, , , A PROBLEM WITH NON-LOCAL BOUNDARY CONDITIONS ON CHARACTERISTICS FOR A MIXED-TYPE EQUATION WITH TWO LINES OF BIRTH // BSU Bulletin. Mathematics, Informatics. - 2024. №1. . - С. 3-17.

A PROBLEM WITH NON-LOCAL BOUNDARY CONDITIONS ON CHARACTERISTICS FOR A MIXED-TYPE EQUATION WITH TWO LINES OF BIRTH

DOI: 10.18101/2304-5728-2024-1-3-17 UDK: 517.95

In this scientific article "A problem with non-local boundary conditions on characteristics for a mixed-type equation with two lines of degeneracy", a proof of the uniqueness and existence of a Cauchy solution for a mixed-type equation with two lines of degeneracy is given. A problem with non-local boundary conditions on the characteristics of the equation is investigated to find a regular solution for a mixed-type equation. Non-local conditions contain fractional operators in the sense of Riem-na- Liouville integro-differentiation. It is proved that when certain conditions are met, there cannot be more than one so- lution. The question of the solvability of the problem is equivalently reduced to the question of the solvability of a system of integral equations, which is a system of singu- lar integral equations. A condition is found that guarantees the existence of a regulator that leads a system of singular integral equations to Fredholm equations of the second kind. The possibility of reducing the problem to equivalent Fredholm integral equations of the second kind and the uniqueness of the desired solution implies the existence of a solution to the problem.

non-local boundary conditions, regular solution of the problem, homo- geneous problem, fractional order derivative, fractional integro-differentiation operator, singular integral equations, regularizer, point affixes, the Jordan curve