Maximum Principle for Space and Time-Space Fractional Partial Differential Equations

In this paper, new estimates of the sequential Caputo fractional derivatives a function at its extremum points are obtained.We derive comparison principles for linear differential equations, then apply these to obtain lower and upper bounds solutions nonlinear equations. The principle is applied show that initial-boundary-value problem anomalous diffusion admits most one classical solution depends continuously on initial boundary data. This answers positively open about maximum space time-space PDEs posed by Y. Luchko \[Fract. Calc. Appl. Anal. 14 (2011)]. an elliptic equation with derivative Laplace also proved.