Figure 1.
(Example 6.1) The exact solution with $ \alpha = 1 $ and numerical solution with $ \alpha = 0.99 $
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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 |
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 and 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
R. M. Ganji and H. Jafari,
Numerical solution of variable order integro–differential equations, Advanced Math. Models & Applications, 4 (2019), 64-69.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [23] |
P. Roache, Computational Fluid Dynamics, Hermosa Press, Albuquerque, NM, 1972.
|
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
R. M. Ganji and H. Jafari,
Numerical solution of variable order integro–differential equations, Advanced Math. Models & Applications, 4 (2019), 64-69.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [23] |
P. Roache, Computational Fluid Dynamics, Hermosa Press, Albuquerque, NM, 1972.
|
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
Figure 2.
(Example 6.1) The exact and approximate solutions with different values of $ \alpha $ and $ M = 5 $ at $ t = 0.5 $
Figure 3.
(Example 6.2) The exact solution with $ \alpha = 1 $ and numerical solution with $ \alpha = 0.99 $
Figure 5.
(Example 6.2) The exact solution with $ \alpha = 1 $ and numerical solution with $ \alpha = 0.99 $
Table 1.
Numerical results of the absolute error when $ \alpha = 0.99 $ and $ t = 1 $ for Example 6.1
| M=3 | M=6 | |
| M=3 | M=6 | |
| M=3 | M=6 | |
| M=3 | M=6 | |
Table 3.
Numerical results of the absolute error when $ \alpha = 0.98 $ and $ t = 1 $ for Example 6.3
| M=3 | M=6 | |
| M=3 | M=6 | |
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