Numerical Solutions of One-Dimensional Space-Fractional Diffusion Equation Using Least-Squares-Petrov-Galerkin Approach
DOI:
https://doi.org/10.56714/bjrs.51.2.6Keywords:
One-dimension space-fractional diffusion equation, Least-squares method, Petrov-Galerkin method, Caputo-Fabrizio fractional derivative, hebyshev polynomial, Laguerre polynomialAbstract
In this work, we proposed a novel method for solving one-dimensional space fractional diffusion equations (SFDE) based on combining the least-squares method with Petrov-Galerkin approach, utilizing orthogonal polynomials as basis functions, with the fractional derivative considered in the Caputo-Fabrizio sense. This method is to express the unknown function as a series of orthogonal polynomials that are linearly combined. By using this approach, we can turn the problem into a system of linear algebraic equations that can be solved using MATLAB R2023a for the unknown constants associated with the approximate solution. We provide two examples that illustrate the accuracy of our method and its ability to be applied effectively. The graphs and error tables support the proposed approach's effectiveness and efficiency. The results indicate that the proposed method yields more accurate solutions than others for solving similar problems.
Downloads
References
[1] M. Vellappandi, S., Lee, " Physics-informed neural fractional differential equations, " Appl. Mathe. Model., vol. 145, 116127, 2025. DOI: https://doi.org/10.1016/j.apm.2025.116127. DOI: https://doi.org/10.1016/j.apm.2025.116127
[2] Z. Afzal, M. Alshehri, " Fractional-order SIR model for ADHD as a neurobiological and genetic disorder, " Scient. Repor., vol. 15, no. 1, pp. 22992, 2025. DOI: https://doi.org/10.1038/s41598-025-07646-7. DOI: https://doi.org/10.1038/s41598-025-07646-7
[3] G. Huang, H. Y. Qin, Q. Chen, Z. Shi, S. Jiang, C. Huang, " Research on application of fractional calculus operator in image underlying processing, " Frac. Fracti., vol. 8, no. 1, pp. 37, 2024. DOI: https://doi.org/10.3390/fractalfract8010037. DOI: https://doi.org/10.3390/fractalfract8010037
[4] M. Dalir and M. Bashour, " Applications of fractional calculus," Appl. Math. Sci., vol. 4, no. 21, pp. 1021–1032, 2010.
[5] L. Zada, I. Aziz, " Numerical solution of fractional partial differential equations via Haar wavelet, " Numer. Methods Partial. Differ. Equ., vol. 38, no. 2, pp. 222-242, 2022. DOI: https://doi.org/10.1002/num.22658 DOI: https://doi.org/10.1002/num.22658
[6] H. Bazgir and B. Ghazanfari, " Spectral solution of fractional fourth-order partial integro-differential equations," Comput. Methods Differ. Equ., vol. 7, no. 2, pp. 289–301, 2019. DOI: 20.1001.1.23453982.2019.7.2.11.5.
[7] M. Aychluh, D. L. Suthar, S. D. Purohit, " Analysis of the nonlinear Fitzhugh-Nagumo equation and its derivative based on the Rabotnov fractional exponential function," Par. Differ. Equ. Appl. Math., vol. 11, pp. 100764, 2024, DOI: https://doi.org/10.1016/j.padiff.2024.100764. DOI: https://doi.org/10.1016/j.padiff.2024.100764
[8] M. Aychluh, M. Ayalew, D. L. Suthar, S. D. Purohit, " Numerical analysis of multi-dimensional Navier–Stokes equation based on Yang–Abdel–Cattani fractional operator," Int. J. Math. Ind., vol. 27, no. 2440001, pp. 1-11, 2024, DOI:10.1142/S266133522440001X DOI: https://doi.org/10.1142/S266133522440001X
[9] J. Liu, M. Nadeem, A. Islam, S. Mureşan, L. F. Iambor, " A modified residual power series method for the approximate solution of two-dimensional fractional Helmholtz equations, " Symm., vol. 15, no. 12, pp. 2152, 2023. DOI: https://doi.org/10.3390/sym15122152. DOI: https://doi.org/10.3390/sym15122152
[10] P. Roul and V. Rohil, " An accurate numerical method and its analysis for time-fractional Fisher's equation," Soft Comput., vol. 28, pp. 11495–11514, 2024. DOI: https://doi.org/10.1007/s00500-024-09885-8. DOI: https://doi.org/10.1007/s00500-024-09885-8
[11] H. O. Al-Humedi, F. A. Al-Saadawi, A. M. Abdulhussein, " Solving the Nonlinear Time-Fractional Zakharov-Kuznetsov Equation with the Chebyshev Spectral Method," Ira. J. Scie., vol. 65, no. 11, pp. 6589-6602, 2024. DOI: https://doi.org/10.24996/ijs.2024.65.11.33. DOI: https://doi.org/10.24996/ijs.2024.65.11.33
[12] M. Caputo and M. Fabrizio, " A new definition of fractional derivative without singular kernel," Progr. Fract. Differ. Appl., vol. 1, no. 2, pp. 73-85, 2015. DOI: 10.12785/pfda/010201.
