A nonlocal problem with fractional derivatives for third-order equations

The unique solvability of a nonlocal boundary value problem for the equation of mixed hyperbolic-parabolic type of the third order is investigated. The boundary condition of the problem contains a linear combination of fractional, in the sense of Riemann–Liouville, operators of integro-differentiation with hypergeometric Gauss function on the values of the solution on the characteristics pointwise associated with the values of the solution and its derivative on the degeneration line. Theorems of existence and uniqueness of the solution to the problem in various cases of the exponent in the equation under consideration are formulated and proved. The uniqueness of the solution of the problem, under certain restrictions of the inequality type on the given functions and orders of fractional derivatives in the boundary condition, is proved by the method of energy integrals. Functional relations between the trace of the desired solution and its derivative brought to the degeneration line from the hyperbolic and parabolic parts of the mixed domain are written. Under the conditions of the uniqueness theorem, we prove the existence of a solution to the problem by equivalent reduction to Fredholm integral equations of the second kind with respect to the derivative of the trace of the desired solution. Also, the intervals of changing the orders of fractional integro-differentiation operators are determined, at which the solution of the problem exists and is unique. The effect of the coefficient at the lower derivative in the original equation on the unique solvability of the problem is established.