Fractional diffusion-wave equation with concentrated capacity and its finite difference approximation

[А. Делић]

109 листова

Математика - Нумеричка математика / Mathematics - Numerical Mathematics

Дифузионо-таласна једначина разломљеног реда по веременској променљивој добија се из класичне дифузионе или таласне једначине заменом првог, односно другог извода по временској променљивој изводом разломљеног реда...

The time fractional diusion-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 subdiusion (0 <  < 1), normal diusion ( = 1), superdiusion (1 <  < 2) and ballistic motion ( = 2). Fractional derivatives are non-local operators, which makes it dicult to construct ecient numerical method. The subject of this dissertation is the time fractional diusion-wave equation with coecient which contains a singular distribution, primarily Dirac distribution, and its approximation by nite dierences. 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 dicult to establish convergence of the nite dierence schemes using the classical Taylor's expansion. The existence of generalized solutions of this initial-boundary value problem has been proved. Some nite dierence 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.

изводи разломљеног реда, субдифузија, супердифузија, проблем с интерфејсом, простори Собољева, слаба решења, априорна оцена, коначне резлике, брзина конвергенције

fractional derivatives, subdiusion, superdiusion, interface problems, Sobolev spaces, weak solutions, a priori estimate, nite dierence, rate of convergance

[517.962.8:519.63]:517.958(043.3)

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