  For fractional derivatives and time-fractional differential equations, we
construct a framework on the basis of the operator theory in fractional Sobolev
spaces. Our framework provides a feasible extension of the classical Caputo and
the Riemann-Liouville derivatives within Sobolev spaces of fractional orders
including negative ones. Our approach enables a unified treatment for
fractional calculus and time-fractional differential equations. We formulate
initial value problems for fractional ordinary differential equations and
initial boundary value problems for fractional partial differential equations
to prove the well-posedness and other properties.