[13] C. Li, D. Qian, Y. Chen, " On Riemann‐Liouville and Caputo derivatives," Disc. Dyn. Nature Soci., vol. 2011, no. 1, pp. 1-15, 2011. DOI: https://doi.org/10.1155/2011/562494. DOI: https://doi.org/10.1155/2011/562494
[14] R. Abu-Gdairi, S. Hasan, S. Al-Omari, M. Al-Smadi, S. Momani, " Attractive multistep reproducing kernel approach for solving stiffness differential systems of ordinary differential equations and some error analysis," Comput. Model. Eng. Sci., vol. 130, no. 1, pp. 129-313, 2022. DOI: 10.32604/cmes.2022.017010. DOI: https://doi.org/10.32604/cmes.2022.017010
[15] E. Abuteen, A. Freihat, M. Al-Smadi, H. Khalil, R. A. Khan, " Approximate series solution of nonlinear, fractional Klein-Gordon equations using fractional reduced differential transform method," J. of Math. and Statis., vol. 12, no. 1, pp. 23-33, 2016. DOI: https://doi.org/10.3844/jmssp.2016.23.33. DOI: https://doi.org/10.3844/jmssp.2016.23.33
[16] A. Mohammadpour, A. Babaei, S. Banihashemi, "A numerical scheme for solving a class of time fractional integro-partial differential equations with Caputo–Fabrizio derivative, " Asian-Euro. J. Math., vol. 15, no. 11, pp. 2250190. 2022. DOI: https://doi.org/10.1142/S179355712250190X DOI: https://doi.org/10.1142/S179355712250190X
[17] S. K. Yadav, M. Purohit, M. M. Gour, L. K. Yadav, M. N. Mishra, "Hybrid technique for multi-dimensional fractional diffusion problems involving Caputo–Fabrizio derivative, " Inter. J. Mathe. Indus., vol. 16, no. 1, pp. 2450020, 2024. DOI: 10.1142/S2661335224500205 DOI: https://doi.org/10.1142/S2661335224500205
[18] H. Azizi and G. B. Loghmani, " Numerical approximation for space fractional diffusion equations via Chebyshev finite difference method," J. Fract. Calc. Appl., vol.4, no. 2, pp. 303–311, 2013.
[19] A. Saadatmandi and M. Dehghan, " A tau approach for solution of the space fractional diffusion equation," Comput. Math. Appl, vol. 62, no. 3, pp. 1135–1142, 2011. DOI: https://doi.org/10.1016/j.camwa.2011.04.014. DOI: https://doi.org/10.1016/j.camwa.2011.04.014
[20] H. B. Jebreen, H. Wang, " On the effective method for the space-fractional advection-diffusion equation by the Galerkin method," AIMS Math., vol. 9, no. 9, pp. 24143-24162, 2024. DOI: 10.3934/math.20241173. DOI: https://doi.org/10.3934/math.20241173
[21] E. Hanert and C. Piret, " A Chebyshev pseudo spectral method to solve the space-time tempered fractional diffusion equation," SIAM J. Sci. Comput., vol. 36, no. 4, pp. A1797–A1812, 2014. DOI: https://doi.org/10.1137/130927292. DOI: https://doi.org/10.1137/130927292
[22]E. Sousa, " Numerical approximations for fractional diffusion equation via splines," Comput. Math. Appl. vol. 62, no. 3, pp. 938–944, 2011. DOI: https://doi.org/10.1016/j.camwa.2011.04.015. DOI: https://doi.org/10.1016/j.camwa.2011.04.015
[23] K. Krishnaveni, K. Kannan, S. R. Balachandar, S. G. Venkatesh, " An efficient fractional polynomial method for space fractional diffusion equations," Ain Shams Eng. J., vol. 9,no. 4, pp. 767–776, 2016. DOI: https://doi.org/10.1016/j.asej.2016.03.014. DOI: https://doi.org/10.1016/j.asej.2016.03.014
[24] Z. Soori, A. Aminataei, " Numerical solution of space fractional diffusion equation by spline method combined with Richardson extrapolation," Comp. Appl. Math, vol. 39, no. 2, pp. 1-18, 2020. DOI: https://doi.org/10.1007/s40314-020-01160-4. DOI: https://doi.org/10.1007/s40314-020-01160-4
