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Abstract:
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In recent years there has been increasing interest in modeling the physical and
chemical processes with equations involving fractional derivatives and integrals. One
of such equations is the subdi usion equation which is obtained from the di usion
equation by replacing the classical rst order time derivative by a fractional derivative
of order with 0 < < 1:
The subject of this dissertation is the initial-boundary value problem for the
subdi usion equation and its approximation by nite di erences. At the beginning,
the one-dimensional equation is observed. The existence and the uniqueness of weak
solution is proved. The stability and the convergence rate estimates for implicite
and the weighted scheme are obtained.
The main focus is on two-dimensional subdi usion problem with Laplace operator
as well as problem with general second-order partial di erential operator. It is
assumed that the coe cients of the di erential operator satisfy standard ellipticity
conditions that guarantees existence of solution in appropriate spaces of Sobolev
type. In that case, apart from above mensoned, we constructed the additive and the
factorized di erence schemes. We investigated their stability and convergence rate
depending on the smoothness of the input data and of generalized solution. |