Journal
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
Volume 18, Issue 3, Pages 799-820Publisher
SPRINGERNATURE
DOI: 10.1515/fca-2015-0048
Keywords
Riemann-Liouville integral; Caputo fractional derivative; fractional Sobolev spaces; norm equivalences; fractional diffusion equation in Sobolev spaces; norm estimates of the solutions; initialboundary-value problems; weak solution; existence and uniqueness results
Funding
- A3 Foresight Program Modeling and Computation of Applied Inverse Problems by Japan Society of the Promotion of Science
- Grants-in-Aid for Scientific Research [15H05740, 15K13455] Funding Source: KAKEN
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The Caputo time-derivative is usually defined pointwise for well-behaved functions, say, for the continuously differentiable functions. Accordingly, in the publications devoted to the theory of the partial fractional differential equations with the Caputo derivatives, the functional spaces where the solutions are looked for are often the spaces of smooth functions that appear to be too narrow for several important applications. In this paper, we propose a definition of the Caputo derivative on a finite interval in the fractional Sobolev spaces and investigate it from the operator theoretic viewpoint. In particular, some important equivalences of the norms related to the fractional integration and differentiation operators in the fractional Sobolev spaces are given. These results are then applied for proving the maximal regularity of the solutions to some initial-boundary-value problems for the time-fractional diffusion equation with the Caputo derivative in the fractional Sobolev spaces.
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