(untitled) (European Journal of Pure and Applied Mathematics, 2026)
The classical thermal entrance problem in a circular tube is re-examined through two independent formulations of the reduced Graetz equation: an analytical Bessel series solution and a finite-difference Crank–Nicolson (CN) numerical scheme. The analytical formulation represents the temperature field as a superposition of exponentially decaying radial eigenmodes, providing a compact and physically transparent description of thermal development.The numerical secheme employs a second-order radial discretisation combined with an unconditionally stable CN marching strategy, enabling accurate resolution of steep temperature gradients near the inlet. A detailed comparison between the analytical and numerical solutions demonstrates excellent agreement for radial temperature profiles, centreline evolution, and near-wall behaviour, with a maximum centreline discrepancy of approximately 0.2%. The results confirm the accuracy, stability, and low numerical dissipation of the CN scheme, while the analytical solution clearly captures the axial decay of radial temperature gradients during boundary-layer development. The combined framework therefore provides a reliable reference solution for the reduced Graetz problem and a consistent mathematical basis for further investigations of parabolic partial differential equations arising in internal flow models.