Abstract
Fractional differential equations describe nature adequately because of the symmetry properties that describe physical and biological processes. In this paper, a new approximation is found for the variable-order (VO) Riemann–Liouville fractional derivative (RLFD) operator; on that basis, an efficient numerical approach is formulated for VO time-fractional modified subdiffusion equations (TFMSDE). Complete theoretical analysis is performed, such as stability by the Fourier series, consistency, and convergence, and the feasibility of the proposed approach is also discussed. A numerical example illustrates that the proposed scheme demonstrates high accuracy, and that the obtained results are more feasible and accurate.
1. Introduction
VO-FDEs are generalizations of constant-order fractional differential equations (FDEs). Sometime, constant-order FDEs cannot describe the complex processes in porous media and medium structures because of changes with time []. Samko et al. first discussed the concept of VO integral and differential operators of the RLFD-type formula and investigated their mathematical properties []. Lorenzo and Hartley [] discussed the basic concept of variable- and distributed-order fractional operators. They introduced definitions based on the Riemann–Liouville definition and discussed the behavior of new operators. Many researchers have worked on VO-FDEs with various numerical methods. For example, Younes [] considered the moving least-squares method and finite difference method to find the solution of 2D VO fractional diffusion-wave equations. They solved some nonlinear numerical examples with complex geometries and the obtained results confirmed the accuracy and efficiency of the proposed method. A numerical scheme was proposed for 1D VO anomalous subdiffusion equations by Chen et al. []. They used fourth-order approximation for spatial and first-order temporal approximation and also proposed an improved numerical scheme for better accuracy. Xu et al. [] developed a numerical scheme for multiterm FDE of VO, and reported a stability and convergence investigation through mathematical induction. The tested examples showed a strong level of accuracy. Wang et al. [] formulated the numerical inversion of inverse problems with the homotopy regularization method by utilizing Legendre polynomials. Bhrawy and Zaky [] formulated the Jacobi spectral collocation method on the basis of Legendre polynomials, and the Cahebyshev method for VO fractional 1D and 2D Cable equations. The numerical solution demonstrated a powerful method with a high level of accuracy for VO-FDE. The two new approximations were derived for the VO time derivative from the first to the second order by Zhao et al. []. They utilized superdiffusion and subdiffusion problems to demonstrate the effectiveness of the proposed approximation. Shen et al. [] used the Caputo definition for the diffusion equation of VO and developed a numerical scheme, discussing the theoretical analysis via the von Neumann method. Sun et al. [] presented a recent survey on VO-FDEs. They discussed initial existing definitions, numerical methods, and a summary on physical models and their applications. The Chebbyshev cardinal functional was constructed for VO delay factional models by Avazzaddeh et al. []. They obtained an algebraic system of equations with an operational matrix that reduced the cost of computation. Zayernouri and Karniadakis [] proposed a spectral collocation method for linear and nonlinear VO space and time FDEs. They obtained differentiation matrices by collocating the fractional VO, and successfully solved many problems, such as time and space fractional advection, advection–diffusion, and Burger’s equations to confirm the efficiency of the proposed method. Ali [] investigated the new approximation for fractional Riemann–Liouville derivatives and also successfully applied the same approximation to VO-FDEs. They analyzed the theoretical analysis of the Fourier series method and showed that the proposed schemes were unconditionally stable and convergent. Ma et al. [] considered the system of equations of VO and solved by Adams–Bashforth–Moulton algorithm. The numerical results reported that the suggested method was powerful and efficient. VOs and with variable coefficient FDEs were solved by Katsikadelis []. They contributed both implicit and explicit numerical approaches for linear and nonlinear VO-FDEs. Akgül et al. [] successfully discussed the reproducing kernel method for VO-FDEs, which is more efficient and workable. In another study, Jia et al. [] studied the simplified reproducing kernel technique for VO-FDE. They proved the theoretical analysis for the proposed scheme. Dehghan [] worked on a numerical method on the basis of shifted Legendre polynomials for VO-FDEs, and reduced the equation into a system of algebraic equations by using Legendre polynomials. They compared the numerical method with two existing methods, and the results showed a great level of performance. Chen [] solved a two-dimensional VO-TFMSDE for the first time. They used the Grünwald-Letnikov formula for VO time-fractional derivatives and central difference approximation for second-order space derivatives, and developed an