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Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation. (English) Zbl 1202.35339
Summary: Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation over an open bounded domain G×(0,T), G 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:
35R11Fractional partial differential equations
26A33Fractional derivatives and integrals (real functions)
35B50Maximum principles (PDE)
35B45A priori estimates for solutions of PDE
35A01Existence problems for PDE: global existence, local existence, non-existence
33E12Mittag-Leffler functions and generalizations
35B30Dependence of solutions of PDE on initial and boundary data, parameters
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