Equivalent definitions of Caputo derivatives and applications to subdiffusion equations

An equivalent definition of the fractional Caputo derivative $mathbf{D}_{t}^{ u}g$, for $ uin(0,1)$, is found, within suitable assumptions on $g$. Some applications to the fractional calculus and to the theory of fractional partial differential equations are then discussed. In particular, this alternative definition is used to prove the maximum principle for the classical solutions to the linear subdiffusion equation subject to nonhomogeneous boundary conditions. This approach also allows to construct numerical solutions to the initial-boundary value problem for the subdiffusion equation with memory.