implicit scheme. They found stability, convergence, and solvability via the Fourier series method. A numerical method for improving temporal accuracy was also developed for the proposed scheme. Ali et al. [] derived an approximation for a VO fractional integral operator and solved the diffusion equation of VO. They investigated the theoretical analysis and compared the numerical results with an existing method that had shown better accuracy. Hijaz et al. [] formulated a novel numerical approach for noninteger-order nonlinear differential equations. The noninteger order derivative was in a Caputo sense, and the numerical results were powerful and suitable for differential equations of noninteger order. The explicit difference scheme was considered for VO fractional heat equations with a linear forcing term by Sweilam and Mrawm []. They discretized the VO Caputo fractional derivative and discussed the stability by means of the Gerschgorin theorem. The results confirmed the effectiveness of the developed scheme via some VO heat equation models. Babaei et al. [] developed the Chebyshev collocation technique of the sixth kind for the solution of nonlinear VO fractional integrodifferential equations. They reduced the VO fractional equation into a system of algebraic equations, and established the convergence and rate of convergence for the proposed approach. They discussed the robustness of the method and found it to be faster and more accurate. Wang and Zheng [] proved the well-posedness and singularity of the solutions for VO FDEs. They used the FDS and discussed different cases of singularity of VO at and the theoretical results. Kaur et al. [] formulated a new approach and solved the fractional-order advection problem, the fractional derivative in the sense of a conformable time sense. The solution was obtained with the differential transform method, and the obtained results were compared with the existing literature. Abbas [] considered the eigenvalue approach and found the exact solution in the Laplacian domain without any restriction. They discussed the effect of all parameters by increasing the value of , decreasing the values of all studied fields. Alzahrani et al. [] used Laplacian Fourier transformation with the eigenvalue method for 2D porous material and the obtained outcomes for different types of conductivity. Further related studies can be found in [,,,,,,].
The above cited literature shows that more efficient and better numerical schemes are needed to investigate VO-FDS. In this paper, our aim is to formulate a new numerical approach for VO-RLFD operators that is approximated through Jumrie properties, and the partial derivative with respect to time is replaced by backward difference approximation. The obtained approximation of the fractional derivative is used, and the space derivatives are approximated with the finite difference method for VO-TFMSDE. Theoretical analysis from the aspects of stability, consistency, and convergence is discussed, and a numerical experiment for 2D RSP-HGSGF with a fractional derivative is presented for further confirmation.
Consider the 2D VO-TFMSDE []:
The initial and boundary conditions are:
where and are defined values, and are VO fractional partial derivatives with respect to time, and u represents the quantity of concentration function.
The remaining paper is organized as follows: Section 2, briefly discusses the preliminaries. Section 3 explains the implicit difference scheme and, Section 3.1 and Section 3.2 provide the complete theoretical analysis of stability, consistency, and convergence. In Section 4, the numerical experiments are presented. Lastly, the conclusion is discussed in Section 5.
2. Preliminaries
In VO fractional theory, many definitions have been developed. Here, we introduce the VO-RLFD property and its approximations.
Definition 1.
The VO-RLFD of order , for function can be written as follows:
For the VO-RLFD, we have an important property, :
Definition 2.
The VO Riemann–Liouville fractional integral operator can be defined as follows:
To formulate a new approximation for VO-RLFD at grid point :
and
Applying the Jumarie property from [] to Equation (7) above, we obtain
Here, using the following Jumarie property in the above equation:
and
Lemma 1.
The -order RLFD of function on can be defined in discretized form as follows:
Lemma 2.
Coefficients satisfy the following properties []:
- (i)
- (ii)
- (iii)
- There exists a positive constantsuch that
- (vi)
3. Implicit Difference Scheme
To construct the IDS for VO-TFMSDE, we used Lemma 1 for the VO fractional derivative part and the difference approximation for the space derivatives. We obtain
From Equation (10), the IDS for VO-TFMSDE (1), as
with
Here,
and
3.1. Stability Analysis
Using the Fourier series to find the stability of the VO-TFMSDE following similar analysis in [], let represent the exact solution for Equation (11). We have
The error can be defined as follows:
Error satisfies (13). We have
In addition,
Here, following the same approach as that in [,], suppose that
where , ; putting Equation (16) in Equation (15) and dividing both sides by and then put , and , we have
where
Proposition 1.
If Equation (17) is satisfied. Then, it must be proven that
Proof.
