On a mixed problem for Hilfer type differential equation of higher order

Abstract

The study considers the solvability of a mixed problem for a Hilfer type partial differential equation of the even order with initial value conditions and small positive parameters in mixed derivatives in threedimensional domain. It studies the solution to this fractional differential equation of higher order in the class of regular functions. The case, when the order of fractional operator is 1 <α< 2, is examined. During this study the authors use the Fourier series method and obtain a countable system of ordinary differential equations. The initial value problem is integrated as an ordinary differential equation and the integrated constants find by the aid of given initial value conditions. Using the Cauchy-Schwarz inequality and the Bessel inequality, it is proved the absolute and uniform convergence of the obtained Fourier series. The stability of the solution to the mixed problem on the given functions is studied.