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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 and Continuous Dynamical Systems - S, 2021, 14 (10) : 3747-3761. doi: 10.3934/dcdss.2020435
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##### 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 $0.0$ $9.58422e-3$ $0.00000$ $0.1$ $9.47883e-3$ $1.32252e-3$ $0.2$ $1.1959e-2$ $2.51929e-3$ $0.3$ $7.30440e-3$ $3.47139e-3$ $0.4$ $2.89100e-3$ $4.07656e-3$ $0.5$ $3.24315e-3$ $4.28296e-3$ $0.6$ $9.53273e-3$ $4.07656e-3$ $0.7$ $1.94066e-2$ $3.47139e-3$ $0.8$ $2.71593e-2$ $2.51929e-3$ $0.9$ $2.42335e-2$ $1.32252e-3$ $1.0$ $4.19753e-16$ $4.50522e-17$

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