical results corroborated our theoretical findings and demonstrated that the suggested method is fast, convergent, and viable. This method can also be extended to different types of models arising in the realm of mathematical physics.
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary material; further inquiries can be directed to the corresponding authors.
Author contributions
All authors listed made a substantial, direct, and intellectual contribution to the work and approved it for publication.
Funding
The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work under grant code 22UQU4310396DSR68.
Acknowledgments
The authors would like to thank the referee for the valuable comments.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
Nomenclature
| FDWE | Fractional diffusion-wave equation |
| RL | Riemann–Liouville |
| NL-FDEs | Non-local fractional-order differential equations |
| 2D | Two dimensional |
| FEM | Finite element method |
| FDM | Finite difference method |
| MLS | Moving least squares |
| TFDE | Time-fractional diffusion equation |
| PDEs | Partial differential equations |
| FO | Fractional order |
| ADI | Alternating direct implicit |
| IDS | Implicit difference scheme |
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Summary
Keywords
fractional-order diffusion-wave equation, implicit scheme, Riemann–Liouville fractional integral operator, stability, consistency, convergence
Citation
Rafaqat K, Naeem M, Akgül A, Hassan AM, Abdullah FA and Ali U (2023) Analysis and numerical approximation of the fractional-order two-dimensional diffusion-wave equation. Front. Phys. 11:1199665. doi: 10.3389/fphy.2023.1199665
Received
03 April 2023
Accepted
21 August 2023
Published
21 September 2023
Volume
11 - 2023
Edited by
Emanuel Guariglia, São Paulo State University, Brazil
Reviewed by
Yusif Gasimov, Azerbaijan University, Azerbaijan
Ebenezer Bonyah, University of Education, Winneba, Ghana
Tamilvanan Kandhasamy, Kalasalingam University, India
Updates
Copyright
© 2023 Rafaqat, Naeem, Akgül, Hassan, Abdullah and Ali.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Farah Aini Abdullah, farahaini@usm.my; Umair Ali, umairkhanmath@gmail.com
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.