Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation

Authors:
Yury Luchko
Affiliations:
Technical University of Applied Sciences Berlin, Department of Mathematics II, Berlin, Germany
Venue:
Computers & Mathematics with Applications
Year:
2010

Citing 1
Cited 3

Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies)

Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies)

Numerical simulations of 2D fractional subdiffusion problems

Journal of Computational Physics
Estimation of parameters in fractional subdiffusion equations by the time integral characteristics method

Computers & Mathematics with Applications
A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation

Applied Numerical Mathematics

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Abstract

In this paper, some uniqueness and existence results for the solutions of the initial-boundary-value problems for the generalized time-fractional diffusion equation over an open bounded domain Gx(0,T),G@?R^n are given. To establish the uniqueness of the solution, a maximum principle for the generalized time-fractional diffusion equation is used. In turn, the maximum principle is based on an extremum principle for the Caputo-Dzherbashyan fractional derivative that is considered in the paper, too. Another important consequence of the maximum principle is the continuous dependence of the solution on the problem data. To show the existence of the solution, the Fourier method of the variable separation is used to construct a formal solution. Under certain conditions, the formal solution is shown to be a generalized solution of the initial-boundary-value problem for the generalized time-fractional diffusion equation that turns out to be a classical solution under some additional conditions.