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1

Determination of the stationary thermal state of simple geometry layered structures with themperature dependent heat conductivity factors

Makhorkin M, Makhorkin I., Makhorkina T.

ECONTECHMOD : An International Quarterly Journal on Economics of Technology and Modelling Processes

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2018

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Vol. 7, No 3

23--28

EN

An analytical-numerical method to determine the one-dimensional stationary thermal state of simple geometry multilayer structures for arbitrary dependences of heat-conductivity factors on temperature is proposed (the multilayer bodies of thermosensitive materials, referred to one of the classical orthogonal coordinate systems are considered, the thermal state caused by thermal load is characterized by a onedimensional stationary temperature field. Approbation of the methodology by studying the stationary thermal state of a two-layer cylinder is realized. The cases of existence of a closed-form analytic solutions for the nonlinear heat conduction problem are considered.

2

Multiplication of the distributions (x±i0)z

Franssens G. R

Journal of Applied Analysis

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2014

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Vol. 20, nr 1

15--27

EN

In previous work of the author, a convolution and multiplication product for the set of Associated Homogeneous Distributions (AHDs) with support in ℝ was defined and fully investigated. Here this definition is used to calculate the multiplication product of homogeneous distributions of the form (x±i0)z, for all z∈C. Multiplication products of AHDs generally contain an arbitrary constant if the resulting degree of homogeneity is a negative integer, i.e., if it is a critical product. However, critical products of the forms (x+i0)a.(x+i0)b and (x−i0)a.(x−i0)b, with a+b∈Z−, are exceptionally unique. This fact combined with Sokhotskii–Plemelj expressions then leads to linear dependencies of the arbitrary constants occurring in products like δ(k).δ(l), η(k).δ(l), δ(k).η(l) and η(k).η(l) for all k,l∈N (η≜1πx−1). This in turn gives a unique distribution for products like δ(k).η(l)+η(k).δ(l) and δ(k).δ(l)−η(k).η(l). The latter two products are of interest in quantum field theory and appear for instance in products of the partial derivatives of the zero-mass two-point Wightman distribution.

3

Matrix of fundamental solutions for the coupled parabolic system of partial differential equations

Łazuka J., Szymaniec A.

Biuletyn Wojskowej Akademii Technicznej

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2000

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Vol. 49, nr 1

19-31

EN

The aim of this paper is to construct the matrix of fundamental solutions for linear coupled parabolic system in the three-dilinensional space, consisting of two equations. The matrix of the fundamental solutions has been constructed using the Fourier transform. Basing of the matrix of fundamenal solutions we represent the solution of the Cauchy problem for this system of equations by the convolution type represntation.

PL

Celem pracy jest konstrukcja macierzy rozwiązań podstawowych dla liniowego sprzężonego parabolicznego układu równań w przestrzeni trójwymiarowej zawierającego dwa równania. Macierz rozwiązań podstawowych skonstruowano używając transformacji Fouriera. Bazując na macierzy rozwiązań podstawowych przedstawiono rozwiązanie zagadnienia Cauchy'ego dla takiego układu równań za pomocą splotu z danymi początkowymi.