Abstract
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.
Highlights
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In view of the Young inequality on the convolution, we can directly verify that the classical Caputo derivative (3) can be well-defined for v ∈ W1,1(0, T) and dαt v ∈ L1(0, T)
We extend the approach in Kubica, Ryszewska and Yamamoto [5] to discuss fractional derivatives in fractional Sobolev spaces of arbitrary real number orders and construct a convenient theory for initial value problems and initial boundary value problems for time-fractional differential equations
Summary
One of our main interests is to define a fractional derivative, denoted by ∂αt , and characterize the space of u satisfying ∂αt u ∈ L2(0, T) Such ∂αt should be an extension of dαt and Dtα in a minimum sense in order that important properties of these classical fractional derivatives should be inherited to ∂αt. We will define a time-fractional derivative, denoted by ∂αt , as a suitable extension of dαt satisfying the requirements:. Section 2: Definition of the extended derivative ∂αt : We extend dαt as operator so that it is well-defined as an isomorphism in relevant Sobolev spaces. Equation (20) means that ∂αt coincides with the Riemann–Liouville fractional derivative, provided that we consider Hα(0, T) as the domain of ∂αt and Dtα This extension ∂αt of dαt |0C1[0,T] is not yet complete, and in the subsection, we will continue to extend
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