Space fractional differential operators are used to study long-range interactions, and time differential operators handle memory effects. A semi-infinite circular cylinder is taken into consideration to analyse both effects in a two-dimensional thermoelastic situation where heat conduction is influenced by internal heat generation. A prescribed jump function is applied to the bottom of the semi-infinite circular cylinder, and the time-dependent heat flux happens at the curved edge of the cylinder. The transformative approach of Laplace, Fourier, and Hankel was used to solve the governing equation of heat transfer with Caputo and the finite fractional derivatives of Riesz. The outcomes are expressed in terms of the Bessel function series. The numerical calculations are performed with the material properties of pure copper, and the graphical representations of the thermal distributions are successfully plotted.