Analysis of dual Bernstein operators in the solution of the fractional convection-diffusion equation arising in underground water pollution. (English) Zbl 1472.35441

Summary: The Bernstein operators (BO) are not orthogonal, but they have duals, which are obtained by a linear combination of BO. In recent years dual BO have been adopted in computer graphics, computer aided geometric design, and numerical analysis. This paper presents a numerical method based on the Bernstein operational matrices to solve the time-space fractional convection-diffusion equation. A generalization of the derivative matrix operator of fractional order and the error analysis are discussed. Numerical examples compare the proposed approach with previous works, showing that the method is more accurate and efficient.


35R11 Fractional partial differential equations
35K20 Initial-boundary value problems for second-order parabolic equations
92D40 Ecology
Full Text: DOI


[1] Abbasbandy, S.; Kazem, S.; Alhuthali, M. S.; Alsulami, H. H., Application of the operational matrix of fractional-order Legendre functions for solving the time-fractional convection-diffusion equation, Appl. Math. Comput., 266, 31-40 (2015) · Zbl 1410.65388
[2] Schmoll, O.; Howard, G.; Chilton, G., Protecting Groundwater for Health: Managing the Quality of Drinking-Water (2006), IWA Publishing for World Health Organization
[3] Biazar, J.; Asadi, M. A., Finite integration method with RBFs for solving time-fractional convection-diffusion equation with variable coefficients, Comput. Methods Differ. Equ., 7, 1, 1-15 (2019) · Zbl 1424.65180
[4] Metzler, R.; Jeon, J. H.; Cherstvy, A. G.; Barkai, E., Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking, Phys. Chem. Chem. Phys., 16, 44, 24128-24164 (2014)
[5] Fa, K. S.; Lenzi, E. K., Time-fractional diffusion equation with time dependent diffusion coefficient, Phys. Rev. E, 72, 1, Article 011107 pp. (2005)
[6] Garra, R.; Giusti, A.; Mainardi, F., The fractional Dodson diffusion equation: a new approach, Ricerche Di Mat., 67, 2, 899-909 (2018) · Zbl 1403.35314
[7] Garra, R.; Orsingher, E.; Polito, F., Fractional diffusions with time-varying coefficients, J. Math. Phys., 56, 9, Article 093301 pp. (2015) · Zbl 1337.60064
[8] https://eijournal.com/news/industry-insights-trends/study-maps-hidden-water-pollution-in-u-s-coastal-areas.
[9] Jiang, Y.; Ma, J., High-order finite element methods for time-fractional partial differential equations, J. Comput. Appl. Math., 235, 11, 3285-3290 (2011) · Zbl 1216.65130
[10] Heydari, M. H.; Hooshmandasl, M. R.; Mohammadi, F., Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions, Appl. Math. Comput., 234, 267-276 (2014) · Zbl 1298.65181
[11] Hosseini, V. R.; Koushki, M.; Zou, W. N., The meshless approach for solving 2D variable-order time-fractional advection-diffusion equation arising in anomalous transport, Eng. Comput., 1-19 (2021)
[12] Hosseini, V. R.; Yousefi, F.; Zou, W. N., The numerical solution of high dimensional variable-order time fractional diffusion equation via the singular boundary method, J. Adv. Res. (2021)
[13] Kargar, Z.; Saeedi, H., B-spline wavelet operational method for numerical solution of time-space fractional partial differential equations, Int. J. Wavelets Multiresol. Inf. Process., 15, 4, Article 1750034 pp. (2017) · Zbl 1379.65081
[14] Hong-Xia, G.; Yong-Qing, L.; Rong-Jun, C., Element-free Galerkin (EFG) method for analysis of the time-fractional partial differential equations, Chin. Phys., 21, 1, Article 010206 pp. (2012)
[15] Y. Xing, X. Wu, Z. Xu, Multiclass least squares auto-correlation wavelet support vector machines, in: International Conference on Innovative Computing Information and Control, Vol. 2, No. 4, 2008, pp. 345-350.
[16] Zhuang, P.; Liu, F., An explicit approximation for the space-time fractional diffusion equation, Numer. Math. A: J. Chin. Univ., 27, 223-228 (2005)
[17] Osama, H.; Fadhel, S. F.; Mohammed, G. S., Numerical solution for the time-fractional diffusion-wave equations by using Sinc-Legendre collocation method, Math. Theory Model, 5, 1, 49-57 (2015)
[18] Yang, Y.; Ma, Y.; Wang, L., Legendre polynomials operational matrix method for solving fractional partial differential equations with variable coefficients, Math. Probl. Eng., Article 915195 pp. (2015) · Zbl 1394.65114
[19] Yi, M.; Huang, J.; Wei, J., Block pulse operational matrix method for solving fractional partial differential equation, Appl. Math. Comput., 221, 121-131 (2013) · Zbl 1329.65241
[20] Zhou, Y.; Jiao, F.; Li, J., Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear Anal., 71, 78, 3249-3256 (2009) · Zbl 1177.34084
[21] Zhou, Y.; Jiao, F.; Li, J., Existence and uniqueness for \(p\)-type fractional neutral differential equations, Nonlinear Anal., 71, 8, 2724-2733 (2009) · Zbl 1175.34082
[22] Ditzian, Z.; Ivanov, K., Bernstein-type operators and their derivatives, J. Approx. Theory, 56, 1, 72-90 (1989) · Zbl 0692.41021
[23] Bhatti, M.; Bracken, P., Solutions of differential equations in a Bernstein polynomial basis, Comput. Appl. Math., 205, 272-280 (2007) · Zbl 1118.65087
