Nonstationary contrast structures of the problem of reaction-diffusion with roots of integral sheet in a inhomogeneous medium

A. A. Bykov ; K. E. Ermakova

Matematičeskoe modelirovanie, Tome 31 (2019) no. 9, pp. 101-130.

Résumé

A description is given of contrasting structures arising from the simulation of reaction – diffusion processes in an inhomogeneous medium with a power dependence of the source density on the concentration in the vicinity of the roots. The results obtained earlier for a homogeneous medium are generalized to the case of an inhomogeneous medium, and sufficient conditions for the existence of a solution of the type of contrast structure are strictly substantiated. The exponent of the root function of the right-hand side, in contrast to previously known results, is assumed to be non-integer, including irrational. It is shown that the front (relative to the direction of movement) part of the front is an exponential function, the rear part of the front is a power function, and this is a fundamentally new, previously unknown result. The family of exact solutions of the evolution equation is found. The formal asymptotics of the solution of the initial-boundary value problem for the reaction-diffusion equation is constructed. The substantiation of the correctness of the partial sum of an asymptotic series using the method of differential inequalities is given.

Détail

Keywords: nonlinear differential equations, asymptotic methods, contrast structure, differential inequalities.

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