The method based on quintic B-spline functions for addressing time-fractional advection-dispersion equations

Document Type : Research Article

Authors

1 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Medina, KSA.

2 Department of Applied Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, University of Tabriz, Tabriz, Iran

3 Department of Mathematics

Abstract

This paper introduces a numerical method designed to address the fractional time advection-dispersion equation. Initially, the time dimension is discretized by employing the L1 method. Subsequently, quintic B-spline functions are utilized for the discretization of the spatial dimension. This  approach yields a system of algebraic equations that can be efficiently solved. The proposed method is proven to be unconditionally stable. Numerical experiments provide compelling evidence of the method’s efficiency and effectiveness

Keywords

Main Subjects


[1] A.H.B. Albohiwela, S. Irandoust-Pakchin, A.J. Akbarfam, Optimal collocation method for time fractional advection–dispersion equation using modified generalized Laguerre polynomials and particle swarm optimization algorithm, Filomat 38 (2024) 11453–11475.
[2] H. Azin, F. Mohammadi, M.H. Heydari, A hybrid method for solving time fractional advection-diffusion equation on unbounded space domain, Adv. Differ. Equ. 2020 (2020) 596.
[3] A. Bhardwaj, A. Kumar, A numerical solution of time-fractional mixed diffusion and diffusion–wave equation by an RBF-based meshless method, Eng. Comput. 38 (2022) 1883–1903.
[4] A.H. Bhrawy, A new numerical algorithm for solving a class of fractional advection–dispersion equation with variable coefficients using Jacobi polynomials, Abstr. Appl. Anal. 2013 (2013) Article ID 954983.
[5] S. Cao, J. Jiang, J. Wu, Solving time-fractional advection–dispersion equation by variable weights particle tracking method, J. Stat. Phys. 168 (2017) 1248–1258.
[6] I. Fahimi-Khalilabad, S. Irandoust-Pakchin, S. Abdi-Mazraeh, High-order finite difference method based on linear barycentric rational interpolation for Caputo type sub-diffusion equation, Math. Comput. Simul. 199 (2022) 60–80.
[7] S. Irandoust-Pakchin, S. Abdi-Mazraeh, A. Khani, Numerical solution for a variable-order fractional nonlinear cable equation via Chebyshev cardinal functions, Comput. Math. Math. Phys. 57 (2017) 2047–2056.
[8] S. Irandoust-Pakchin, S. Abdi-Mazraeh, I. Fahimi-Khalilabad, Higher order class of finite difference method for time-fractional Liouville-Caputo and space-Riesz fractional diffusion equation, Filomat 38 (2024) 505–521.
[9] S. Irandoust-Pakchin, Sh. Babapour, M. Lakestani, Image deblurring using adaptive fractional order shock filter, Math. Methods Appl. Sci. 44 (2021) 4907–4922.
[10] S. Irandoust-Pakchin, M. Lakestani, H. Kheiri, Numerical approach for solving a class of nonlinear fractional differential equation, Bull. Iran. Math. Soc. 42 (2016) 1107–1126.
[11] M. Irodotou-Ellina, E.N. Houstis, An O(h^6) quintic spline collocation method for fourth order two-point boundary value problems, BIT 28 (1988) 288–301.
[12] I. Karatay, N. Kale, S.R. Bayramoglu, A new difference scheme for time fractional heat equations based on the Crank–Nicolson method, Fract. Calc. Appl. Anal. 16 (2013) 892–910.
[13] M.M. Khader, N.H. Sweilam, Approximate solutions for the fractional advection-dispersion equation using Legendre pseudo-spectral method, Comp. Appl. Math. 33 (2014) 739–750.
[14] Z. Liu, X. Li, A Crank–Nicolson difference scheme for the time-variable fractional mobile–immobile advection–dispersion equation, Comput. Appl. Math. 56 (2018) 391–410.
[15] C.P. Li, F. Zeng, Numerical Methods for Fractional Calculus, CRC Press, Boca Raton, 2015.
[16] C.E. Mejía, A. Piedrahita, A numerical method for a time-fractional advection–dispersion equation with a nonlinear source term, J. Appl. Math. Comput. 61 (2019) 593–609.
[17] A.S. Moghadam, M. Arabameri, M. Barfeie, Numerical solution of space–time variable fractional order advection–dispersion equation using radial basis functions, J. Math. Model. 10 (2022) 549–562.
[18] A. Mohebbi, M. Abbaszadeh, Compact finite difference scheme for the solution of time-fractional advection–dispersion equation, Numer. Algorithms 63 (2013) 431–452.
[19] S. Momani, Z. Odibat, Numerical solutions of the space–time fractional advection–dispersion equation, Numer. Methods Partial Differ. Equ. 24 (2008) 549–562.
[20] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier, 1998.
[21] A.S.V. Ravi Kanth, S. Deepika, Application and analysis of spline approximation for time-fractional mobile–immobile advection–dispersion equation, Numer. Methods Partial Differ. Equ. 34 (2018) 1799–1819.
[22] P. Roul, A high accuracy numerical method and its convergence for time-fractional Black-Scholes equation governing European options, Appl. Numer. Math. 151 (2020) 472–493.
[23] P. Roul, V.M.K. Prasad Goura, R. Agarwal, A new high order numerical approach for a class of nonlinear derivative dependent singular boundary value problems, Appl. Numer. Math. 145 (2019) 315–341.
[24] P. Roul, K. Thula, V.M.K. Prasad Goura, An optimal sixth-order quartic B-spline collocation method for solving Bratu-type and Lane-Emden type problems, Math. Methods Appl. Sci. 42 (2019) 2613–2630.
[25] V. Saw, S. Kumar, Fourth kind shifted Chebyshev polynomials for solving space fractional order advection–dispersion equation, based on collocation method and finite difference approximation, Int. J. Appl. Comput. Math. 4 (2018) Article 82.
[26] V. Saw, S. Kumar, Second kind Chebyshev polynomials for solving space-fractional advection–dispersion equation using collocation method, Iran. J. Sci. Technol. A Sci. 43 (2019) 1027–1037.
[27] M. Shakeel, I. Hussain, H. Ahmad, I. Ahmad, P. Thounthong, Y.F. Zhang, Meshless technique for the solution of time-fractional partial differential equations having real-world applications, J. Funct. Spaces 2020 (2020) Article ID 8898309.
[28] M.K. Singh, A. Chatterjee, Solution of one-dimensional space- and time-fractional advection–dispersion equation by homotopy perturbation method, Acta Geophys. 65 (2017) 353–361.
[29] M.K. Singh, A. Chatterjee, V.P. Singh, Solution of one-dimensional time-fractional advection–dispersion equation by homotopy analysis method, J. Eng. Mech. 143 (2017) 1–16.
[30] H. Singh, Jacobi collocation method for the fractional advection–dispersion equation arising in porous media, Numer. Methods Partial Differ. Equ. 38 (2020) 636–653.
[31] K. Thula, P. Roul, A high-order B-spline collocation method for solving nonlinear singular boundary value problems arising in engineering and applied science, Mediterr. J. Math. 15 (2018) 176.