Fractional Calculus and Time-Fractional Differential Equations: Revisit and Construction of a Theory

For fractional derivatives and time-fractional differential equations, we construct a framework on the basis of 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 well-posedness and other properties.