Mixed problem for a nonlinear impulsive differential equation of parabolic type

T. K. Yuldashev ; A. K. Fayziyev ; F. D. Rakhmonov

Čelâbinskij fiziko-matematičeskij žurnal, Tome 9 (2024) no. 1, pp. 111-123.

Résumé

In this paper, we consider a nonlinear impulsive parabolic type partial differential equation with nonlinear impulsive conditions. Dirichlet type boundary value conditions with respect to spatial variable is used, and eigenvalues and eigenfunctions of the spectral problem are founded. The Fourier method of the separation of variables is applied. A countable system of nonlinear functional equations is obtained with respect to the Fourier coefficients of the unknown function. A theorem on a unique solvability of the countable system of nonlinear functional equations is proved by the method of successive approximations. A criteria of uniqueness and existence of a solution for the nonlinear impulsive mixed problem is obtained. A solution of the mixed problem is derived in the form of the Fourier series. The absolute and uniform convergence of the Fourier series is proved.

Détail

Keywords: mixed problem, nonlinear parabolic equation, nonlinear impulsive conditions, involution, unique solvability.

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