A new numerical scheme for solving the two dimensional fractional diffusion equation

• Autorzy: Altan Koç Dilara , Gülsu Mustafa

Identyfikatory

DOI

10.17512/jamcm.2021.2.01

• Języki publikacji: EN

Abstrakty

EN

In this study, the locally one dimensional (LOD) method is used to solve the two dimensional time fractional diffusion equation. The fractional derivative is the Caputo fractional derivative of order α. The rate of convergence of the finite difference method is presented. It is seen that this method is in agreement with the obtained numerical solutions with acceptable central processing unit time (CPU time). Error estimates, numerical and exact results are tabulated. The graphics of errors are given.

Słowa kluczowe

EN

finite difference method; fractional calculus; numerical methods

PL

metoda różnic skończonych; rachunek ułamkowy; metody numeryczne

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2021
• Tom: Vol. 20, nr 2
• Strony: 5--16
• Opis fizyczny: Bibliogr. 16 poz. rys., tab

Twórcy

autor

Altan Koç Dilara

• Department Mathematics, Mugla Sitki Kocman University, Turkey

autor

Gülsu Mustafa

• Department Mathematics, Mugla Sitki Kocman University, Turkey

Bibliografia

• [1] Metzler, R., & Klafter, J. (2004). The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional Dynamics. Journal of Physics A, 37, R161-R208.
• [2] Chen, T., & Wang, D. (2020). Combined application of blockchain technology in fractional calculus model of supply chain financial system. Chaos, Solitons & Fractals, 131, 109461.
• [3] Benson, D.A., Meerschaert, M.M., & Revielle, J. (2013). Fractional calculus in hydrologic modelling: a numerical perspective. Advances in Water Resources, 51, 479-497.
• [4] Luchko, Y. (2016). A new fractional calculus model for the two-dimensional anomalous diffusion and its analysis. Mathematical Modelling of Natura Phenımena, 11-3, 1-17.
• [5] Li, Z., Liu, L., Dehghan, S., Chen, Y., & Xue, D. (2017). A review and evaluation of numerical tools for fractional calculus and fractional order controls. International Journal of Controls, 90(6), 1165-1181.
• [6] Wu, G.C., & Baleanu D. (2013). Variational iteration method for fractional calculus – a universal approach by Laplace transform. Advances in Difference Equations, 18, 1-9.
• [7] Langlands, T. (2006). Fractional Dynamics. Physica A, 367, 136.
• [8] Narahari, B.A., & Hanneken J. (2004). Fractional radial diffusion in a cylinder. Journal of Molecular Liquids, 114, 147-151.
• [9] Zhuang, P., & Liu, F. (2007). Finite difference approximation for two-dimensional time fractional diffusion equation. Journal of Algorithms & Computational Technology, 1, 1.
• [10] Tadjeran, C., & Meerschaert, M.M. (2007). A second order accurate numerical method for the two-dimensional fractional diffusion equation. Journal of Computational Physics, 220, 813-823.
• [11] Meerschaert, M.M., Scheffler, H.P., & Tadjeran C. (2006). Finite difference methods for two-dimensional fractional dispersion equation. Journal of Computational Physics, 211, 249-261.
• [12] Dehghan, M. (1999). Implicit locally one-dimensional methods for two-dimensional diffusion with a non-local boundary condition. Mathematics and Computers in Simulation, 49, 331-349.
• [13] Chen, J., & Ge, Y. (2018). High order locally one dimensional methods for solving two-dimensional parabolic equation. Advence in Difference Equations, 361.
• [14] Soori, Z., & Aminataei, A. (2018). Effect of the nodes near boundary points on the stability analysis of sixth-order compact finite difference ADI scheme for the two-dimensional time fractional diffusion-wave equation. Transactions of A. Razmadze Mathematical Institute, 172, 582-605.
• [15] Kutluay, S., & Yağmurlu, N.M. (2012). The modified bi-quintic b-splines for solving the two dimensional unsteady burgers’ equation. European International Journal of Science and Technology, 1, 2.
• [16] Kutluay, S., & Yağmurlu, N.M. (2013). Derivation of the modified bi-quintic b-spline base functions: an application to Poisson equation. American Journal of Computational and Applied Mathematics, 3(1), 26-32.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).