On a nonlocal boundary-value problem for a mixed-type equation of the second kind

B. I. Islomov ; A. A. Abdullayev

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential equations, geometry, and topology, Tome 201 (2021), pp. 65-79.

Résumé

In this paper, we discuss the unique solvability of a nonlocal problem with the Poincaré condition for an equation of elliptic-hyperbolic type of the second kind, i.e., for an equation whose degeneracy line is a characteristic. We develop a new extremum principle for equations of this type. Using this extremum principle, we prove the uniqueness of the problem considered. Using functional relations, we reduce the study of the existence of a solution to the problem for a singular integral equation of the normal type. We find a class of functions that provide the solvability of the singular equation. Using the Carleman–Vekua regularization method, we reduce the singular integral equation to a Fredholm integral equation of the second kind whose solvability is established based on the uniqueness of the solution.

Détail

Keywords: elliptic-hyperbolic equation, equation of the second kind, nonlocal boundary-value problem, extremum principle, regularization method, class of generalized solutions.

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