# American Institute of Mathematical Sciences

• Previous Article
A robust computational framework for analyzing fractional dynamical systems
• DCDS-S Home
• This Issue
• Next Article
Comprehensive analysis of integer-order, Caputo-Fabrizio (CF) and Atangana-Baleanu (ABC) fractional time derivative for MHD Oldroyd-B fluid with slip effect and time dependent boundary condition
October  2021, 14(10): 3747-3761. doi: 10.3934/dcdss.2020435

## Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative

 1 Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran 2 Department of Mathematical Sciences, University of South Africa, UNISA 0003, South Africa 3 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 110122, Taiwan

* Corresponding author: H. Jafari email: jafari.usern@gmail.com

Received  January 2020 Revised  February 2020 Published  October 2021 Early access  November 2020

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.

Citation: S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3747-3761. doi: 10.3934/dcdss.2020435
##### References:

show all references

##### References:
(Example 6.1) The exact solution with $\alpha = 1$ and numerical solution with $\alpha = 0.99$
(Example 6.1) The exact and approximate solutions with different values of $\alpha$ and $M = 5$ at $t = 0.5$
(Example 6.2) The exact solution with $\alpha = 1$ and numerical solution with $\alpha = 0.99$
(Example 6.2) The exact and the approximate solutions for $M = 5$ at $t = 0.5$
(Example 6.2) The exact solution with $\alpha = 1$ and numerical solution with $\alpha = 0.99$
(Example 6.3) The exact and approximate solutions for $M = 5$ at $t = 0.5$
Numerical results of the absolute error when $\alpha = 0.99$ and $t = 1$ for Example 6.1
 $x$ M=3 M=6 $0.0$ $4.39878e-3$ $7.19623e-4$ $0.1$ $2.83179e-3$ $6.56846e-4$ $0.2$ $3.61572e-3$ $1.03531e-3$ $0.3$ $1.44208e-3$ $8.15792e-4$ $0.4$ $8.62515e-4$ $2.26231e-4$ $0.5$ $1.46612e-3$ $5.53976e-4$ $0.6$ $2.43959e-4$ $1.33992e-3$ $0.7$ $3.58188e-3$ $1.91911e-3$ $0.8$ $6.64254e-3$ $2.04788e-3$ $0.9$ $6.52617e-3$ $1.47985e-3$ $1.0$ $3.29430e-4$ $6.37627e-5$
 $x$ M=3 M=6 $0.0$ $4.39878e-3$ $7.19623e-4$ $0.1$ $2.83179e-3$ $6.56846e-4$ $0.2$ $3.61572e-3$ $1.03531e-3$ $0.3$ $1.44208e-3$ $8.15792e-4$ $0.4$ $8.62515e-4$ $2.26231e-4$ $0.5$ $1.46612e-3$ $5.53976e-4$ $0.6$ $2.43959e-4$ $1.33992e-3$ $0.7$ $3.58188e-3$ $1.91911e-3$ $0.8$ $6.64254e-3$ $2.04788e-3$ $0.9$ $6.52617e-3$ $1.47985e-3$ $1.0$ $3.29430e-4$ $6.37627e-5$
Numerical results of the absolute error when $\alpha = 0.99$, $t = 1$ for Example 6.2
 $x$ M=3 M=6 $0.0$ $2.08503e-2$ $3.99527e-5$ $0.1$ $1.05799e-2$ $7.02578e-5$ $0.2$ $1.21467e-2$ $2.89318e-5$ $0.3$ $4.94776e-3$ $1.02674e-4$ $0.4$ $2.35280e-4$ $1.53367e-4$ $0.5$ $2.36604e-3$ $1.16798e-4$ $0.6$ $1.08676e-2$ $5.12801e-5$ $0.7$ $2.18851e-2$ $2.11059e-5$ $0.8$ $2.91950e-2$ $3.51774e-7$ $0.9$ $2.49148e-2$ $7.30602e-5$ $1.0$ $3.02368e-8$ $2.05050e-6$
 $x$ M=3 M=6 $0.0$ $2.08503e-2$ $3.99527e-5$ $0.1$ $1.05799e-2$ $7.02578e-5$ $0.2$ $1.21467e-2$ $2.89318e-5$ $0.3$ $4.94776e-3$ $1.02674e-4$ $0.4$ $2.35280e-4$ $1.53367e-4$ $0.5$ $2.36604e-3$ $1.16798e-4$ $0.6$ $1.08676e-2$ $5.12801e-5$ $0.7$ $2.18851e-2$ $2.11059e-5$ $0.8$ $2.91950e-2$ $3.51774e-7$ $0.9$ $2.49148e-2$ $7.30602e-5$ $1.0$ $3.02368e-8$ $2.05050e-6$
Numerical results of the absolute error when $\alpha = 0.98$ and $t = 1$ for Example 6.3
 $x$ M=3 M=6 $0.0$ $9.58422e-3$ $0.00000$ $0.1$ $9.47883e-3$ $1.32252e-3$ $0.2$ $1.1959e-2$ $2.51929e-3$ $0.3$ $7.30440e-3$ $3.47139e-3$ $0.4$ $2.89100e-3$ $4.07656e-3$ $0.5$ $3.24315e-3$ $4.28296e-3$ $0.6$ $9.53273e-3$ $4.07656e-3$ $0.7$ $1.94066e-2$ $3.47139e-3$ $0.8$ $2.71593e-2$ $2.51929e-3$ $0.9$ $2.42335e-2$ $1.32252e-3$ $1.0$ $4.19753e-16$ $4.50522e-17$
 $x$ M=3 M=6 $0.0$ $9.58422e-3$ $0.00000$ $0.1$ $9.47883e-3$ $1.32252e-3$ $0.2$ $1.1959e-2$ $2.51929e-3$ $0.3$ $7.30440e-3$ $3.47139e-3$ $0.4$ $2.89100e-3$ $4.07656e-3$ $0.5$ $3.24315e-3$ $4.28296e-3$ $0.6$ $9.53273e-3$ $4.07656e-3$ $0.7$ $1.94066e-2$ $3.47139e-3$ $0.8$ $2.71593e-2$ $2.51929e-3$ $0.9$ $2.42335e-2$ $1.32252e-3$ $1.0$ $4.19753e-16$ $4.50522e-17$
