Inverse problems for q-analogue of the heat equation

In this paper we explore the weak solution of a time-dependent inverse source problem and inverse initial problem for $q$-analogue of the heat equation. As an over-determination condition we have used integral type condition on space-variable (in the case of inverse source problem) and final time condition (in the case of inverse initial problem). Series form of considered problems have been obtained via the method of spectral expansions.