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





parabolic equations, Hölder space, Rothe method, a priori estimates, unique solvability, mathematical model, thermodestruction, composite


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.

Author Biography

N.L. Gol’dman

Lomonosov Moscow State University,
Research Computing Center,
Ленинские горы, 119991, Москва
• Leading Researcher


  1. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type (Nauka, Moscow, 1967; SIAM, Providence, 1968).
  2. N. L. Gol’dman, Inverse Stefan Problems (Kluwer, Dordrecht, 1997).
  3. N. L. Gol’dman, Inverse Stefan Problems. Theory and Methods of Solution (Mosk. Gos. Univ., Moscow, 1999) [in Russian].
  4. S. N. Kruzhkov, “A Priori Estimate for the Derivative of a Solution to a Parabolic Equation,” Vestn. Mosk. Univ., Ser. 1: Mat. Mekh., No. 2, 41-48 (1967).
  5. L. V. Kantorovich and G. P. Akilov, Functional Analysis (Nauka, Moscow, 1977; Pergamon, New York, 1982).
  6. A. K. Alekseev, “On the Restoration of the Heating History of a Plate Made of a Thermodestructible Material from the Density Profile in the Final State,” Teplofiz. Vys. Temp. 31 (6), 975-979 (1993) [High Temp. 31 (6), 897-901 (1993)].
  7. A. K. Alekseev, “Heat Memory of Structures with Phase Transitions,” J. Intell. Mater. Syst. Struct. 5 (1), 90-94 (1994).



How to Cite

Гольдман Н.Л. 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 // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2017. 18. 247-266. doi 10.26089/NumMet.v18r322



Section 1. Numerical methods and applications

Most read articles by the same author(s)

1 2 > >>