The problem for a mixed equation with fractional power of the Bessel operator

Recently, of particular interest are partial differential equations containing a fractional order differential operator. Similar equations and problems for them find application in the theory of viscous elasticity, electrochemistry, control theory, modeling of epidemics and pandemics, and in various other areas. The present work is devoted to the solution of differential equations containing the Bessel operator of fractional degree. The article discusses the direct and inverse Meyer transforms, modified for the convenience of working with the Bessel operator of a fractional degree. For the considered Meyer transformation, a convolution is obtained. Using the Laplace and Poisson transformations, factorizations of the direct and inverse Meyer transformations are obtained. Using the considered modified Meyer transform, we find a solution to an ordinary differential equation with a Bessel operator of fractional degree. A nonlocal boundary value problem for a mixed parabolic-hyperbolic equation containing a fractional degree Bessel operator is considered. It is proved that, under certain conditions of smoothness of the input functions of the problem and the condition of conjugation on the dividing line of the regions of hyperbolicity and parabolicity, a regular solution of a nonlocal boundary value problem for a mixed parabolic-hyperbolic equation with a Bessel operator of fractional degree exists and is unique.