Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model based on an efficient hybrid numerical method with stability and convergence analysis. (English) Zbl 1540.65429
Summary: In this paper, we propose a hybrid numerical method based on the weighted finite difference and the quintic Hermite collocation methods. The proposed method is used for solving the variable-order time fractional mobile-immobile advection-dispersion(VOMIM-AD) model, such that the discretization is done by applying collocation method with Hermite splines in the spatial direction and weighted finite difference method in the temporal direction. Provided examples confirm the studied stability and convergence properties of the proposed method. The obtained results from the graphical illustration and numerical simulations, in comparison with other methods in the literature, demonstrate that the reported method is very robust and accurate.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35R11 | Fractional partial differential equations |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
VOMIM-AD model; weighted finite difference method; quintic Hermite collocation method; convergence and stability analysis; variable fractional order mobile-immobile advection-dispersion modelReferences:
| [1] | Abdelkawy, M. A.; Lopes, A. M.; Babatin, M. M., Shifted fractional Jacobi collocation method for solving fractional functional differential equations of variable order, Chaos Solitons Fractals, 134, Article 109721 pp. (2020) · Zbl 1483.65108 |
| [2] | Abdelkawy, M. A.; Zaky, M. A.; Bhrawy, A. H.; Baleanu, D., Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model, Rom. Rep. Phys., 67, 3, 773-791 (2015) |
| [3] | Adel, M., Finite difference approach for variable order reaction-subdiffusion equations, Adv. Difference Equ., 2018, 1, 1-12 (2018) · Zbl 1448.65199 |
| [4] | Agarwal, P.; El-Sayed, A. A.; Tariboon, J., Vieta-fibonacci operational matrices for spectral solutions of variable-order fractional integro-differential equations, J. Comput. Appl. Math., 382, Article 113063 pp. (2021) · Zbl 1471.65066 |
| [5] | Ali, U.; Iqbal, A.; Sohail, M.; Abdullah, F. A.; Khan, Z., Compact implicit difference approximation for time-fractional diffusion-wave equation, Alex. Eng. J., 61, 5, 4119-4126 (2022) |
| [6] | Almeida, R.; Bastos, N. R.; Monteiro, M. T.T., Modeling some real phenomena by fractional differential equations, Math. Methods Appl. Sci., 39, 16, 4846-4855 (2016) · Zbl 1355.34075 |
| [7] | Almeida, R.; Tavares, D.; Torres, D. F.M., The Variable-Order Fractional Calculus of Variations (2019), Springer · Zbl 1402.49002 |
| [8] | Arora, S.; Kaur, I., Applications of quintic Hermite collocation with time discretization to singularly perturbed problems, Appl. Math. Comput., 316, 409-421 (2018) · Zbl 1427.65278 |
| [9] | Atangana, A., On the stability and convergence of the time-fractional variable order telegraph equation, J. Comput. Phys., 293, 104-114 (2015) · Zbl 1349.65263 |
| [10] | Atangana, A., Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A, 1, 505, 688-706 (2018) · Zbl 1514.34009 |
| [11] | Atangana, A.; Gómez-Aguilar, J. F., Fractional derivatives with no-index law property: Application to chaos and statistics, Chaos Solitons Fractals, 114, 516-535 (2018) · Zbl 1415.34010 |
| [12] | Bear, J., Dynamics of Fluids in Porous Media (1972), American Elsevier Publishing Company: American Elsevier Publishing Company NewYork · Zbl 1191.76001 |
| [13] | Benson, D. A.; Schumer, R.; Meerschaert, M. M.; Wheatcraft, S. W., Fractional dispersion, Levy motion, and the MADE tracer tests, Transp. Porous Media, 42, 1-2, 211-240 (2001) |
| [14] | Chen, S.; Liu, F.; Burrage, K., Numerical simulation of a new two-dimensional variable-order fractional percolation equation in non-homogeneous porous media, Comput. Math. Appl., 68, 12, 2133-2141 (2014) · Zbl 1369.35105 |
