Skip to main content
banner image
No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
The full text of this article is not currently available.
MS Mohamed and YS Hamed, “Solving the convection-diffusion equation by means of the optimal q-homotopy analysis method (Oq-HAM),” Results in Physics 6, 20-25 (2016).
A Mojtabi and MO Deville, “One-dimensional linear advection-diffusion equation: Analytical and finite element solutions,” Computers Fluids 107, 189-195 (2015).
J Grant and M Wilkinson, “Advection-diffusion equation with absorbing boundary,” J. Stat. Phys. 160, 622-635 (2015).
KSM Essa, AA Marouf, MS El-Otaify, AS Mohamed, and G Ismail, “New technique for solving the advection-diffusion equation in three dimensions using Laplace and Fourier transforms,” J. Appl. Computat. Math. 4(6), 1000272 (2015).
D Buske, B Bodmann, MTMB Vilhena, and RS de Quadros, “On the solution of the coupled advection-diffusion and Navier-Stokes equations,” Am. J. Environ. Enging. 5(1A), 1-8 (2015).
JM Stockie, “The mathematics of atmospheric dispersion modeling,” SIAM Rev. 53, 349-372 (2011).
VS Aswin, A Awasthi, and C Anu, “A comparative study of numerical schemes for convection-diffusion equation,” Procedia Engineering 127, 621-627 (2015).
M Mazaheri, JMV Samani, and HMV Samani, “Analytical solution to one-dimensional advection-diffusion equation with several point sources through arbitrary time-dependent emission rate patterns,” J. Agr. Sci. Tech. 15, 1231-1245 (2013).
DM Moreira, MT Vilhena, D Buske, and T Tirabassi, “The state-of-art of the GILTT method to simulate pollutant dispersion in the atmosphere,” Atmosheric research 92, 1-17 (2009).
RL Magin, Fractional calculus in bioengineering (Begell House Publishers, Redding, CT, USA, 2006).
VV Uchaikin, Fractional derivatives for Physicists and Engineers (Springer, Berlin Germany, 2013).
R Caponetto, Fractional Order Systems (Modelling and Control Applications) (World Scientific, 2010).
D Baleanu, ZB Guvenc, and JAT Machado, New Trends in Nanotechnology and Fractional Calculus Applications, XI, 1st ed. (ISBN Springer, 2010), Vol. 531.
L Feng, P Zhuang, F Liu, I Turner, and Q Yang, “Second oreder approximation for the space fractional diffusion equation with variable coefficient,” Progr. Fract. Differ. Appl. 1(1), 23-35 (2015).
S Momani and Z Obidat, “Numerical solutions of the space-time fractional advection-dispersion equation,” Numer. Meth. Part. D. E. 24, 1416-1429 (2008).
P Zhuang, F Liu, V Anh, and I Turner, “Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term,” SIAM J. Numer. Anal. 47, 1760-1781 (2009).
XJ Yang, D Baleanu, Y Khan, and ST Mohyud-Din, “Local fractional variational iteration method for diffusion and wave equatios on Cantor sets,” Rom. J. Phys. 59, 36-48 (2014).
Y Povstenko, “Generalized boundary conditions for the time-fractional advection diffusion equation,” Entropy 17, 4028-4039 (2015).
Y Povstenko and J Klekot, “The Dirichlet problem for the time-fractional advection-diffusion equation in a half-space,” J. Appl. Math. Comutat. Mech. 14(2), 73-83 (2015).
M Caputo and M Fabrizio, “A new definition of fractional derivative without singular kernel,” Progr. Fract. Differ. Appl. 1(2), 73-85 (2015).
XJ Yang, D Baleanu, and WP Zhong, “Approximate solutions for diffusion equation on Cantor-space time,” Proc. Romanian Acad. A 14, 127-133 (2013).
J Hristov, “Heat-balance integral to fractional (half-time) heat diffusion sub-model,” Therm. Sci. 14, 291-316 (2010).
A Atangana and JJ Nieto, “Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel,” Avances Mech. Enging. 7(10), 1-7 (2015).
GSF Frederico and MJ Lazo, “Fractional Noether’s theorem with classical and Caputo derivatives: constants of motion for non-conservative systems,” Nonlinear Dyn., DOI 10.10007/s110171-016-2727-z (2016).
A Shakeel, S Ahmad, H Khan, and D Vieru, “Solutions with Wright functions for time fractional convection flow near a heated vertical plate,” Advances in Difference. Equations Article 51, DOI: 10.1186/s13662-016-0775-9 (2016).
K Wang and S Liu, “Analytical study of time –fractional Navier-Stokes equation by using transform methods,” Advances in Difference. Equations Article 61, DOI: 10.1186/s13662-016-0783-9 (2016).
AP Prudnikov, YA Brychkov, and OI Marichev, Integrals and Series, Elementary functions (Gordon and Breach, Amsterdam, 1986), Vol. 1.

Data & Media loading...


Article metrics loading...



The time-fractional advection-diffusion equation with Caputo-Fabrizio fractional derivatives (fractional derivatives without singular kernel) is considered under the time-dependent emissions on the boundary and the first order chemical reaction. The non-dimensional problem is formulated by using suitable dimensionless variables and the fundamental solutions to the Dirichlet problem for the fractional advection-diffusion equation are determined using the integral transforms technique. The fundamental solutions for the ordinary advection-diffusion equation, fractional and ordinary diffusion equation are obtained as limiting cases of the previous model. Using Duhamel’s principle, the analytical solutions to the Dirichlet problem with time-dependent boundary pulses have been obtained. The influence of the fractional parameter and of the drift parameter on the solute concentration in various spatial positions was analyzed by numerical calculations. It is found that the variation of the fractional parameter has a significant effect on the solute concentration, namely, the memory effects lead to the retardation of the mass transport.


Full text loading...


Access Key

  • FFree Content
  • OAOpen Access Content
  • SSubscribed Content
  • TFree Trial Content
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd