Research article

On the effective method for the space-fractional advection-diffusion equation by the Galerkin method

  • Received: 22 June 2024 Revised: 19 July 2024 Accepted: 22 July 2024 Published: 14 August 2024
  • MSC : 26A33, 54A25, 65M70

  • The present work is dedicated to a study that focuses on solving space-fractional advection-diffusion equations (SFADEs) using the Galerkin method. Through our analysis, we demonstrate the effectiveness of this approach in solving the considered equations. After introducing the Chebyshev cardinal functions (CCFs), the Caputo fractional derivative (CFD) was represented based on these bases as an operational matrix. Applying the Galerkin method reduces the desired equation to a system of algebraic equations. We have proved that the method converges analytically. By solving some numerical examples, we have demonstrated that the proposed method is effective and yields superior outcomes compared to existing methods for addressing this problem.

    Citation: Haifa Bin Jebreen, Hongzhou Wang. On the effective method for the space-fractional advection-diffusion equation by the Galerkin method[J]. AIMS Mathematics, 2024, 9(9): 24143-24162. doi: 10.3934/math.20241173

    Related Papers:

  • The present work is dedicated to a study that focuses on solving space-fractional advection-diffusion equations (SFADEs) using the Galerkin method. Through our analysis, we demonstrate the effectiveness of this approach in solving the considered equations. After introducing the Chebyshev cardinal functions (CCFs), the Caputo fractional derivative (CFD) was represented based on these bases as an operational matrix. Applying the Galerkin method reduces the desired equation to a system of algebraic equations. We have proved that the method converges analytically. By solving some numerical examples, we have demonstrated that the proposed method is effective and yields superior outcomes compared to existing methods for addressing this problem.



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    [1] J. Liu, F. Geng, An explanation on four new definitions of fractional operators, Acta Math. Sci., 44 (2024), 1271–1279. https://doi.org/10.1007/s10473-024-0405-7 doi: 10.1007/s10473-024-0405-7
    [2] M. Arif, F. Ali, I. Khan, K. S. Nisar, A time fractional model with non-singular kernel the generalized couette flow of couple stress nanofluid, IEEE Access, 8 (2020), 77378–77395. https://doi.org/10.1109/ACCESS.2020.2982028 doi: 10.1109/ACCESS.2020.2982028
    [3] A. Chang, H. Sun, C. Zheng, B. Lu, C. Lu, R. Ma, et al., A time fractional convection-diffusion equation to model gas transport through heterogeneous soil and gas reservoirs, Physica A, 502 (2018), 356–369. https://doi.org/10.1016/j.physa.2018.02.080 doi: 10.1016/j.physa.2018.02.080
    [4] J. A. Tenreiro Machado, M. F. Silva, R. S. Barbosa, I. S. Jesus, C. M. Reis, M. G. Marcos, et al., Some applications of fractional calculus in engineering, Math. Probl. Eng., 2010 (2010), 1–34. https://doi.org/10.1155/2010/639801 doi: 10.1155/2010/639801
    [5] F. Mainardi, Fractional calculus and waves in linear viscoelasticity, Imperial College Press, 2010. https://doi.org/10.1142/p614
    [6] M. Asadzadeh, B. N. Saray, On a multiwavelet spectral element method for integral equation of a generalized Cauchy problem, BIT Numer. Math., 62 (2022), 1383–1416. https://doi.org/10.1007/s10543-022-00915-1 doi: 10.1007/s10543-022-00915-1
    [7] A. K. Gupta, S. Saha Ray, Wavelet methods for solving fractional order differential equations, Math. Probl. Eng., 2014 (2014), 140453. https://doi.org/10.1155/2014/140453 doi: 10.1155/2014/140453
    [8] L. Shi, B. N. Saray, F. Soleymani, Sparse wavelet Galerkin method: Application for fractional Pantograph problem, J. Comput. Appl. Math., 451 (2024), 116081. https://doi.org/10.1016/j.cam.2024.116081 doi: 10.1016/j.cam.2024.116081
    [9] P. Thanh Toan, T. N. Vo, M. Razzaghi, Taylor wavelet method for fractional delay differential equations, Eng. Comput., 37 (2021), 231–240. https://doi.org/10.1007/s00366-019-00818-w doi: 10.1007/s00366-019-00818-w
    [10] V. Daftardar-Gejji, A. Jafari, Adomian decomposition: A tool for solving a system of fractional differential equations, J. Math. Anal. Appl., 301 (2005), 508–518. https://doi.org/10.1016/j.jmaa.2004.07.039 doi: 10.1016/j.jmaa.2004.07.039
    [11] B. Benkerrouche, D. Baleanu, M. S. Souid, A. Hakem, Boundary value problem for nonlinear fractional differential equations of variable order via Kuratowski MNC technique, Adv. Differ. Equ., 2021 (2021), 365. https://doi.org/10.1186/s13662-021-03520-8 doi: 10.1186/s13662-021-03520-8
    [12] M. Lakestani, M. Dehghan, The use of Chebyshev cardinal functions for the solution of a partial differential equation with an unknown time-dependent coefficient subject to an extra measurement, J. Comput. Appl. Math., 235 (2010), 669–678. https://doi.org/10.1016/j.cam.2010.06.020 doi: 10.1016/j.cam.2010.06.020
    [13] G. J. Fix, J. P. Roop, Least squares finite element solution of a fractional order two-point boundary value problem, Comput. Math. Appl., 48 (2004), 1017–1033. https://doi.org/10.1016/j.camwa.2004.10.003 doi: 10.1016/j.camwa.2004.10.003
    [14] S. Maji, S. Natesan, Adaptive-grid technique for the numerical solution of a class of fractional boundary-value-problems, Comput. Methods Differ. Equ., 12 (2010), 338–349. https://doi.org/10.22034/CMDE.2023.55266.2296 doi: 10.22034/CMDE.2023.55266.2296
    [15] R. Garrappa, On some explicit Adams multistep methods for fractional differential equations, J. Comput. Appl. Math., 229 (2009), 392–399. https://doi.org/10.1016/j.cam.2008.04.004 doi: 10.1016/j.cam.2008.04.004
    [16] Y. L. Zhao, P. Zhu, X. M. Gu, X. Zhao, H. Y. Jian, An implicit integration factor method for a kind of spatial fractional diffusion equations, J. Phys. Conf. Ser., 1324 (2019), 012030. https://doi.org/10.1088/1742-6596/1324/1/012030 doi: 10.1088/1742-6596/1324/1/012030
    [17] Z. Lin, D. Wang, D. Qi, L. Deng, A Petrov-Galerkin finite element-meshfree formulation for multi-dimensional fractional diffusion equations, Comput. Mech., 66 (2020), 323–350. https://doi.org/10.1007/s00466-020-01853-x doi: 10.1007/s00466-020-01853-x
    [18] H. Y. Jian, T. Z. Huang, X. L. Zhao, Y. L. Zhao, A fast second-order accurate difference schemes for time distributed-order and Riesz space fractional diffusion equations, J. Appl. Anal. Comput., 9 (2019), 1359–1392. https://doi.org/10.11948/2156-907X.20180247 doi: 10.11948/2156-907X.20180247
