Abstract. In the paper the problem of explicit FDM scheme stability for the bio-heat transfer equation (the Pennes equation) is discussed. To formulate the appropriate condition the von Neumann approach is applied. The first chapter contains the known derivation of FDM stability condition for the Fourier equation. In the second part, a similar condition is found for the case of the bio-heat transfer equation containing both the perfusion and metabolic heat sources.
Abstract. 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.
Abstract. In the paper the problems connected with the mathematical modeling of alloys solidification (macro scale) are presented. The similar lecture was delivered by the authors during the IV Conference on Mathematical Modelling in Physics and Technique, Kleszczów, 28-30 June 2012. In particular the one domain method (fixed domain approach) in which the parameter called 'a substitute thermal capacity' appears will be discussed. The one domain method bases on the Fourier equation describing the thermal processes in the whole conventionally homogeneous casting domain. The substitute thermal capacity (STC) can be defined in the different ways. The first group of definitions results from the differentiation of the function determining the local volumetric solid state fraction in the neigborhood of the point considered, while the second one is based on the direct assumption of the STC form, at the same time this function should fulfill the condition resulting from the change of alloy physical enthalpy during the phase change. The details of these approaches will be presented in the Sections 2 and 3.
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