Hilfer-type fractional differential equations with variable coefficients

Abstract

• We give the explicit solution of aHilfer-type fractional differential equation with continuous variable coefficients. • The obtained representation of the solution can be used effectively for computational and analytic purposes. • Our results become to different types as: Riemann-Lioville, Caputo, Hadamard, Weyl, Erdélyi-Kober, Prahbakar, etc. • For constant coefficients, the solution is given by the Riemann-Liouville fractional integral of multivariate Mittag-Leffler function. In this paper, we give a representation of the solution of Hilfer-type fractional differential equations with continuous variable coefficients. The solution is represented by convergent infinite series involving composition of Riemann–Liouville fractional integral operators. The obtained representation of the solution can be used effectively for computational and analytic purposes. For the case of constant coefficients, the solution is given by the Riemann-Liouville fractional integral of the multivariate Mittag-Leffler function.