Dirichlet boundary value problem in half-strip for fractional differential equation with Bessel operator and Riemann–Liouville partial derivative

In the work we study the Dirichlet boundary value problem in a half-strip for a fractional differential equations with the Bessel operator and the Riemann–Liouville partial derivatives. We formulate the unique solvability theoresm for the considered problem. We find the representations for the solutions in terms of the integral transform with the Wright function in the kernel. The proof of the existence theorem is made on the base of the mentioned integral transform and the modified Bessel function of first kind. The uniqueness of the solutions is shown in the class of the functions satisfying an analogue of Tikhonov equation. In the case, when the considered equations is the fractional order diffusion equation, we show that the obtained solutions coincides with the known solution to the Dirichlet problem for the corresponding equation. We also consider the case when the initial function is power in the spatial variable. In this case the solution to the problem is written out in terms of the Fox $H$-function.