Finite difference Scheme with exact spectrum for the diffusion equation with piecewise constant coefficients in a layered medium

Abstract

Maģistra darba mērķis - izstrādāt galīgo diferenču shēmu (metodi) ar precīzo spektru difūzijas vienādojumam. Darba ietvaros tika apskatīta taišņu metode Puasona vienādojumam. Tika iegūta taišņu metode ar precīzo spektru, kā arī pierādīta teorēma par precīzo taišņu metodi. Tika risināti daži piemēri, izmantojot iegūto teoriju. Difūzijas vienādojumam tika izstrādāta gan parastā taišņu metode, gan precīza spektra taišņu metode. Tika pierādīta teorēma par precīzo spektru, kā arī risināts viens piemērs.The aim of this Master's Thesis is to work out finite difference scheme (method) with exact spectrum for the diffusion equation. In this Thesis, method of lines for Poisson's equation is studied. Method of lines with exact spectrum was obtained, as well as the theorem about the method of lines with exact spectrum was proved. Using the obtained theory, Some problems were solved. Both: ordinary method of lines and method of lines with exact spectrum for the diffusion equation were obtained. The theorem on the exact spectrum was proved, as well as one preoblem were solved