Fractional derivatives and time-fractional ordinary differential equations in $L^p$-space

Abstract

We define fractional derivatives ∂ α t in Sobolev spaces based on L p (0, T ) by an operator theory, and characterize the domain of ∂ α t in subspaces of the Sobolev-Slobodecki spaces W α,p (0, T ). Moreover we define ∂ α t u for u ∈ L p (0, T ) in a sense of distribution. Then we discuss initial value problems for linear fractional ordinary differential equations by means of such ∂ α t and establish several results on the unique existence of solutions within specified classes according to the regularity of the coefficients and the non-homogeneous terms in the equations.

Initial-boundary value problems to semilinear multi-term fractional differential equations

“…where C ν = C(ν, ν), C νi = C(ν i , ν 0 ), C µj = C(µ j , µ 0 ), are positive constants defined in (i) of Theorem 2.2 [43]. Gathering (6.29) and (6.30) and setting…”

Fractional calculus and time-fractional differential equations: revisit and construction of a theory

“…This is not the best possible way for gaining L p -regularity in time with p = 2, and we should construct the corresponding L p -theory for ∂ α t . We can refer to Yamamoto [35] as for a similar work discussing some fundamental properties in the L p -case with 1 ≤ p < ∞.…”