 [1] Editorial Office. WITHDRAWN: Fractional diffusion equation described by the Atangana-Baleanu fractional derivative and its approximate solution. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2020173 [2] Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Ebraheem O. Alzahrani. A fractional model for the dynamics of tuberculosis (TB) using Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 937-956. doi: 10.3934/dcdss.2020055 [3] Ilknur Koca. Numerical analysis of coupled fractional differential equations with Atangana-Baleanu fractional derivative. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 475-486. doi: 10.3934/dcdss.2019031 [4] G. M. Bahaa. Generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's derivatives and application. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 485-501. doi: 10.3934/dcdss.2020027 [5] Kashif Ali Abro, Ilyas Khan. MHD flow of fractional Newtonian fluid embedded in a porous medium via Atangana-Baleanu fractional derivatives. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 377-387. doi: 10.3934/dcdss.2020021 [6] Muhammad Bilal Riaz, Syed Tauseef Saeed. Comprehensive analysis of integer-order, Caputo-Fabrizio (CF) and Atangana-Baleanu (ABC) fractional time derivative for MHD Oldroyd-B fluid with slip effect and time dependent boundary condition. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3719-3746. doi: 10.3934/dcdss.2020430 [7] Qing Tang. On an optimal control problem of time-fractional advection-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 761-779. doi: 10.3934/dcdsb.2019266 [8] Assyr Abdulle. Multiscale methods for advection-diffusion problems. Conference Publications, 2005, 2005 (Special) : 11-21. doi: 10.3934/proc.2005.2005.11 [9] Lena-Susanne Hartmann, Ilya Pavlyukevich. Advection-diffusion equation on a half-line with boundary Lévy noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 637-655. doi: 10.3934/dcdsb.2018200 [10] Michael Taylor. Random walks, random flows, and enhanced diffusivity in advection-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1261-1287. doi: 10.3934/dcdsb.2012.17.1261 [11] Alexandre Caboussat, Roland Glowinski. A Numerical Method for a Non-Smooth Advection-Diffusion Problem Arising in Sand Mechanics. Communications on Pure & Applied Analysis, 2009, 8 (1) : 161-178. doi: 10.3934/cpaa.2009.8.161 [12] Patrick Henning, Mario Ohlberger. The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift. Networks & Heterogeneous Media, 2010, 5 (4) : 711-744. doi: 10.3934/nhm.2010.5.711 [13] Patrick Henning, Mario Ohlberger. A-posteriori error estimate for a heterogeneous multiscale approximation of advection-diffusion problems with large expected drift. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1393-1420. doi: 10.3934/dcdss.2016056 [14] Janos Kollar. Polynomials with integral coefficients, equivalent to a given polynomial. Electronic Research Announcements, 1997, 3: 17-27. [15] Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic & Related Models, 2011, 4 (4) : 1159-1191. doi: 10.3934/krm.2011.4.1159 [16] Iman Malmir. Caputo fractional derivative operational matrices of Legendre and Chebyshev wavelets in fractional delay optimal control. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021013 [17] Joshua Du, Jun Ji. An integral representation of the determinant of a matrix and its applications. Conference Publications, 2005, 2005 (Special) : 225-232. doi: 10.3934/proc.2005.2005.225 [18] Kaouther Bouchama, Yacine Arioua, Abdelkrim Merzougui. The Numerical Solution of the space-time fractional diffusion equation involving the Caputo-Katugampola fractional derivative. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021026 [19] Carlos Fresneda-Portillo. A new family of boundary-domain integral equations for the diffusion equation with variable coefficient in unbounded domains. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5097-5114. doi: 10.3934/cpaa.2020228 [20] Gennaro Infante. Positive and increasing solutions of perturbed Hammerstein integral equations with derivative dependence. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 691-699. doi: 10.3934/dcdsb.2019261

2020 Impact Factor: 2.425