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 Num. Meth. Prog., 2017, Volume 18, Issue 3, Pages 247–266 (Mi vmp877)

A nonlinear problem for a parabolic equation with an unknown coefficient at the time derivative and its applications in mathematical models of physico-chemical processes

N. L. Gol'dman

Lomonosov Moscow State University, Research Computing Center

Abstract: We consider conditions of unique solvability in a class of smooth functions for a nonlinear system with an unknown coefficient at the time derivative in a parabolic equation. To this end, the Rothe method is applied, which provides not only the proof of solvability but also the constructive solution of the considered system. A priori estimates in the grid-continuous Hölder spaces are established for the corresponding differential-difference nonlinear system that approximates the initial parabolic system by the Rothe method. Such estimates allow one to prove the existence of the smooth solution of this parabolic system and to obtain the error estimates for the Rothe method. This study is connected with the mathematical modelling of physico-chemical processes where the inner characteristics of materials are subjected to changes. As an example, the problem on the destruction of a heat-protective composite under the effect of high-temperature heating is discussed.

Keywords: parabolic equations, Hölder spaces, Rothe method, a priori estimates, unique solvability, mathematical model, thermodestruction, composite material.

Full text: PDF file (342 kB)
UDC: 517.958

Citation: N. L. Gol'dman, “A nonlinear problem for a parabolic equation with an unknown coefficient at the time derivative and its applications in mathematical models of physico-chemical processes”, Num. Meth. Prog., 18:3 (2017), 247–266

Citation in format AMSBIB
\Bibitem{Gol17} \by N.~L.~Gol'dman \paper A nonlinear problem for a parabolic equation with an unknown coefficient at the time derivative and its applications in mathematical models of physico-chemical processes \jour Num. Meth. Prog. \yr 2017 \vol 18 \issue 3 \pages 247--266 \mathnet{http://mi.mathnet.ru/vmp877} 

• http://mi.mathnet.ru/eng/vmp877
• http://mi.mathnet.ru/eng/vmp/v18/i3/p247

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