Journal
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
Volume 19, Issue 3, Pages 676-695Publisher
WALTER DE GRUYTER GMBH
DOI: 10.1515/fca-2016-0036
Keywords
general fractional derivative; general time-fractional diffusion equation; initial-boundary-value problems; maximum principle; a priori estimates; Fourier method of variables separation; generalized solution
Funding
- Japan Society for the Promotion of Science
- [15H05740]
- Grants-in-Aid for Scientific Research [26220702] Funding Source: KAKEN
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In this paper, we deal with the initial-boundary-value problems for a general time-fractional diffusion equation which generalizes the single- and the multi-term time-fractional diffusion equations as well as the time-fractional diffusion equation of the distributed order. First, important estimates for the general time-fractional derivatives of the Riemann-Liouville and the Caputo type of a function at its maximum point are derived. These estimates are applied to prove a weak maximum principle for the general time-fractional diffusion equation. As an application of the maximum principle, the uniqueness of both the strong and the weak solutions to the initial-boundary-value problem for this equation with the Dirichlet boundary conditions is established. Finally, the existence of a suitably defined generalized solution to the the initial-boundary-value problem with the homogeneous boundary conditions is proved.
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