Logarithmic convexity of non-symmetric time-fractional diffusion equations

Abstract

We consider a class of diffusion equations with the Caputo time-fractional derivative ${\partial }_t^{\alpha }u=Lu$ subject to the homogeneous Dirichlet boundary conditions. Here, we consider a fractional order $0<\alpha <1$ and a second-order operator $L$ which is elliptic and non-symmetric. In this paper, we show that the logarithmic convexity extends to this non-symmetric case provided that the drift coefficient is given by a gradient vector field. Next, we perform some numerical experiments to validate the theoretical results in both symmetric and non-symmetric cases. Finally, some conclusions and open problems will be mentioned.

12 pages, 6 figures