Numerical simulation of advection–diffusion equation with caputo-fabrizio time fractional derivative in cylindrical domains: Applications of pseudo-spectral collocation method

Abstract

The study of the unsteady fractional advection–diffusion equation (ADE) is carried out in cylindrical geometry along with time-exponential concentration on a cylindrical surface. We have used the Caputo-Fabrizio time-fractional derivative for the fractional model of the advection–diffusion. The analytical solutions for the solute concentration are determined by using integral transformations. For comparison, we also present a numerical scheme to get a numerical solution for different parameters' values. For time ordinary derivative approximation, we use the finite difference method. For space derivatives, we use a pseudo-spectral collocation method for higher-order accuracies. The advection-diffusion’ classical model is obtained by taking the fractional parameter to be unit. The influence of the memory, namely, the Caputo-Fabrizio time-fractional derivative on the solute concentration is studied and compared with the ordinary case. Also, the impact of the drift velocity is analyzed by employing the Peclet number. It has been found that the concentration is decreasing with the Peclet number and is increasing with the radial coordinate. The present investigation will be helpful in future research to use a higher-order approximation for ordinary derivatives. As the derivatives in space are ordinary derivatives, we use highly accurate pseudo-spectral collocation approximation for them.

Keywords

• CF (Caputo-Fabrizio) fractional derivative
• Advection-diffusion equation (ADE)
• Cylindrical coordinates
• Integral transform, memory effects