American Institute of Mathematical Sciences

Advanced
  • 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

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 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  (Example 6.1) The exact and approximate solutions with different values of $ \alpha $ and $ M = 5 $ at $ t = 0.5 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  (Example 6.2) The exact solution with $ \alpha = 1 $ and numerical solution with $ \alpha = 0.99 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  (Example 6.2) The exact and the approximate solutions for $ M = 5 $ at $ t = 0.5 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  (Example 6.2) The exact solution with $ \alpha = 1 $ and numerical solution with $ \alpha = 0.99 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  (Example 6.3) The exact and approximate solutions for $ M = 5 $ at $ t = 0.5 $
Figure Options

Download full-size image

Download as PowerPoint slide

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 $
Table Options

Download as excel

$ 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 $
Table Options

Download as excel

$ 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 $
Table Options

Download as excel

$ 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

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences