In a rectangular domain, a boundary-value problem is considered for a mixed equation with a regularized Caputo-like counterpart of hyper-Bessel differential operator and the bi-ordinal Hilfer's fractional derivative. By using the method of separation of variables a unique solvability of the considered problem has been established. Moreover, we have found the explicit solution of initial-boundary problems for the heat equation with the regularized Caputo-like counterpart of the hyper-Bessel differential operator with the non-zero starting point.

In this paper, we consider a non-local boundary-value problem for a mixed-type equation involving the bi-ordinal Hilfer fractional derivative in rectangular domain. The main target of this work is to analyze the uniqueness and the existence of the solution of the considered problem by means of eigenfunctions. Moreover, we construct the solution of the ordinary fractional differential equation with the right-sided bi-ordinal Hilfer derivative by the method of reduction to the Volterra integral equation. Then, we present sufficient conditions for given data in order to show the existence of the solution.

The non-local problem is considered for the partial differential equation of mixed-type with Bessel operator and fractional order. Explicit solution is represented by Fourier-Bessel series in the given domain. It is established the connection between the given data and the uniquely solvability of the problem.

The m -point non-local problem is considered for the partial differential equation of mixed-type with singular coefficients, namely fractional wave equation involving the right-hand side bi-ordinal Hilfer derivative and sub-diffusion equation with the regularized Caputo-like counterpart hyper-Bessel differential operator. The main technique of solving the problem is based on the method of separation variables. Also, the connection between the given data and the uniquely solvability of the problem is established and an explicit solution is represented by Fourier-Bessel series in the given domain.