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Zbl 1202.35339
Luchko, Yury
Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation.
(English)
[J] J. Math. Anal. Appl. 374, No. 2, 538-548 (2011). ISSN 0022-247X

Summary: Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation over an open bounded domain $G\times (0,T)$, $G \in \Bbb R^n$ are considered. Based on an appropriate maximum principle that is formulated and proved in the paper, some a priori estimates for the solution and then its uniqueness are established. To show the existence of the solution, first a formal solution is constructed using the Fourier method of separation of variables. The time-dependent components of the solution are given in terms of the multinomial Mittag-Leffler function. Under certain conditions, the formal solution is shown to be a generalized solution of the initial-boundary-value problem for the generalized time-fractional multi-term diffusion equation that turns out to be a classical solution under some additional conditions. Another important consequence from the maximum principle is a continuously dependence of the solution on the problem data (initial and boundary conditions and a source function) that -- together with the uniqueness and existence results -- makes the problem under consideration to a well-posed problem in the Hadamard sense.
MSC 2000:
*35R11
26A33 Fractional derivatives and integrals (real functions)
35B50 Maximum principles (PDE)
35B45 A priori estimates
35A01
33E12 Mittag-Leffler functions and generalizations
35B30 Dependence of solutions of PDE on initial and boundary data

Keywords: Caputo fractional derivative; multi-term time-fractional diffusion equation; initial-boundary-value problems; maximum principle; uniqueness theorem; spectral method; multinomial Mittag-Leffler function; well-posed problems

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