    [19] P. Biler, W. A. Woyczynski, Global and exploding solutions for nonlocal quadratic evolution problems, SIAM J. Appl. Math., 59 (1998), 845–869. https://doi.org/10.1137/S003613999631344 doi: 10.1137/S003613999631344
    [20] W. Y. Tian, W. Deng, Y. Wu, Polynomial spectral collocation method for space fractional advection-diffusion equation, Numer. Meth. Partial Differ. Equ., 30 (2014), 514–535. https://doi.org/10.1002/num.21822 doi: 10.1002/num.21822
    [21] R. Metzler, J. Klafter, The random walks guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1–77. https://doi.org/10.1016/S0370-1573(00)00070-3 doi: 10.1016/S0370-1573(00)00070-3
    [22] A. Pablo, F. Quirós, A. Rodríguez, J. L. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), 1378–1409. https://doi.org/10.1016/j.aim.2010.07.017 doi: 10.1016/j.aim.2010.07.017
    [23] Y. Zheng, C. Li, Z. Zhao, A note on the finite element method for the space-fractional advection diffusion equation, Comput. Math. Appl., 59 (2010), 1718–1726. https://doi.org/10.1016/j.camwa.2009.08.071 doi: 10.1016/j.camwa.2009.08.071
    [24] H. Hejazi, T. Moroney, F. Liu, Stability and convergence of a finite volume method for the space fractional advection-dispersion equation, J. Comput. Appl. Math., 255 (2014), 684–697. https://doi.org/10.1016/j.cam.2013.06.039 doi: 10.1016/j.cam.2013.06.039
    [25] A. Jannelli, M. Ruggieri, M. P. Speciale, Numerical solutions of space-fractional advection-diffusion equations with nonlinear source term, Appl. Numer. Math., 155 (2020), 93–102. https://doi.org/10.1016/j.apnum.2020.01.016 doi: 10.1016/j.apnum.2020.01.016
    [26] H. Y. Jian, T. Z. Huang, X. M. Gu, Y. L. Zhao, Compact implicit integration factor method for two-dimensional space-fractional advection-diffusion-reaction equations, J. Phys. Conf. Ser., 1592 (2020), 012048. https://doi.org/10.1088/1742-6596/1592/1/012048 doi: 10.1088/1742-6596/1592/1/012048
    [27] H. Y. Jian, T. Z. Huang, A. Ostermann, X. M. Gu, Y. L. Zhao, Fast IIF-WENO method on non-uniform meshes for nonlinear space-fractional convection-diffusion-reaction equations, J. Sci. Comput., 89 (2021), 13. https://doi.org/10.1007/s10915-021-01622-9 doi: 10.1007/s10915-021-01622-9
    [28] J. P. Boyd, Chebyshev and fourier spectral methods, Dover Publications, 2001.
    [29] A. Afarideh, F. D. Saei, M. Lakestani, B. N. Saray, Pseudospectral method for solving fractional Sturm-Liouville problem using Chebyshev cardinal functions, Phys. Scr., 96 (2021), 125267. https://doi.org/10.1088/1402-4896/ac3c59 doi: 10.1088/1402-4896/ac3c59
    [30] A. Afarideh, F. D. Saei, B. N. Saray, Eigenvalue problem with fractional differential operator: Chebyshev cardinal spectral method, J. Math. Model., 11 (2021), 343–355. https://doi.org/10.22124/JMM.2023.24239.2169 doi: 10.22124/JMM.2023.24239.2169
    [31] F. Tchier, I. Dassios, F. Tawfiq, L. Ragoub, On the approximate solution of partial integro-differential equations using the pseudospectral method based on Chebyshev cardinal functions, Mathematics, 9 (2021), 286. https://doi.org/10.3390/math9030286 doi: 10.3390/math9030286
    [32] M. Shahriari, B. N. Saray, B. Mohammadalipour, S. Saeidian, Pseudospectral method for solving the fractional one-dimensional Dirac operator using Chebyshev cardinal functions, Phys. Scr., 98 (2023), 055205. https://doi.org/10.1088/1402-4896/acc7d3 doi: 10.1088/1402-4896/acc7d3
    [33] K. Sayevand, H. Arab, An efficient extension of the Chebyshev cardinal functions for differential equations with coordinate derivatives of non-integer order, Comput. Methods Differ. Equ., 6 (2018), 339–352. https://doi.org/20.1001.1.23453982.2018.6.3.6.5
    [34] A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006.
    [35] Y. Saad, M. H. Schultz, GMRES: A generalized minimal residual method for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986), 856–869. https://doi.org/10.1137/090705 doi: 10.1137/090705
    [36] G. Dahlquist, A. Björck, Numerical methods, Englewood Cliffs: Prentice-Hall, 1974.
    [37] B. N. Saray, M. Lakestani, M. Dehghan, On the sparse multiscale representation of 2‐D Burgers equations by an efficient algorithm based on multiwavelets, Numer. Math. Partial Differ. Equ., 39 (2023), 1938–1961. https://doi.org/10.1002/num.22795 doi: 10.1002/num.22795
    [38] G. Pang, W. Chen, Z. Fu, Space-fractional advection-dispersion equations by the Kansa method, J. Comput. Phys., 293 (2015), 280–296. https://doi.org/10.1016/j.jcp.2014.07.020 doi: 10.1016/j.jcp.2014.07.020
    [39] M. M. Meerschaert, H. P. Scheffler, C. Tadjeran, Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys., 211 (2006), 249–261. https://doi.org/10.1016/j.jcp.2005.05.017 doi: 10.1016/j.jcp.2005.05.017
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