[24] Bernstein, S., Demonstration of a theorem of Weierstrass based on the calculus of probabilities, Commun. Kharkov Math. Soc., 13, 1-2 (1912)
[25] 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)
[26] Doha, E. H.; Bhrawy, A. H.; Saker, M. A., On the derivatives of Bernstein polynomials: an application for the solution of high even-order differential equations, Bound. Value Probl., 1, Article 829543 pp. (2011) · Zbl 1220.33006
[27] Juttler, B., The dual basis functions for the Bernstein polynomials, Adv. Comput. Math., 8, 345-352 (1998) · Zbl 0913.41004
[28] Ghanbari, B., On novel nondifferentiable exact solutions to local fractional Gardner’s equation using an effective technique, Math. Methods Appl. Sci., 44, 6, 4673-4685 (2021) · Zbl 07376945
[29] K.S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, New York, 1993. · Zbl 0789.26002
[30] Karimi, M.; Zallani, F.; Sayevand, K., Wavelet regularization strategy for the fractional inverse diffusion problem, Numer. Algorithms, 1-27 (2020)
[31] Erfanifar, R.; Sayevand, K.; Esmaeili, H., On modified two-step iterative method in the fractional sense: some applications in real world phenomena, Int. J. Comput. Math., 97, 10, 2109-2141 (2020)
[32] Sayevand, K.; Erfanifar, R.; Ghanbari, S.; Esmaeili, H., A modified Chebyshev-weighted Crank-Nicolson method for analyzing fractional sub-diffusion equations, Numer. Methods Partial Differential Equations, 37, 1, 614-625 (2020)
[33] Ghanbari, B.; Atangana, A., Some new edge detecting techniques based on fractional derivatives with non-local and non-singular kernels, Adv. Difference Equ., 2020, 1, 1-19 (2020)
[34] J.H. He, Nonlinear oscillation with fractional derivative and its applications, in: International Conference on Vibrating Engineering, Vol. 98, 1998, pp. 288-291.
[35] Mainardi, F., Fractional Calculus, in Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer Science and Business Media: Springer Science and Business Media Vienna · Zbl 0917.73004
[36] Hosseini, V. R.; Shivanian, E.; Chen, W., Local integration of 2-D fractional telegraph equation via local radial point interpolant approximation, Eur. Phys. J. Plus, 130, 2, 1-21 (2015)
[37] Hosseini, V. R.; Shivanian, E.; Chen, W., Local radial point interpolation (MLRPI) method for solving time fractional diffusion-wave equation with damping, J. Comput. Phys., 312, 307-332 (2016) · Zbl 1352.65348
[38] Ghanbari, B., On approximate solutions for a fractional prey-predator model involving the Atangana-Baleanu derivative, Adv. Difference Equ., 679 (2020), 2020
[39] Ghanbari, B., A new model for investigating the transmission of infectious diseases in a prey-predator system using a non-singular fractional derivative, Math. Methods Appl. Sci. (2021)
[40] Ghanbari, B., Chaotic behaviors of the prevalence of an infectious disease in a prey and predator system using fractional derivatives, Math. Methods Appl. Sci. (2021)
[41] Ghanbari, B., On the modeling of the interaction between tumor growth and the immune system using some new fractional and fractional-fractal operators, Adv. Difference Equ., 585 (2020), 2020
[42] Ghanbari, B., A fractional system of delay differential equation with nonsingular kernels in modeling hand-foot-mouth disease, Adv. Difference Equ., 536, 2020 (2020)
[43] Baleanu, D.; Guvenc, Z. B.; Machado, J. T., New Trends in Nanotechnology and Fractional Calculus Applications (2010), Springer Science and Business Media: Springer Science and Business Media New York
[44] Sayevand, K.; Pichaghchi, K., A novel operational matrix method for solving singularly perturbed boundary value problems of fractional multi-order, Int. J. Comput. Math., 95, 767-796 (2018) · Zbl 1390.34035
[45] Badr, M.; Yazdani, A.; Jafari, H., Stability of a finite volume element method for the time-fractional advection-diffusion equation, Numer. Methods Partial Differential Equations, 34, 5, 1459-1471 (2018) · Zbl 1407.65138
[46] Sayevand, K.; Machado, J. T.; Masti, I., On dual Bernstein polynomials and stochastic fractional integro-differential equations, Math. Methods Appl. Sci., 43, 17, 9928-9947 (2020) · Zbl 1456.60167
[47] Chen, Y.; Wu, Y.; Cui, Y.; Wang, Z.; Jin, D., Wavelet method for a class of fractional convection-diffusion equation with variable coefficients, J. Comput. Sci., 1, 3, 146-149 (2010)
[48] Yousefi, S. A.; Behroozifar, M.; Dehghan, M., Numerical solution of the nonlinear age-structured population models by using the operational matrices of Bernstein polynomials, Appl. Math. Model., 36, 3, 945-963 (2012) · Zbl 1243.65127
[49] Li, W.; Bai, L.; Chen, Y.; Santos, S.; Li, B., Solution of linear fractional partial differential equations based on the operator matrix of fractional Bernstein polynomials and error correction, Int. J. Innovative Comput. Appl., 14, 211-226 (2018)
[50] Gasca, M.; Sauer, T., On the history of multivariate polynomial interpolation, numerical analysis: Historical developments in the 20th century, 135-147 (2001)
[51] Ferras, L. L.; Ford, N. J.; Morgado, M. L.; Rebelo, M., A numerical method for the solution of the time-fractional diffusion equation, Comput. Sci. Appl., 14, 117-131 (2014)
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