A problem with M. Saigo operator in the boundary condition for a loaded heat conduction equation

The existence of a unique solution of the non-classical boundary value problem for the heat equation, the loaded value of the desired function $u(x,y)$ on the boundary $x=0$ of the rectangular area $ \Omega = \{ (x,t): 0 < x < l, 0 < t < T \}$ was proved. One of the boundary conditions of the problem has a generalized operator of fractional integro-differentiation in the sense of Saigo. Using the properties of the Green function of the mixed boundary value problem and the specified boundary condition, the problem reduces to an integral equation of Volterra type with respect to the trace of the desired function $u(0, t)$. It is shown that the equation is Volterra integral equation of the second kind with weak singularity in the kernel, which is unambiguously and unconditionally solvable. The main result is given in the form of the theorem. The special case is considered, where the generalized operator of fractional integro-differentiation of M. Saigo in the boundary condition reduces to the operator of Kober–Erdeyi. In this case, the existence of an unique solution of the boundary value problem is justified.