On initial-boundary value problem for the Burgers equation in nonlinearly degenerating domain

Abstract

In this paper, we study the solvability of one initial-boundary value problem for the Burgers equation with periodic boundary conditions in a nonlinearly degenerating domain. In this paper, we found an orthonormal basis for domains with time-varying boundaries. On this basis, we use the Faedo–Galerkin method to prove theorems about the unique solvability of the problem under consideration. We also present some numerical results in the form of graphs of solutions to the problem under study for various initial data.

Keywords:

• Burgers equation
• periodic boundary conditions
• degenerating domain
• Galerkin method

Mathematics Subject Classifications:

• 35K55
• 35K10
• 35R37

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research of the second author was supported by the grant of the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan, Project AP13067805. The research of the first author was supported by the grant of the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan, Project AP09258892.