3.2. Consistency
The consistency of 2D VO-TFMSDE is investigated in this section. Let the approximate and exact solutions be represented by w and W, respectively. If the approximated solution of TFMSDE is , then the local truncation error is at point .
Theorem 1.
The truncation error of the finite difference scheme is:
Proof.
With the Taylor series expansion, we can write
After simplification, we obtain
Hence,
This theorem shows that this method is consistent because , , . □
Theorem 2.
((Lax equivalence theorem (see [,])).If the method is consistent and stable, then it is convergent.
4. Numerical Experiments
In this section, we solve a numerical example as a 2D VO-TFMSDE for different values of VO to check the feasibility of the proposed IDS. The error between the numerical and exact solutions follows the references in equations,
i.e., the error formula is as follows:
Example 1.
Consider the following 2D VO-TFMSDE [].
where and the closed form solution is .
Example 2.
Consider the two-dimensional RSP-HGSGF with fractional derivative
where
The exact solution is given by
In the present study, the IDS was developed on the basis of a new approximation for RLFD operator that was derived in Lemma 1 for 2D VO-TFMSDE. The numerical results were compared with the exact solution and with the results in []. Chen [] numerically solved the VO-TFMSDE and treated the VO fractional derivative by Grünwald-Letnikov. In Table 1, the proposed IDS shows better accuracy at various values of space and time steps and derivative of VO , and Table 2 shows the accuracy with the exact solution. For more confirmation, Table 3 shows the even better accuracy of the suggested scheme for two-dimensional RSP-HGSGF with fractional derivatives. Figure 1 and Figure 2 represent the exact and approximate solutions, respectively, at , which were more similar, and the approximated solution converged to the exact solution. Figure 3 and Figure 4 represent the comparison of IDS with the exact solution, which showed high accuracy at and respectively.

Table 1.
Comparison of IDS (11) at , and for different values of , .

Table 2.
Numerical results of IDS (11) at and for different values of , .

Figure 1.
Exact solution.

Figure 2.
Approximate solution.

Figure 3.
Comparison of the IDS and the exact solution of Equation (23) at , and .

Figure 4.
Comparison of the IDS and the exact solution of Equation (23) at , and .
The comparison plot of the exact and approximate solutions of Equation (23) at , , and .
5. Conclusions
In this study, we formulated the numerical approximation of a VO-RLFD operator and developed a new IDS. The proposed scheme was successfully applied to a 2D VO-TFMSDE and RSP-HGSGF with a fractional derivative. Theoretical analysis regarding stability on a Fourier approach, consistency, and convergence was studied for 2D VO-TFMSDE. The numerical results are reported in the form of error tables, and 3D and 2D plots. Moreover, the solution was tested with a numerical example, and the theoretical analysis was confirmed with the numerical results. The proposed approach is highly effective and efficient for VO-FDEs. This scheme can also be extended to other types of VO fractional differential equations.
Author Contributions
Conceptualization, U.A.; Methodology, D.F. and U.A.; Software, D.F. and U.A.; Validation, A.H.G.; Formal analysis, M.N.; Investigation, M.N.; Resources, A.H.G.; Writing—original draft, A.H.G.; Writing—review & editing, D.F. and F.A.A.; Visualization, M.N.; Supervision, U.A. and F.A.A.; Project administration, D.F. and F.A.A. All authors have read and agreed to the published version of the manuscript.
Funding
The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work under grant code: 22UQU4310396DSR35.
Data Availability Statement
All the required data are cited in the manuscript.
Acknowledgments
The authors are thankful to the reviewers for the useful suggestions to improve the quality of this manuscript. The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work under grant code: 22UQU4310396DSR35.
Conflicts of Interest
The authors declare no conflict of interest.
