On a boundary value problem for a nonlocal mixed-type equation with the Hilfer operator

In this paper, we consider a boundary value problem for a fourth-order differential equation of mixed type with involution and with Hilfer operator of fractional integro-differentiation in a rectangular domain. The mixed type differential equation under consideration is a fourth-order differential equation with respect to the second variable. Regarding the first variable, this equation is a fractional differential equation in the positive part of the segment and is a second-order differential equation in the negative part of the segment. Using the spectral method of separation of variables, the solution of the problem is constructed in the form of a Fourier series. Theorems on the existence and uniqueness of the problem are proved.

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