On a non-local boundary problem for a parabolic–hyperbolic equation involving a Riemann–Liouville fractional differential operator

In the present work, a non-local boundary value problem with special gluing conditions for a mixed parabolic–hyperbolic equation with parameter is considered. The parabolic part of this equation is a fractional analogue of heat equation and the hyperbolic part is the telegraph equation. The considered problem is reduced, for positive values of the parameter, to an equivalent system of the second kind Volterra integral equations. Due to the influence of the fractional diffusion equation, the looked for solution belongs to a specific class of functions. The method of the Green functions and the properties of integro-differential operators are on the basis of the investigation.