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Abstract:
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The time fractional di usion-wave equation can be obtained from the classical diffusion
or wave equation by replacing the rst or second order time derivative, respectively,
by a fractional derivative of order 0 < 2. In particular, depending
on the value of the parameter , we distinguish subdi usion (0 < < 1), normal
di usion ( = 1), superdi usion (1 < < 2) and ballistic motion ( = 2).
Fractional derivatives are non-local operators, which makes it di cult to construct
e cient numerical method.
The subject of this dissertation is the time fractional di usion-wave equation with
coe cient which contains a singular distribution, primarily Dirac distribution, and
its approximation by nite di erences. Initial-boundary value problems of this type
are usually called interface problems. Solutions of such problems have discontinuities
or non-smoothness across the interface, i.e. on support of Dirac distribution, making
it di cult to establish convergence of the nite di erence schemes using the classical
Taylor's expansion.
The existence of generalized solutions of this initial-boundary value problem has
been proved. Some nite di erence schemes approximating the problem are proposed
and their stability and estimates for the rate of convergence compatible with
the smoothness of the solution are obtained. The theoretical results are con rmed
by numerical examples. |