References
- Cao, J.; Qiu, Y.; Song, G. A compact fnite difference scheme for variable order subdiffusion equation. Commun. Nonlinear Sci. Numer. Simul. 2017, 48, 140–149. [Google Scholar] [CrossRef]
- Sun, H.; Chen, W.; Chen, Y. Variable-order fractional differential operators in anomalous diffusion modeling. Phys. A Stat. Mech. Its Appl. 2009, 388, 4586–4592. [Google Scholar] [CrossRef]
- Lorenzo, C.F.; Hartley, T.T. Variable order and distributed order fractional operators. Nonlinear Dyn. 2002, 29, 57–98. [Google Scholar] [CrossRef]
- Shekari, Y.; Tayebi, A.; Heydari, M.H. A meshfree approach for solving 2D variable-order fractional nonlinear diffusion-wave equation. Comput. Methods Appl. Mech. Eng. 2019, 350, 154–168. [Google Scholar] [CrossRef]
- Chen, C.M.; Liu, F.; Anh, V.; Turner, I. Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation. SIAM J. Sci. Comput. 2010, 32, 1740–1760. [Google Scholar] [CrossRef]
- Xu, T.; Lü, S.; Chen, W.; Chen, H. Finite difference scheme for multi-term variable-order fractional diffusion equation. Adv. Differ. Equ. 2018, 2018, 103. [Google Scholar] [CrossRef]
- Wang, S.; Wang, Z.; Li, G.; Wang, Y. A simultaneous inversion problem for the variable-order time fractional differential equation with variable coefficient. Math. Probl. Eng. 2019, 2019, 13. [Google Scholar] [CrossRef]
- Bhrawy, A.H.; Zaky, M.A. Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation. Nonlinear Dyn. 2015, 80, 101–116. [Google Scholar] [CrossRef]
- Zhao, X.; Sun, Z.Z.; Karniadakis, G.E. Second-order approximations for variable order fractional derivatives: Algorithms and applications. J. Comput. Phys. 2015, 293, 184–200. [Google Scholar] [CrossRef]
- Shen, S.; Liu, F.; Chen, J.; Turner, I.; Anh, V. Numerical techniques for the variable order time fractional diffusion equation. Appl. Math. Comput. 2012, 218, 10861–10870. [Google Scholar] [CrossRef]
- Sun, H.; Chang, A.; Zhang, Y.; Chen, W. A review on variable-order fractional differential equations: Mathematical foundations, physical models, numerical methods and applications. Fract. Calc. Appl. Anal. 2019, 22, 27–59. [Google Scholar] [CrossRef]
- Avazzadeh, Z.; Heydari, M.H.; Mahmoudi, M.R. An approximate approach for the generalized variable-order fractional pantograph equation. Alex. Eng. J. 2020, 59, 2347–2354. [Google Scholar] [CrossRef]
- Zayernouri, M.; Karniadakis, G.E. Fractional spectral collocation methods for linear and nonlinear variable order FPDEs. J. Comput. Phys. 2015, 293, 312–338. [Google Scholar] [CrossRef]
- Ali, U. Numerical Solutions for Two Dimensional Time-Fractional Differential Sub-Diffusion Equation. Ph.D. Thesis, University Sains Malaysia, Gelugor, Malaysia, 2019; pp. 1–200. [Google Scholar]
- Ma, S.; Xu, Y.; Yue, W. Numerical solutions of a variable-order fractional financial system. J. Appl. Math. 2012, 2012, 417942. [Google Scholar] [CrossRef]
- Katsikadelis, J.T. Numerical solution of variable order fractional differential equations. arXiv 2018, arXiv:1802.00519. [Google Scholar]
- Akgül, A.; Baleanu, D. On solutions of variable-order fractional differential equations. Int. J. Optim. Control Theor. Appl. (IJOCTA) 2017, 7, 112–116. [Google Scholar] [CrossRef]
- Jia, Y.T.; Xu, M.Q.; Lin, Y.Z. A numerical solution for variable order fractional functional differential equation. Appl. Math. Lett. 2017, 64, 125–130. [Google Scholar] [CrossRef]
- Dehghan, R. A numerical solution of variable order fractional functional differential equation based on the shifted Legendre polynomials. SeMA J. 2019, 76, 217–226. [Google Scholar] [CrossRef]
- Chen, C.M. Numerical methods for solving a two-dimensional variable-order modified diffusion equation. Appl. Math. Comput. 2013, 225, 62–78. [Google Scholar] [CrossRef]
- Ali, U.; Sohail, M.; Abdullah, F.A. An Efficient Numerical Scheme for Variable-Order Fractional Sub-diffusion Equation. Symmetry 2020, 12, 1437. [Google Scholar] [CrossRef]
