Unique solvability of a Bitsadze--Samarskiy type problem for equations with discontinuous coefficient

A. G. Ezaova

Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 4, pp. 50-58

Résumé

In this paper, unique solvability of a Bitsadze–Samarsky type problem for a third-order equation with discontinuous coefficients in a simply connected domain is investigated. The boundary condition of the problem contains the fractional integro-differentiation operator with the Gauss hypergeometric function. Under certain inequality type constraints on given functions and orders of fractional derivatives in the boundary condition, the energy integrals method enables one to proved the uniqueness of the solution of the problem. The functional relations between the trace of the desired solution and its derivative are obtained, which are brought to the degeneration line from the hyperbolic and parabolic parts of the mixed region. Under the conditions of the uniqueness theorem the existence of a solution to the problem is proved by equivalent reduction to the second kind Fredholm integral equations with the derivative of the sought function as an unknown, the unconditional solvability of which is deduced from the uniqueness of the solution of the problem. The limits of the change of orders of fractional integro-differential operators in which the solution of the problem exists and is unique are also determined. The effect of the coefficient of the lowest derivative in the equation on the solvability of the problem is established.