This work utilizes the recently developed coupled constitutive relations (CCR) for closing the balance equations. Together, these are referred to as the CCR model. The model circumvents the shortcomings of well-established continuum theories, such as the Navier–Stokes–Fourier equations, in capturing rarefaction effects. The CCR model turns out to be adept for describing gas flows with a moderate degree of rarefaction accurately. The present work deals with harnessing the applicability and effectiveness of a numerical framework built upon the method of fundamental solutions applied on the CCR model, to describe thermally induced quasi two-dimensional rarefied gas flows. The work pertains specifically to the flow occurring between coaxially arranged elliptic and circular cylinders having different temperatures