Asymptotically Self-Similar Solution for the Convection-Diffusion Equation

2014 ◽  
Vol 623 ◽  
pp. 97-100 ◽  
Author(s):  
Xue Yu ◽  
Qiu Sheng Zhang
Keyword(s):  
Diffusion Equation ◽  
Global Solutions ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Self Similar Solution ◽  
The Cauchy Problem ◽  
Self Similar ◽  
Diffusion And Convection ◽  
Physical Phenomena ◽  
Physical Quantities

We study the Cauchy problem for the convection-diffusion equation, which describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to diffusion and convection processes. For, we show the continuous dependence upon the initial data. Moreover, asymptotically self–similar global solutions are investigated with nonhomogeneous initial date.

scholarly journals Analytical solution of the time fractional diffusion equation and fractional convection-diffusion equation

2018 ◽  
Vol 65 (1) ◽  
pp. 82 ◽  
Author(s):  
Francisco Gomez ◽  
Victor Morales ◽  
Marco Taneco
Keyword(s):  
Diffusion Equation ◽  
Analytical Solutions ◽  
Fourier Transforms ◽  
Time Derivative ◽  
Fractional Diffusion ◽  
Transport Processes ◽  
Diffusion Equations ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Diffusion And Convection

In this paper, we obtain analytical solutions for the time-fractional diffusion and time-fractional convection-diffusion equations. These equations are obtained from the standard equations by replacing the time derivative with a fractional derivative of order $\alpha$. Fractional operators of type Liouville-Caputo, Atangana-Baleanu-Caputo, fractional conformable derivative in Liouville-Caputo sense and Atangana-Koca-Caputo were used to model diffusion and convection-diffusion equation. The Laplace and Fourier transforms were applied to obtain the analytical solutions for the fractional order diffusion and convection-diffusion equations. The solutions obtained can be useful to understand the modeling of anomalous diffusive, subdiffusive systems and super-diffusive systems, transport processes, random walk and wave propagation phenomenon.

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