Direct and inverse problems for time-fractional heat equation generated by Dunkl operator

Abstract

Abstract In this paper, we study non–local in time evolution type equations generated by the Dunkl operator. Direct and inverse problems are investigated with the Caputo time-fractional heat equation with the parameter 0 < γ ≤ 1 {0<\gamma\leq 1} . In particular, well-posedness properties are established for the forward problem. To adopt techniques of the harmonic analysis, we solve the problems in the Sobolev type spaces associated with the Dunkl operator. Our special interest is an inverse source problem for the Caputo–Dunkl heat equation. As additional data, the final time measurement is taken. Since our inverse source problem is ill-posed, we also show the stability result. Moreover, as an advantage of our calculus used here, we derive explicit formulas for the solutions of the direct and inverse problems.