| [15] | Chen, C.; Liu, H.; Zheng, X.; Wang, H., A two-grid MMOC finite element method for nonlinear variable-order time-fractional mobile/immobile advection-diffusion equations, Comput. Math. Appl., 79, 9, 2771-2783 (2020) · Zbl 1437.65180 |
| [16] | Chen, Z.; Qian, J.; Zhan, H.; Chen, L.; Luo, S., Mobile-immobile model of solute transport through porous and fractured media, Manag. Groundw. Environ., 341, 154-158 (2011) |
| [17] | Dehestani, H.; Ordokhani, Y., A novel direct method based on the Lucas multiwavelet functions for variable-order fractional reaction-diffusion and subdiffusion equations, Numer. Linear Algebra Appl., 28, 2, Article e2346 pp. (2021) · Zbl 07332759 |
| [18] | Dehestani, H.; Ordokhani, Y.; Razzaghi, M., Pseudo-operational matrix method for the solution of variable-order fractional partial integro-differential equations, Eng. Comput., 1-16 (2020) |
| [19] | Deng, Z. Q.; Bengtsson, L.; Singh, V. P., Parameter estimation for fractional dispersion model for rivers, Environ. Fluid Mech., 6, 451-475 (2006) |
| [20] | Derakhshan, M. H.; Aminataei, A., A new approach for solving multi-variable orders differential equations with prabhakar function, J. Math. Model., 8, 2, 139-155 (2020) · Zbl 1488.65200 |
| [21] | Doha, E. H.; Abdelkawy, M. A.; Amin, A. Z.M., Spectral technique for solving variable-order fractional Volterra integro-differential equations, Numer. Methods Partial Differential Equations, 34, 5, 1659-1677 (2018) · Zbl 1407.65211 |
| [22] | Du, R.; Alikhanov, A. A.; Sun, Z. Z., Temporal second order difference schemes for the multi-dimensional variable-order time fractional sub-diffusion equations, Comput. Math. Appl., 79, 10, 2952-2972 (2020) · Zbl 1447.65020 |
| [23] | Dyksen, W. R.; Lynch, R. E., A new decoupling technique for the Hermite cubic collocation equations arising from boundary value problems, Math. Comput. Simulation, 54, 4-5, 359-372 (2000) · Zbl 0985.65088 |
| [24] | Ganaie, I. A.; Arora, S.; Kukreja, V., Cubic Hermite collocation solution of Kuramoto-Sivashinsky equation, Int. J. Comput. Math., 93, 1, 223-235 (2016) · Zbl 1339.35266 |
| [25] | Gharian, D.; Ghaini, F. M.; Heydari, M. H.; Avazzadeh, Z., A meshless solution for the variable-order time fractional nonlinear Klein-Gordon equation, Int. J. Appl. Comput. Math., 6, 5, 1-16 (2020) · Zbl 1461.65245 |
| [26] | Habibirad, A.; Hesameddini, E.; Heydari, M. H.; Roohi, R., An efficient meshless method based on the moving kriging interpolation for two-dimensional variable-order time fractional mobile/immobile advection-diffusion model, Math. Methods Appl. Sci., 44, 4, 3182-3194 (2021) · Zbl 1473.65195 |
| [27] | Hall, C., On error bounds for spline interpolation, J. Approx. Theory, 1, 2, 209-218 (1968) · Zbl 0177.08902 |
| [28] | Hansen, S. K., Effective ADE models for first-order mobile-immobile solute transport: Limits on validity and modeling implications, 86, 184-192 (2015) |
| [29] | Heydari, M. H.; Atangana, A., An optimization method based on the generalized Lucas polynomials for variable-order space-time fractional mobile-immobile advection-dispersion equation involving derivatives with non-singular kernels, Chaos Solitons Fractals, 132, Article 109588 pp. (2020) · Zbl 1434.35274 |
| [30] | Hilfer, R., Applications of Fractional Calculus in Physics (2000), World Scientific: World Scientific New Jersey · Zbl 0998.26002 |
| [31] | Jiang, W.; Liu, N., A numerical method for solving the time variable fractional order mobile-immobile advection-dispersion model, Appl. Numer. Math., 119, 18-32 (2017) · Zbl 1432.65155 |
| [32] | Kim, S.; Kavvas, M. L., Generalized ficks law and fractional ADE for pollutant transport in a river: Detailed derivation, J. Hydrol. Eng., 11, 1, 80-83 (2006) |
