Analysis of the fractional diffusion equations with fractional derivative of non-singular kernel

In this paper we study linear and nonlinear fractional diffusion equations with the Caputo fractional derivative of non-singular kernel that has been launched recently (Caputo and Fabrizio in Prog. Fract. Differ. Appl. 1(2):73-85, 2015). We first derive simple and strong maximum principles for the linear fractional equation. We then implement these principles to establish uniqueness and stability results for the linear and nonlinear fractional diffusion problems and to obtain a norm estimate of the solution. In contrast with the previous results of the fractional diffusion equations, the obtained maximum principles are analogous to the ones with the Caputo fractional derivative; however, extra necessary conditions for the existence of a solution of the linear and nonlinear fractional diffusion models are imposed. These conditions affect the norm estimate of the solution as well.