Wyniki wyszukiwania - Biblioteka Nauki

1

A system of two conservation laws with flux conditions and small viscosity

100%

Joseph K. T.

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tom Vol. 20, nr 1

97

EN

This is an erratum to [J. Appl. Anal. 15 (2009), no. 2, 247–267], http://dx.doi.org/10.1515/JAA.2009.247. In this note the author would like to make a correction in formulas (2.12) and (2.13) for the limit limϵ→0(uϵ(x,t),vϵ(x,t)) stated in Theorem (2.1); there is no change in the proof.

2

Existence of traveling profiles solutions to porous medium equation

88%

Arioua Y. , Benhamidouche N.

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2023

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tom Vol. 46

149--162

EN

In this paper, we shall study the existence and uniqueness of solutions called "traveling profiles solutions" to the porous medium equation in one dimension. By these solutions, we generalize the results obtained by Gilding and Peletier who proved the existence of self similar solutions of type I, II and III to the same equation. The principal idea of our work is to convert the porous media equation in to an equivalent nonlinear differential equation, and to prove the existence and uniqueness of these new solutions under certain conditions.

3

Symmetry reductions and exact solutions to the Sharma–Tasso–Olever equation

75%

Zhang Y.

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tom Vol. 22, nr 2

107--114

EN

In the present paper, the Sharma–Tasso–Olever (STO) equation is considered by the Lie symmetry analysis. All of the geometric vector fields to the STO equation are obtained, and then the symmetry reductions and exact solutions of the equation are investigated. Our results witness that symmetry analysis is a very efficient and powerful technique in finding the solutions of the proposed equation.

4

Mathematical modelling of heat transfer in liquid flat-plate solar collector tubes

63%

Zima W. , Dziewa P.

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2010

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tom Vol. 31, no. 2

45-62

EN

The paper presents a one-dimensional mathematical model for simulating the transient processes which occur in the liquid flat-plate solar collector tubes. The proposed method considers the model of collector tube as one with distributed parameters. In the suggested method one tube of the collector is taken into consideration. In this model the boundary conditions can be time-dependent. The proposed model is based on solving the equation describing the energy conservation on the fluid side. The temperature of the collector tube wall is determined from the equation of transient heat conduction. The derived differential equations are solved using the implicit finite difference method of iterative character. All thermo-physical properties of the operating fluid and the material of the tube wall can be computed in real time. The time-spatial heat transfer coefficient at the working fluid side can be also computed on-line. The proposed model is suitable for collectors working in a parallel or serpentine tube arrangement. As an illustration of accuracy and effectiveness of the suggested method the computational verification was carried out. It consists in comparing the results found using the presented method with results of available analytic solutions for transient operating conditions. Two numerical analyses were performed: for the tube with temperature step function of the fluid at the inlet and for the tube with heat flux step function on the outer surface. In both cases the conformity of results was very good. It should be noted, that in real conditions such rapid changes of the fluid temperature and the heat flux of solar radiation, as it was assumed in the presented computational verification, do not occur. The paper presents the first part of the study, which aim is to develop a mathematical model for simulating the transient processes which occur in liquid flat-plate solar collectors. The experimental verification of the method is a second part of the study is not presented in this paper. In order to perform this verification, the mathematical model would be completed with additional energy conservation equations. The experimental verification will be carry out in the close future.

5

Differential Equation Chazy of the Third Order with Exact Solutions Expressed via the Elliptic Functions

51%

Chichurin A.

Studia i Materiały / Europejska Uczelnia Informatyczno-Ekonomiczna w Warszawie

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2017

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tom Nr 2(14)

29--46

EN

The procedure of integrating the third-order Chazy differential equation with six singularities is presented, when the first six coefficients are equal to ±1/2, ± 1, ± 2. For five third-order differential equations it is shown that their general solutions are expressed via the elliptic functions. Two-parameter families of solutions in elliptic functions and one-parameter families of solutions in elementary functions are presented. The proposed integration method can be realized for the differential Chazy equation in those cases when the first six coefficients have values for which it is possible to solve a special algebraic equation of the fifth degree in radicals.

PL

W artykule rozpatrzona została metoda rozwiązania równania różniczkowego Chazy’ego trzeciego rzędu, zawierającego sześć osobliwości w przypadku, gdy sześć pierwszych współczynników równa się ± 1/2, ± 1, ± 2. Dla pięciu równań różniczkowych trzeciego rzędu znaleziono ich ogólne rozwiązania za pomocą funkcji eliptycznych. Przedstawione zostały rodziny rozwiązań dwuparametrycznych za pomocą funkcji eliptycznych oraz rodziny rozwiązań jednoparametrycznych za pomocą funkcji elementarnych. Proponowana metoda rozwiązania może być zrealizowana dla równania różniczkowego Chazy’ego w tych przypadkach, gdy pierwsze sześć współczynników ma wartości, dla których istnieje możliwość rozwiązania szczególnego równania algebraicznego piątego stopnia za pomocą pierwiastków.