Abstract

AbstractIn this article, we formulate two kinds of time fractional derivatives of the Caputo type with order \(\alpha \) in fractional Sobolev spaces and prove that they are isomorphisms between the corresponding Sobolev space of order \(\alpha \) and the \(L^2\)-space. On the basis of such fractional derivatives, we formulate initial value problems for time fractional ordinary differential equations and prove the well-posedness.KeywordsTime fractional ordinary differential equationFractional Sobolev spaceInitial value problemWell-posedness

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