On a new nonlocal boundary value problem for an equation of the mixed parabolic-hyperbolic type

Abstract

In the present work a new nonlocal boundary value problem for an equation of the mixed type
is formulated. This equation is parabolic-hyperbolic and belongs to the first kind because the
line of type change is not a characteristic of the equation. Nonlocal condition links points on
boundaries of the parabolic and hyperbolic parts of the domain with each other. This problem
is generalization of the well-known problems of Frankl type. A boundary value problem for the
heat equation with conditions of the Samarskii-Ionlin type arises in solving this problem. Unlike
the existing publications of the other authors related to the theme it is necessary to note that
in this papers nonlocal problems were considered in rectangular domains. But in our formulation
of the problem the hyperbolic part of the domain coincides with a characteristical triangle. The
formulated problem is equivalently reduced to an integral Volterra equation of the second kind.
Unique strong solvability of the formulated problem is proved.