Assessment of the convergence of solutions of integro-differential equations of heat conduction in the conditions of the relaxation system

Authors

  • Тімур Муртазовіч Босенко Oles Gonchar Dnepropetrovsk National University Gagarina Av, 72, Dnepropetrovsk, Ukraine, 49010, Ukraine

DOI:

https://doi.org/10.15587/1729-4061.2013.19148

Keywords:

relaxation, integro-differential equation, thermal memory, convergence, localization time

Abstract

Modeling of relaxation processes is possible in the presence of relaxation components of a system, leading to consideration of integro-differential equations (IDE) of heat conduction taking into account relaxation functions and definition of the estimates of the convergence of their solutions. In this paper, we build solutions in times of system relaxation by the method of successive approximations.

This method allows explicitly specify the time interval on which there is the solution, unlike the existing methods, involving the introduction of relative time. The function, which defines the scope of application of IDE of heat conduction only at intervals of system relaxation, is given. As appears from the definition of boundedness of input functions and their derivatives, the equation has bounded solutions at all critical points of relaxation. The conducted analysis determines the boundedness, uniformity and existence of solutions under the influence of relaxation, allowed to carry out the evaluation of the integral terms of the heat conduction equation.

In this paper, the boundaries of existence and uniformity of solutions of heat conduction problems for IDE taking into account the thermal memory are defined. Theorems on the existence of uniformity and convergence of solutions of the integrodifferential equation on the interval 0<FO≤FOr and the boundedness of functions |Q(X, FO), |FO Q (X, FO)| at FO, which are responsible for thermophysical properties of material, are proved.

This allows to perform calculations at times of relaxation of the system of locally-nonequilibrium state of material in problems of heat and mass transfer

Author Biography

Тімур Муртазовіч Босенко, Oles Gonchar Dnepropetrovsk National University Gagarina Av, 72, Dnepropetrovsk, Ukraine, 49010

Candidate of technical Sciences

Assistant professor of Department of Differential Equations

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Published

2013-12-16

How to Cite

Босенко, Т. М. (2013). Assessment of the convergence of solutions of integro-differential equations of heat conduction in the conditions of the relaxation system. Eastern-European Journal of Enterprise Technologies, 6(4(66), 4–9. https://doi.org/10.15587/1729-4061.2013.19148

Issue

Vol. 6 No. 4(66) (2013): Mathematics and cybernetics - applied aspects

Section

Mathematics and Cybernetics - applied aspects

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Copyright (c) 2014 Тімур Муртазовіч Босенко

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