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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

S. Sadeghi 1, , H. Jafari 1,2,3,, and  S. Nemati 1,

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.

Keywords: Atangana-Baleanu derivative, Atangana-Baleanu integral, Advection-Diffusion equation, Operational matrix, Genocchi polynomials.
Mathematics Subject Classification: Primary: 35R11; Secondary: 34K28.
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:
[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. doi: 10.1155/2016/4626940.  Google Scholar

[2]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non–singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A.  Google Scholar

[3]

A. Atangana and D. Baleanu, Caputo–Fabrizio derivative applied to groundwater flow within a confined aquifer, J. Eng. Mech., 143 (2016). doi: 10.1061/(ASCE)EM.1943-7889.0001091.  Google Scholar

[4]

A. Atangana and Z. Hammouch, Fractional calculus with power law: The cradle of our ancestors, Eur. Phys. J. Plus, 134 (2019), 429. doi: 10.1140/epjp/i2019-12777-8.  Google Scholar

[5]

A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order, Chaos, Solitons and Fractals, 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012.  Google Scholar

[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). doi: 10.1186/s13662-020-02544-w.  Google Scholar

[7]

M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract. Differ. Appl., 2 (2016), 1-11.  doi: 10.18576/pfda/020101.  Google Scholar

[8]

Y. Chatibi, E. H. El Kinani and A. Ouhadan, Variational calculus involving nonlocal fractional derivative with Mittag–Leffler kernel, Chaos, Solitons and Fractals, 118 (2019), 117-121.  doi: 10.1016/j.chaos.2018.11.017.  Google Scholar

[9]

M. Dehghan, Weighted finite difference techniques for the one–dimensional advection–diffusion equation, Appl. Math. Comput., 147 (2004), 307-319.  doi: 10.1016/S0096-3003(02)00667-7.  Google Scholar

[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). doi: 10.1007/s40819-019-0610-6.  Google Scholar

[11]

R. M. Ganji and H. Jafari, Numerical solution of variable order integro–differential equations, Advanced Math. Models & Applications, 4 (2019), 64-69.   Google Scholar

[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. doi: 10.2298/TSCI190729371M.  Google Scholar

[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. doi: 10.1016/j.chaos.2019.109405.  Google Scholar

[14]

M. M. Ghalib, A. A. Zafar, Z. Hammouch, M. B. Riaz and 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 (2020), 683-693.  doi: 10.3934/dcdss.2020037.  Google Scholar

[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. doi: 10.1016/j.physa.2019.123941.  Google Scholar

[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). doi: 10.1016/j.physa.2019.122524.  Google Scholar

[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). doi: 10.3389/fphy.2019.00196.  Google Scholar

[18]

I. Koca, Numerical analysis of coupled fractional differential equations with Atangana–Baleanu fractional derivative, Discrete and Continuous Dynamical Systems - S, 12 (2018), 475-486.  doi: 10.3934/dcdss.2019031.  Google Scholar

[19]

A. Mohebbi and M. Dehghan, High-order compact solution of the one–dimensional heat and advection–diffusion equations, Appl Math Model, 34 (2010), 3071-3084.  doi: 10.1016/j.apm.2010.01.013.  Google Scholar

[20]

S. Nemati, P. M. Lima and Y. Ordokhani, Numerical solution of a class of two–dimensional nonlinear Volterra integral equations using Legendre polynomials, Journal of Computational and Applied Mathematics, 242 (2013), 53-69.  doi: 10.1016/j.cam.2012.10.021.  Google Scholar

[21]

F. Ozpinar and F. B. M. Belgacem, The discrete homotopy perturbation Sumudu transform method for solving partial difference equations, Discrete and Continuous Dynamical Systems - S, 12 (2019), 615-624.  doi: 10.3934/dcdss.2019039.  Google Scholar

[22]

K. M. Owolabi and Z. Hammouch, Spatiotemporal patterns in the Belousov–Zhabotinskii reaction systems with Atangana–Baleanu fractional order derivative, Physica A: Statistical Mechanics and its Applications, 523 (2019), 1072-1090.  doi: 10.1016/j.physa.2019.04.017.  Google Scholar

[23] P. Roache, Computational Fluid Dynamics, Hermosa Press, Albuquerque, NM, 1972.   Google Scholar
[24]

S.S. Roshan, H. Jafari and D. Baleanu, Solving FDEs with Caputo–Fabrizio derivative by operational matrix based on Genocchi polynomials, Mathematical Methods in the Applied Sciences, 4 (2018), 9134-9141.  doi: 10.1002/mma.5098.  Google Scholar

[25]

H. M. Srivastava and K. M. Saad, Some new models of the time-fractional gas dynamics equation, Advanced Math. Models & Applications, 3 (2018), 5-17.   Google Scholar

[26]

H. Tajadodi, A Numerical approach of fractional advection–diffusion equation with Atangana–Baleanu derivative, Chaos, Solitons and Fractals, 130 (2020), 109527. doi: 10.1016/j.chaos.2019.109527.  Google Scholar

[27]

S. Ucar, E. Ucar, N. Ozdemir and Z. Hammouch, Mathematical analysis and numerical simulation for a smoking model with Atangana–Baleanu derivative, Chaos, Solitons and Fractals, 118 (2019), 300-306.  doi: 10.1016/j.chaos.2018.12.003.  Google Scholar

[28]

M. Zerroukat, K. Djidjeli and A. Charafi, Explicit and implicit meshless methods for linear advection–diffusion–type partial differential equations, Int. J. Numer. Meth. Eng., 48 (2000), 19-35.  doi: 10.1002/(SICI)1097-0207(20000510)48:1<19::AID-NME862>3.0.CO;2-3.  Google Scholar

show all references

References:
[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. doi: 10.1155/2016/4626940.  Google Scholar

[2]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non–singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A.  Google Scholar

[3]

A. Atangana and D. Baleanu, Caputo–Fabrizio derivative applied to groundwater flow within a confined aquifer, J. Eng. Mech., 143 (2016). doi: 10.1061/(ASCE)EM.1943-7889.0001091.  Google Scholar

[4]

A. Atangana and Z. Hammouch, Fractional calculus with power law: The cradle of our ancestors, Eur. Phys. J. Plus, 134 (2019), 429. doi: 10.1140/epjp/i2019-12777-8.  Google Scholar

[5]

A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order, Chaos, Solitons and Fractals, 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012.  Google Scholar

[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). doi: 10.1186/s13662-020-02544-w.  Google Scholar

[7]

M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract. Differ. Appl., 2 (2016), 1-11.  doi: 10.18576/pfda/020101.  Google Scholar

[8]

Y. Chatibi, E. H. El Kinani and A. Ouhadan, Variational calculus involving nonlocal fractional derivative with Mittag–Leffler kernel, Chaos, Solitons and Fractals, 118 (2019), 117-121.  doi: 10.1016/j.chaos.2018.11.017.  Google Scholar

[9]

M. Dehghan, Weighted finite difference techniques for the one–dimensional advection–diffusion equation, Appl. Math. Comput., 147 (2004), 307-319.  doi: 10.1016/S0096-3003(02)00667-7.  Google Scholar

[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). doi: 10.1007/s40819-019-0610-6.  Google Scholar

[11]

R. M. Ganji and H. Jafari, Numerical solution of variable order integro–differential equations, Advanced Math. Models & Applications, 4 (2019), 64-69.   Google Scholar

[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. doi: 10.2298/TSCI190729371M.  Google Scholar

[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. doi: 10.1016/j.chaos.2019.109405.  Google Scholar

[14]

M. M. Ghalib, A. A. Zafar, Z. Hammouch, M. B. Riaz and 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 (2020), 683-693.  doi: 10.3934/dcdss.2020037.  Google Scholar

[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. doi: 10.1016/j.physa.2019.123941.  Google Scholar

[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). doi: 10.1016/j.physa.2019.122524.  Google Scholar

[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). doi: 10.3389/fphy.2019.00196.  Google Scholar

