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2 Journal of Applied Mathematics and Computational Mechanics

2 2016

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Dual-phase lag equation. Stability conditions of a numerical algorithm based on the explicit scheme of the finite difference method

Majchrzak E., Mochnacki B.

Journal of Applied Mathematics and Computational Mechanics

2016

Vol. 15, nr 3

89--96

The dual-phase lag equation (DPLE) is considered. This equation belongs to the group of hyperbolic PDE, contains a second order time derivative and higher order mixed derivative in both time and space. From the engineer’s point of view, the DPLE results from the generalized form of the Fourier law. It is applied as a mathematical model of thermal processes proceeding in the micro-scale and also in the case of bio-heat transfer problem analysis. At the stage of numerical computations the different approximate methods of the PDE solving can be used. In this paper, the authors present the considerations concerning the stability conditions of the explicit scheme of finite difference method (FDM). The appropriate conditions have been found using the von Neumann analysis. In the final part of the paper, the results of testing computations are shown.

Cattaneo-Vernotte bioheat transfer equation. Stability conditions of numerical algorithm based on the explicit scheme of the finite difference method

Mochnacki B., Tuzikiewicz W.

Journal of Applied Mathematics and Computational Mechanics

2016

Vol. 15, nr 4

137-144

The Cattaneo-Vernotte (CVE) equation is considered. This equation belongs to the group of hyperbolic PDE. Supplementing this equation by two additional terms corresponding to perfusion and metabolic heat sources one can apply the CVE as a mathematical model describing the heat transfer processes proceeding in domain of the soft tissue. Such an approach is recently often preferred substituting the classical Pennes model. At the stage of numerical computations the different numerical methods of the PDE solving can be used. In this paper the problems of stability conditions for the explicit scheme of the finite difference method (FDM) are discussed. The appropriate condition limiting the admissible time step have been found using the von Neumann analysis.