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1

Applications of the fractional Sturm-Liouville difference problem to the fractional diffusion difference equation

Malinowska Agnieszka B., Odzijewicz Tatiana, Poskrobko Anna

International Journal of Applied Mathematics and Computer Science

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2023

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Vol. 33, no. 3

349--359

EN

This paper deals with homogeneous and non-homogeneous fractional diffusion difference equations. The fractional operators in space and time are defined in the sense of Grünwald and Letnikov. Applying results on the existence of eigenvalues and corresponding eigenfunctions of the Sturm-Liouville problem, we show that solutions of fractional diffusion difference equations exist and are given by a finite series.

2

Numerical scheme for one phase 1D fractional Stefan problem using the similarity variable technique

Błasik M.

Journal of Applied Mathematics and Computational Mechanics

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2014

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Vol. 13, nr 1

13--21

EN

In this paper we present a numerical method to solve a one-dimensional, one-phase extended Stefan problem with fractional time derivative described in the Caputo sense. The proposed method is based on applying a similarity variable for the anomalous-diffusion equation and the finite difference method. In the final part, examples of numerical results are discussed.

3

Numerical solutions to integral equations equivalent to differential equations with fractional time

Bandrowski B., Karczewska A., Rozmej P.

International Journal of Applied Mathematics and Computer Science

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2010

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Vol. 20, no 2

261-269

EN

This paper presents an approximate method of solving the fractional (in the time variable) equation which describes the processes lying between heat and wave behavior. The approximation consists in the application of a finite subspace of an infinite basis in the time variable (Galerkin method) and discretization in space variables. In the final step, a large-scale system of linear equations with a non-symmetric matrix is solved with the use of the iterative GMRES method.

4

Komputerowa analiza matematycznego modelu dielektrycznej dyspersji w tkance

Kotulska M.

Acta Bio-Optica et Informatica Medica. Inżynieria Biomedyczna

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1998

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Vol. 4, nr 2

57-62

PL

Praca poświęcona jest własnościom dielektrycznym tkanki w obszarze dyspersji α. Przedstawiony został model teoretyczny zjawiska odpowiedzialnego za obserwowany tu przebieg charakterystyki, którą cechuje występowanie dyspersji anomalnej. Przeprowadzone zostały symulacje komputerowe, które potwierdzają wyniki rozważań teoretycznych. W pracy wskazano też na możliwości zastosowania odpowiedzi dielektrycznej tkanki w diagnostyce medycznej.

EN

Dielectric properties of tissue in the region α are analysed. It is introduced a theoretical model of the phenomenon underlying the characteristics with anomalous dispersion. The analysis is supported by computer simulations, which confirm the theoretical results. Also, an application of tissue dielectric characteristics in medical diagnostics is indicated.