[18]

I. Koca, Numerical analysis of coupled fractional differential equations with Atangana–Baleanu fractional derivative, Discrete and Continuous Dynamical Systems - S, 12 (2018), 475-486.  doi: 10.3934/dcdss.2019031.  Google Scholar

[19]

A. Mohebbi and M. Dehghan, High-order compact solution of the one–dimensional heat and advection–diffusion equations, Appl Math Model, 34 (2010), 3071-3084.  doi: 10.1016/j.apm.2010.01.013.  Google Scholar

[20]

S. Nemati, P. M. Lima and Y. Ordokhani, Numerical solution of a class of two–dimensional nonlinear Volterra integral equations using Legendre polynomials, Journal of Computational and Applied Mathematics, 242 (2013), 53-69.  doi: 10.1016/j.cam.2012.10.021.  Google Scholar

[21]

F. Ozpinar and F. B. M. Belgacem, The discrete homotopy perturbation Sumudu transform method for solving partial difference equations, Discrete and Continuous Dynamical Systems - S, 12 (2019), 615-624.  doi: 10.3934/dcdss.2019039.  Google Scholar

[22]

K. M. Owolabi and Z. Hammouch, Spatiotemporal patterns in the Belousov–Zhabotinskii reaction systems with Atangana–Baleanu fractional order derivative, Physica A: Statistical Mechanics and its Applications, 523 (2019), 1072-1090.  doi: 10.1016/j.physa.2019.04.017.  Google Scholar

[23] P. Roache, Computational Fluid Dynamics, Hermosa Press, Albuquerque, NM, 1972.   Google Scholar
[24]

S.S. Roshan, H. Jafari and D. Baleanu, Solving FDEs with Caputo–Fabrizio derivative by operational matrix based on Genocchi polynomials, Mathematical Methods in the Applied Sciences, 4 (2018), 9134-9141.  doi: 10.1002/mma.5098.  Google Scholar

[25]

H. M. Srivastava and K. M. Saad, Some new models of the time-fractional gas dynamics equation, Advanced Math. Models & Applications, 3 (2018), 5-17.   Google Scholar

[26]

H. Tajadodi, A Numerical approach of fractional advection–diffusion equation with Atangana–Baleanu derivative, Chaos, Solitons and Fractals, 130 (2020), 109527. doi: 10.1016/j.chaos.2019.109527.  Google Scholar

[27]

S. Ucar, E. Ucar, N. Ozdemir and Z. Hammouch, Mathematical analysis and numerical simulation for a smoking model with Atangana–Baleanu derivative, Chaos, Solitons and Fractals, 118 (2019), 300-306.  doi: 10.1016/j.chaos.2018.12.003.  Google Scholar

[28]

M. Zerroukat, K. Djidjeli and A. Charafi, Explicit and implicit meshless methods for linear advection–diffusion–type partial differential equations, Int. J. Numer. Meth. Eng., 48 (2000), 19-35.  doi: 10.1002/(SICI)1097-0207(20000510)48:1<19::AID-NME862>3.0.CO;2-3.  Google Scholar

Figure 1.  (Example 6.1) The exact solution with $ \alpha = 1 $ and numerical solution with $ \alpha = 0.99 $
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Figure 2.  (Example 6.1) The exact and approximate solutions with different values of $ \alpha $ and $ M = 5 $ at $ t = 0.5 $
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Figure 3.  (Example 6.2) The exact solution with $ \alpha = 1 $ and numerical solution with $ \alpha = 0.99 $
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Figure 4.  (Example 6.2) The exact and the approximate solutions for $ M = 5 $ at $ t = 0.5 $
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Figure 5.  (Example 6.2) The exact solution with $ \alpha = 1 $ and numerical solution with $ \alpha = 0.99 $
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Figure 6.  (Example 6.3) The exact and approximate solutions for $ M = 5 $ at $ t = 0.5 $
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Table 1.  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 $
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$ 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 $
Table 2.  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 $
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$ 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 $
Table 3.  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 $
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$ 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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