Study of Caputo time fractional transport equation

Arindam Sutradhar; Aghalaya S Vatsala

Arindam Sutradhar University of Louisiana at Lafayette,\\ Lafayette, Louisiana 70504, USA
Aghalaya S Vatsala University of Louisiana at Lafayette,\\ Lafayette, Louisiana 70504, USA

Abstract

In this article, we have obtained an approximate solution for the linear one-dimensional non-homogeneous Caputo time fractionaltransport equation with initial conditions. The Caputo time fractional
derivative is of order $q$, where $ 0 <q <1$. In order to solve the Caputo fractional derivative, we have used the Laplace transform method for the time derivative and the Fourier transform method for
the spatial derivative since the standard method of characteristics cannot be used. In the process of solving we have obtained the inverse Fourier transform of a convolution integral
of the product function involving the Mittag-Leffler function by numerical methods. This provides a good approximate solution of the Caputo fractional transport equation which tends to the solution of the
standard integer transport equation as $ q \rightarrow 1.$ As an application of our main results, we have provided two numerical examples.