An accurate numerical scheme for three-dimensional variable-order time-fractional partial differential equations in two types of space domains
Abstract
We consider the discretization method for solving three-dimensional variable-order (3D-VO) time-fractional partial differential equations. The proposed method is developed based on discrete shifted Hahn polynomials (DSHPs) and their operational matrices. In the process of method implementation, the modified operational matrix (MOM) and complement vector (CV) of integration and pseudooperational matrix (POM) of VO fractional derivative plays an important role in the accuracy of the method. Further, we discuss the error of the approximate solution. At last, the methodology is validated by well test examples in two types of space domains. In order to evaluate the accuracy and applicability of the approach, the results are compared with other methods.
Keyword : discrete shifted Hahn polynomials, variable-order Caputo fractional derivative, operational matrix, three-dimensional partial differential equations
How to Cite
Dehestani, H., Ordokhani, Y., & Razzaghi, M. (2024). An accurate numerical scheme for three-dimensional variable-order time-fractional partial differential equations in two types of space domains. Mathematical Modelling and Analysis, 29(3), 406–425. https://doi.org/10.3846/mma.2024.18535
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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M.D. Bramson. Maximal displacement of branching brownian motion. Commun. Pure Appl. Math., 31(5):531–581, 1978. https://doi.org/10.1016/j.camwa.2017.02.034
J. Canosa. Diffusion in nonlinear multiplicative media. J. Math. Phys., 10(10):1862–1868, 1969. https://doi.org/10.1063/1.1664771
H. Dehestani and Y. Ordokhani. An efficient approach based on Legendre–Gauss–Lobatto quadrature and discrete shifted Hahn polynomials for solving Caputo–Fabrizio fractional Volterra partial integrodifferential equations. J. Comput. Appl. Math., 403:113851, 2022. https://doi.org/10.1016/j.cam.2021.113851
H. Dehestani, Y. Ordokhani and M. Razzaghi. The novel operational matrices based on 2d-Genocchi polynomials: solving a general class of variable-order fractional partial integro-differential equations. Comput. Appl. Math., 39(4):1–32, 2020. https://doi.org/10.1007/s40314-020-01314-4
H. Dehestani, Y. Ordokhani and M. Razzaghi. Modified wavelet method for solving multitype variable-order fractional partial differential equations generated from the modeling of phenomena. Math. Sci., 16:343–359, 2022. https://doi.org/10.1007/s40096-021-00425-1
D.A. Frank-Kamenetskii. Diffusion and heat exchange in chemical kinetics. In Diffusion and Heat Exchange in Chemical Kinetics. Princeton University Press, 2015.
Y. Gu and H.G. Sun. A meshless method for solving three-dimensional time fractional diffusion equation with variable-order derivatives. Appl. Math. Model., 78:539–549, 2020. https://doi.org/10.1016/j.apm.2019.09.055
M.H. Heydari, M.R. Mahmoudi, A. Shakiba and Z. Avazzadeh. Chebyshev cardinal wavelets and their application in solving nonlinear stochastic differential equations with fractional Brownian motion. Commun. Nonlinear Sci. Numer. Simul., 64:98–121, 2018. https://doi.org/10.1016/j.cnsns.2018.04.018
M. Hosseininia, M.H. Heydari, Z. Avazzadeh and F.M.M. Ghaini. Twodimensional Legendre wavelets for solving variable-order fractional nonlinear advection-diffusion equation with variable coefficients. Int. J. Nonlinear Sci. Numer. Simul., 19(7-8):793–802, 2018.
J. Liu, X. Li and X. Hu. A RBF-based differential quadrature method for solving two-dimensional variable-order time fractional advection-diffusion equation. J. Comput. Phys., 384:222–238, 2019. https://doi.org/10.1016/j.jcp.2018.12.043
Willy Malfliet. Solitary wave solutions of nonlinear wave equations. Am. J. Phys., 60(7):650–654, 1992. https://doi.org/10.1119/1.17120
B.P. Moghaddam and J.A.T. Machado. A stable three-level explicit spline finite difference scheme for a class of nonlinear time variable order fractional partial differential equations. Comput. Math. Appl., 73(6):1262–1269, 2017. https://doi.org/10.1016/j.camwa.2016.07.010
F. Mohammadi, L. Moradi and J.A. Tenreiro Machado. A discrete polynomials approach for optimal control of fractional Volterra integro-differential equations. J. Vib. Control, 28(1-2):72–82, 2022. https://doi.org/10.1177/1077546320971156
P. Pandey, S. Das, E.M. Craciun and T. Sadowski. Two-dimensional nonlinear time fractional reaction–diffusion equation in application to sub-diffusion process of the multicomponent fluid in porous media. Meccanica, 56(1):99–115, 2021. https://doi.org/10.1007/s11012-020-01268-1
H.T.C. Pedro, M.H. Kobayashi, J.M.C. Pereira and C.F.M. Coimbra. Variable order modeling of diffusive-convective effects on the oscillatory flow past a sphere. J. Vib. Control, 14(9-10):1659–1672, 2008. https://doi.org/10.1177/1077546307087397
L.E. Ramirez and C.F. Coimbra. On the variable order dynamics of the nonlinear wake caused by a sedimenting particle. Physica D: nonlinear phenomena, 240(13):1111–1118, 2011. https://doi.org/10.1016/j.physd.2011.04.001
L.E.S. Ramirez and C.F.M. Coimbra. On the selection and meaning of variable order operators for dynamic modeling. Int. J. Differ. Equ., 2010, 2010. https://doi.org/10.1155/2010/846107
Y. Shan, W. Liu and B. Wu. Space–time Legendre–Gauss–Lobatto collocation method for two-dimensional generalized sine-Gordon equation. Appl. Numer. Math., 122:92–107, 2017. https://doi.org/10.1016/j.apnum.2017.08.003
Y. Shekari, A. Tayebi and M.H. Heydari. A meshfree approach for solving 2D variable-order fractional nonlinear diffusion-wave equation. Comput. Methods Appl. Mech. Eng., 350:154–168, 2019. https://doi.org/10.1016/j.cma.2019.02.035
H. Sheng, H.G. Sun, C. Coopmans, Y.Q. Chen and G.W. Bohannan. A physical experimental study of variable-order fractional integrator and differentiator. Eur. Phys. J. Spec. Top., 193(1):93–104, 2011. https://doi.org/10.1140/epjst/e201101384-4
I. Singh and S. Kumar. Wavelet methods for solving three-dimensional partial differential equations. Math. Sci., 11(2):145–154, 2017. https://doi.org/10.1007/s40096-017-0220-6
H.G. Sun, W. Chen, H. Wei and Y.Q. Chen. A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems. Eur. Phys. J. Spec. Top., 193(1):185–192, 2011. https://doi.org/10.1140/epjst/e2011-01390-6
T.N. Vo, M. Razzaghi and P.T. Toan. A numerical method for solving variable-order fractional diffusion equations using fractional-order Taylor wavelets. Numer. Methods Partial Differ. Equ., 37(3):2668–2686, 2021. https://doi.org/10.1002/num.22761
S. Yüzbaşı and M. Karaçayır. A Galerkin-like scheme to solve twodimensional telegraph equation using collocation points in initial and boundary conditions. Comput. Math. Appl., 74(12):3242–3249, 2017. https://doi.org/10.1016/j.camwa.2017.08.020