General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems

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...

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Bibliographic Details
Published inFractional calculus & applied analysis Vol. 19; no. 3; pp. 676 - 695
Main Authors Luchko, Yuri, Yamamoto, Masahiro
Format Journal Article
LanguageEnglish
Published Warsaw Versita 01.06.2016
De Gruyter
Subjects
35B30
35B50
35C05
35E05
35L05
45K05
60E99
a priori estimates
Abstract Harmonic Analysis
Analysis
Fourier method of variables separation
Functional Analysis
general fractional derivative
general time-fractional diffusion equation
generalized solution
initial-boundary-value problems
Integral Transforms
Mathematics
maximum principle
Operational Calculus
Primary 26A33
Research Paper
Secondary 35A05
35C05
Fourier method of variables separation
general fractional derivative
Primary 26A33
60E99
general time-fractional diffusion equation
initial-boundary-value problems
35L05
Secondary 35A05
maximum principle
35E05
a priori estimates
35B30
45K05
35B50
generalized solution
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Summary: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.
ISSN:1311-0454
1314-2224
DOI:10.1515/fca-2016-0036