| [33] | Kumar, D.; Singh, J.; Baleanu, D., Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel, Physica A, 492, 155-167 (2018) · Zbl 1514.35463 |
| [34] | Kumar, D.; Singh, J.; Baleanu, D., A new analysis of the Fornberg-Whitham equation pertaining to a fractional derivative with Mittag-Leffler type kernel, Eur. Phys. J. Plus, 1, 133, 1-21 (2018) |
| [35] | Liu, Z.; Li, X., A Crank-Nicolson difference scheme for the time variable fractional mobile-immobile advection-dispersion equation, J. Appl. Math. Comput., 56, 1, 391-410 (2018) · Zbl 1448.65112 |
| [36] | Luo, Z.; Zhang, X.; Wang, S.; Yao, L., Numerical approximation of time fractional partial integro-differential equation based on compact finite difference scheme, Chaos Solitons Fractals, 161, Article 112395 pp. (2022) · Zbl 1504.65291 |
| [37] | Ma, H.; Yang, Y., Jacobi spectral collocation method for the time variable-order fractional mobile-immobile advection-dispersion solute transport model, East Asian J. Appl. Math., 6, 3, 337-352 (2016) · Zbl 1499.65570 |
| [38] | Michelsen, M. L.; Villadsen, J., A convenient computational procedure for collocation constants, Chem. Eng. J., 4, 1, 64-68 (1972) |
| [39] | Miller, K. S.; Ross, B., An Introductional the Fractional Calculus and Fractional Differential Equations (1974), Academic Press: Academic Press New York, London |
| [40] | Moallem, G. R.; Jafari, H.; Adem, A. R., A numerical scheme to solve variable order diffusion-wave equations, Therm. Sci., 23, 6, 2063-2071 (2019) |
| [41] | Moghaddam, B. P.; Machado, J. A.T., 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) · Zbl 1412.65084 |
| [42] | Oldham, K. B.; Spanier, J., The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order (1974), Academic Press · Zbl 0292.26011 |
| [43] | Pandey, R. K.; Singh, O. P.; Baranwal, V. K.; Tripathi, M. P., An analytic solution for the space-time fractional advection-dispersion equation using the optimal homotopy asymptotic method, Comput. Phys. Comm., 183, 10, 2098-2106 (2012) · Zbl 1296.76004 |
| [44] | Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010 |
| [45] | Saffarian, M.; Mohebbi, A., An efficient numerical method for the solution of 2D variable order time fractional mobile-immobile advection-dispersion model, Math. Methods Appl. Sci. (2021) · Zbl 1473.65243 |
| [46] | Singh, J.; Kumar, D.; Hammouch, Z.; Atangana, A., A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Appl. Math. Comput., 316, 504-515 (2018) · Zbl 1426.68015 |
| [47] | Sweilam, N. H.; Ahmed, S. M.; Adel, M., A simple numerical method for two-dimensional nonlinear fractional anomalous sub-diffusion equations, 2914-2933 (2021), 44 (4) · Zbl 1490.65161 |
| [48] | Tian, X.; Reutskiy, S. Y.; Fu, Z. J., A novel meshless collocation solver for solving multi-term variable-order time fractional PDEs, Eng. Comput., 1-12 (2021) |
| [49] | Tuan, N. H.; Nemati, S.; Ganji, R. M.; Jafari, H., Numerical solution of multi-variable order fractional integro-differential equations using the Bernstein polynomials, Eng. Comput., 1-9 (2020) |
| [50] | W.K. Zahra, M.M. Hikal, Non standard finite difference method for solving variable order fractional optimal control problems, J. Vib. Control 23 (6) 948-958. · Zbl 1387.93095 |
| [51] | Zhang, H.; Liu, F.; Phanikumar, M. S.; Meerschaert, M. M., A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model, Comput. Math. Appl., 66, 693-701 (2013) · Zbl 1350.65092 |
| [52] | Zhang, Q.; Liu, L.; Zhang, C., Compact scheme for fractional diffusion-wave equation with spatial variable coefficient and delays, Appl. Anal., 101, 6, 1911-1932 (2022) · Zbl 1490.65163 |
| [53] | Zheng, G. H.; Wei, T., Spectral regularization method for the time fractional inverse advection-dispersion equation, Math. Comput. Simulation, 81, 37-51 (2010) · Zbl 1205.65280 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
