Solving fractional advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. (English) Zbl 1473.35635

Summary: In recent years, a new definition of fractional derivative which has a nonlocal and non-singular kernel has been proposed by Atangana and Baleanu. This new definition is called the Atangana-Baleanu derivative. In this paper, we present a new technique to obtain the numerical solution of advection-diffusion equation containing Atangana-Baleanu derivative. For this purpose, we use the operational matrix of fractional integral based on Genocchi polynomials. An error bound is given for the approximation of a bivariate function using Genocchi polynomials. Finally, some examples are given to illustrate the applicability and efficiency of the proposed method.


35R11 Fractional partial differential equations
35A35 Theoretical approximation in context of PDEs
35K15 Initial value problems for second-order parabolic equations
Full Text: DOI


[1] A. Alsaedi, D. Baleanu, S. Etemad and S. Rezapour, On coupled systems of time-fractional differential problems by using a new fractional derivative, Journal of Function Spaces, 2016 (2015), 8 pp. · Zbl 1367.34006
[2] A. Atangana; D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20, 763-769 (2016)
[3] A. Atangana and D. Baleanu, Caputo-Fabrizio derivative applied to groundwater flow within a confined aquifer, J. Eng. Mech., 143 (2016).
[4] A. Atangana and Z. Hammouch, Fractional calculus with power law: The cradle of our ancestors, Eur. Phys. J. Plus, 134 (2019), 429.
[5] A. Atangana; I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos, Solitons and Fractals, 89, 447-454 (2016) · Zbl 1360.34150
[6] D. Baleanu, H. Mohammadi and S. Rezapour, Analysis of the model of HIV-1 infection of CD4+ T-cell with a new approach of fractional derivative, Adv. Differ. Equ., 2020 (2020).
[7] M. Caputo; M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract. Differ. Appl., 2, 1-11 (2016)
[8] Y. Chatibi; E. H. El Kinani; A. Ouhadan, Variational calculus involving nonlocal fractional derivative with Mittag-Leffler kernel, Chaos, Solitons and Fractals, 118, 117-121 (2019) · Zbl 1442.49026
[9] M. Dehghan, Weighted finite difference techniques for the one-dimensional advection-diffusion equation, Appl. Math. Comput., 147, 307-319 (2004) · Zbl 1034.65069
[10] R. M. Ganji and H. Jafari, A numerical approach for multi-variable orders differential equations using Jacobi polynomials, International Journal of Applied and Computational Mathematics, 5 (2019). · Zbl 1411.65100
[11] R. M. Ganji; H. Jafari, Numerical solution of variable order integro-differential equations, Advanced Math. Models & Applications, 4, 64-69 (2019)
[12] R. M. Ganji and H. Jafari and A. R. Adem, A numerical scheme to solve variable order diffusion-wave equations, Thermal Science, (2019), 371-371.
[13] R. M. Ganji and H. Jafari and D. Baleanu, A new approach for solving multi variable orders differential equations with Mittag-Leffler kernel, Chaos, Solitons and Fractals, 130 (2020), 109405.
[14] M. M. Ghalib; A. A. Zafar; Z. Hammouch; M. B. Riaz; K. Shabbir, Analytical results on the unsteady rotational flow of fractional-order non-Newtonian fluids with shear stress on the boundary, Discrete and Continuous Dynamical Systems - S, 13, 683-693 (2020) · Zbl 1434.76049
[15] M. M. Ghalib, A. A. Zafar, M. B. Riaz, Z. Hammouch and K. Shabbir, Analytical approach for the steady MHD conjugate viscous fluid flow in a porous medium with nonsingular fractional derivative, Physica A: Statistical Mechanics and its Applications, (2020), 123941.
[16] A. Jajarmi, S. Arshad and D. Baleanu, A new fractional modelling and control strategy for the outbreak of dengue fever, Physica A: Statistical Mechanics and its Applications, 535 (2019).
[17] A. Jajarmi, D. Baleanu, S. S. Sajjadi and J. H. Asad, A new feature of the fractional Euler-Lagrange equations for a coupled oscillator using a nonsingular operator approach, Frontiers in Physics, 7 (2019).
[18] I. Koca, Numerical analysis of coupled fractional differential equations with Atangana-Baleanu fractional derivative, Discrete and Continuous Dynamical Systems - S, 12, 475-486 (2018) · Zbl 1423.34013
[19] A. Mohebbi; M. Dehghan, High-order compact solution of the one-dimensional heat and advection-diffusion equations, Appl Math Model, 34, 3071-3084 (2010) · Zbl 1201.65183
[20] S. Nemati; P. M. Lima; Y. Ordokhani, Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials, Journal of Computational and Applied Mathematics, 242, 53-69 (2013) · Zbl 1255.65248
[21] F. Ozpinar; F. B. M. Belgacem, The discrete homotopy perturbation Sumudu transform method for solving partial difference equations, Discrete and Continuous Dynamical Systems - S, 12, 615-624 (2019) · Zbl 1417.39029
[22] K. M. Owolabi; Z. Hammouch, Spatiotemporal patterns in the Belousov-Zhabotinskii reaction systems with Atangana-Baleanu fractional order derivative, Physica A: Statistical Mechanics and its Applications, 523, 1072-1090 (2019)
[23] S. S. Roshan; H. Jafari; D. Baleanu, Solving FDEs with Caputo-Fabrizio derivative by operational matrix based on Genocchi polynomials, Mathematical Methods in the Applied Sciences, 4, 9134-9141 (2018) · Zbl 1406.34017
[24] H. M. Srivastava; K. M. Saad, Some new models of the time-fractional gas dynamics equation, Advanced Math. Models & Applications, 3, 5-17 (2018)
[25] H. Tajadodi, A Numerical approach of fractional advection-diffusion equation with Atangana-Baleanu derivative, Chaos, Solitons and Fractals, 130 (2020), 109527.
[26] S. Ucar; E. Ucar; N. Ozdemir; Z. Hammouch, Mathematical analysis and numerical simulation for a smoking model with Atangana-Baleanu derivative, Chaos, Solitons and Fractals, 118, 300-306 (2019) · Zbl 1442.92074
[27] M. Zerroukat; K. Djidjeli; A. Charafi, Explicit and implicit meshless methods for linear advection-diffusion-type partial differential equations, Int. J. Numer. Meth. Eng., 48, 19-35 (2000) · Zbl 0968.65053
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.