- Ahmad, H.; Akgül, A.; Khan, T.A.; Stanimirović, P.S.; Chu, Y.M. New Perspective on the Conventional Solutions of the Nonlinear Time-Fractional Partial Differential Equations. Complexity 2020, 2020, 8829017. [Google Scholar] [CrossRef]
- Sweilam, N.H.; AL-Mrawm, H.M. On the numerical solutions of the variable order fractional heat equation. Stud. Nonlinear Sci. 2011, 2, 31–36. [Google Scholar]
- Babaei, A.; Jafari, H.; Banihashemi, S. Numerical solution of variable order fractional nonlinear quadratic integro-differential equations based on the sixth-kind Chebyshev collocation method. J. Comput. Appl. Math. 2020, 377, 112908. [Google Scholar] [CrossRef]
- Wang, H.; Zheng, X. Analysis and numerical solution of a nonlinear variable-order fractional differential equation. Adv. Comput. Math. 2019, 45, 2647–2675. [Google Scholar] [CrossRef]
- Kaur, S.; Kumar, A.; Pathak, A.K. October. A new computational procedure for the solution of the time-fractional advection problem. AIP Conf. Proc. 2022, 2451, 020058. [Google Scholar]
- Abbas, I.A. Eigenvalue approach on fractional order theory of thermoelastic diffusion problem for an infinite elastic medium with a spherical cavity. Appl. Math. Model. 2015, 39, 6196–6206. [Google Scholar] [CrossRef]
- Alzahrani, F.; Hobiny, A.; Abbas, I.; Marin, M. An eigenvalues approach for a two-dimensional porous medium based upon weak, normal and strong thermal conductivities. Symmetry 2020, 12, 848. [Google Scholar] [CrossRef]
- Ahmad, I.; Ahmad, H.; Thounthong, P.; Chu, Y.M.; Cesarano, C. Solution of multi-term time-fractional PDE models arising in mathematical biology and physics by local meshless method. Symmetry 2020, 12, 1195. [Google Scholar] [CrossRef]
- Ali, U.; Abdullah, F.A. December. Modified implicit difference method for one-dimensional fractional wave equation. AIP Conf. Proc. 2019, 2184, 060021. [Google Scholar]
- Ahmad, I.; Khan, M.N.; Inc, M.; Ahmad, H.; Nisar, K.S. Numerical simulation of simulate an anomalous solute transport model via local meshless method. Alex. Eng. J. 2020, 59, 2827–2838. [Google Scholar] [CrossRef]
- Mahmood, A.; Md Basir, M.F.; Ali, U.; Mohd Kasihmuddin, M.S.; Mansor, M. Numerical Solutions of Heat Transfer for Magnetohydrodynamic Jeffery-Hamel Flow Using Spectral Homotopy Analysis Method. Processes 2019, 7, 626. [Google Scholar] [CrossRef]
- Ahmad, I.; Ahmad, H.; Abouelregal, A.E.; Thounthong, P.; Abdel-Aty, M. Numerical study of integer-order hyperbolic telegraph model arising in physical and related sciences. Eur. Phys. J. Plus 2020, 135, 759. [Google Scholar] [CrossRef]
- Srivastava, M.H.; Ahmad, H.; Ahmad, I.; Thounthong, P.; Khan, N.M. Numerical simulation of three-dimensional fractional-order convection-diffusion PDEs by a local meshless method. Therm. Sci. 2020, 25, 347–358. [Google Scholar] [CrossRef]
- Ali, U.; Abdullah, F.A.; Ismail, A.I. Crank-Nicolson finite difference method for two-dimensional fractional sub-diffusion equation. J. Interpolat. Approx. Sci. Comput. 2017, 2017, 18–29. [Google Scholar] [CrossRef]
- Ali, U.; Abdullah, F.A.; Mohyud-Din, S.T. Modified implicit fractional difference scheme for 2D modified anomalous fractional sub-diffusion equation. Adv. Differ. Equ. 2017, 2017, 185. [Google Scholar] [CrossRef]
- Jumarie, G. Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions further results. Comput. Math. Appl. 2006, 51, 1367–1376. [Google Scholar] [CrossRef]
- Ali, U.; Sohail, M.; Usman, M.; Abdullah, F.A.; Khan, I.; Nisar, K.S. Fourth-Order Difference Approximation for Time-Fractional Modified Sub-Diffusion Equation. Symmetry 2020, 12, 691. [Google Scholar] [CrossRef]
- Aslefallah, M.; Rostamy, D.; Hosseinkhani, K. Solving time-fractional differential diffusion equation by theta-method. Int. J. Appl. Math. Mech. 2014, 2, 1–8. [Google Scholar]
- Sanz-Serna, J.; Palencia, C. A general equivalence theorem in the theory of discretization methods. Math. Comput. 1985, 45, 143–152. [Google Scholar] [CrossRef]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).