MULTI-TERM TIME-FRACTIONAL DERIVATIVE HEAT EQUATION FOR ONE-DIMENSIONAL DUNKL OPERATOR

Abstract

In this paper, we investigate the well-posedness for Cauchy problem for multi-term time-fractional heat equation associated with Dunkl operator. The equation under consideration includes a linear combination of Caputo derivatives in time with decreasing orders in (0, 1) and positive constant coefficients and one-dimensional Dunkl operator.  To show solvability of this problem  we use several important properties of multinomial Mittag-Leffler functions and Dunkl transforms, since various estimates follow from the explicit solutions in form of these special functions and transforms. Then we prove the uniqueness and existence results. To achieve our goals, we use methods corresponding to the different areas of mathematics such as the theory of partial differential equations, mathematical physics, hypoelliptic operators theory and functional analysis. In particular, we use the direct and inverse Dunkl transform to establish the existence and uniqueness of solutions to this problem on the Sobolev space. The generalized solutions of this problem are studied.