[25] H. Safdari, H. Mesgarani, M. Javidi, Y. E. Aghdam, "Convergence analysis of the space fractional-order diffusion equation based on the compact finite difference scheme", Comp. Appl. Math., vol. 39, no. 62, pp. 1-15, 2020. DOI: https://doi.org/10.1007/s40314-020-1078-z. DOI: https://doi.org/10.1007/s40314-020-1078-z
[26] N. H. Tuan, Y. E. Aghdam, H. Jafari, H. Mesgarani, "A novel numerical manner for two-dimensional space fractional diffusion equation arising in transport phenomena," Numer. Meth. Part. DE, vol. 37, no. 2, pp. 1397–1406, 2021. DOI:https://doi.org/10.1002/num.22586 DOI: https://doi.org/10.1002/num.22586
[27] K. Issa, B. M. Yisa, J. Biazar, " Numerical solution of space fractional diffusion equation using shifted Gegenbauer polynomials," Comput. Methods Differ. Equ., vol. 10, no. 2, pp. 431-444, 2022. DOI: 20.1001.1.23453982.2022.10.2.11.6.
[28] F. S. M. Nasrudin, S. Mahadi, N. N. A. Hassan, " A Direct Method to Approximate Solution of the Space-fractional Diffusion Equation," MJFAS, vol. 20, no. 4, pp. 862-870, 2024. DOI: https://doi.org/10.11113/mjfas.v20n4.3533. DOI: https://doi.org/10.11113/mjfas.v20n4.3533
[29] H. Hajinezhad, A. R. Soheili, "A finite difference approximation for the solution of the space fractional diffusion equation, " J. Math. Exten., vol. 18, no. 1, pp. 1-14, 2024. DOI: https://doi.org/10.30495/JME.2024.2714
[30] M. Ayalew, M. Ayalew, M. Aychluh, " Numerical approximation of space-fractional diffusion equation using Laguerre spectral collocation method," Inter. J. Math. Indus., p. 2450029, 2025. DOI: https://doi.org/10.1142/S2661335224500291. DOI: https://doi.org/10.1142/S2661335224500291
[31] H. L. Liao, P. Lyu, S. Vong, " Second-order BDF time approximation for Riesz space-fractional diffusion equations," Inter. J. Comput. Appl. Math., vol. 95, no. 1, pp. 144-158, 2009. DOI: http://dx.doi.org/10.1016/j.cam.2008.08.038 DOI: https://doi.org/10.1080/00207160.2017.1366461
[32] I. Podlubny, Fractional Differential Equations: an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications. Academic Press, San Diego, CA, USA, 1998.
[33] W. W. Bell, Special Functions for Scientists and Engineers, Van Nostrand, London, 1968.
[34] J. C. Mason and D. C. Handscamb, Chebyshev Polynomials, 1st Edition, Chapman and Hall/CRC, New York, 2002. DOI: https://doi.org/10.1201/9781420036114. DOI: https://doi.org/10.1201/9781420036114
[35] W. Tian, H. Zhou, W. Deng, " A class of second-order difference approximations for solving space fractional diffusion equations," Math. Comput., vol.. 84, no. 294, pp. 1703-1727, 2015, DOI: https://doi.org/10.1090/S0025-5718-2015-02917-2. DOI: https://doi.org/10.1090/S0025-5718-2015-02917-2
[36] K. K. Ali, A. Mukheimer, J. A. Younis, M. A. Abd El Salam, H. Aydi, " Spectral collocation approach with shifted Chebyshev sixth-kind series approximation for generalized space fractional partial differential equations," AIMS Math., vol. 7, no. 5, pp. 8622-8644, 2022. DOI: 10.3934/math.. 20022482. DOI: https://doi.org/10.3934/math.2022482
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Basrah Researches Sciences
This work is licensed under a Creative Commons Attribution 4.0 International License.
