A boundary-value problem for a mixed type equation involving hyper-Bessel fractional differential operator and Hilfer's bi-ordinal fractional derivative

TL;DR

This paper investigates a boundary-value problem involving a mixed-type equation with hyper-Bessel and Hilfer's fractional derivatives, establishing unique solvability and providing explicit solutions using advanced mathematical methods.

Contribution

It introduces a novel boundary-value problem with hyper-Bessel and Hilfer's derivatives and solves it explicitly, advancing fractional differential equations theory.

Findings

Proved unique solvability of the boundary-value problem.

Derived explicit solutions for initial problems.

Applied separation of variables and Laplace transform methods.

Abstract

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