Time-fractional diffusion equation in the fractional Sobolev spaces

Abstract

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. MSC 2010 : Primary 26A33; Secondary 35C05, 35E05, 35L05, 45K05, 60E99 Key Words and Phrases: 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 c 2015 Diogenes Co., Sofia pp. 799-820 ,

Case Studies and a Pitfall for Nonlinear Variational Regularization Under Conditional Stability

“…Due to (9) the norm x δ α 1 is uniformly bounded by a finite constant for α(δ) from (15). This yields the rate (14) and completes the proof.…”

Inverse problems of determining sources of the fractional partial differential equations

“…More precisely, the proof of Theorem 3.5 is based on the following lemma which one can prove also by using [9]. (9), where we assume ρ ∈ C[0, ∞) and (11). Then Lemma 2.2 still holds, that is, u allows the same representation (5), where v solves the initial value problem (14) for the homogeneous equation.…”

“…(b) Let g satisfy (11). Then there exists a classical solution to (14), which takes the form v(x, t) =…”

Maximum principles for time-fractional Caputo-Katugampola diffusion equations

“…On the other hand, using those maximum principles mentioned above, a formal solution of a Fourier series in regard to the eigenfunctions of Sturm-Liouville eigenvalue problems were achieved in [13], and non-axisymmetric solutions to time-fractional diffusionwave equation in an infinite cylinder were established by Povstenko [18]. For more details on this topics readers are suggested to consult [6,7,14] and the references therein.…”