Analytical and Numerical Solution of the Heat Conduction Equation in a Cylindrical Domain: Convergence Analysis and Bessel Collocation Method

Abstract

This paper presents both analytical and numerical approaches for solving the non-homogeneous heat conduction equation in a cylindrical domain with time-dependent polynomial and harmonic boundary conditions. The analytical solution is constructed using the superposition principle and Fourier-Bessel series expansion, with rigorous proofs of convergence in the \(L_2\) norm and asymptotic stability. For numerical implementation, we develop a Bessel Collocation Method that achieves spectral accuracy with moderate computational cost. Theoretical error estimates are derived, showing exponential convergence in space and second-order accuracy in time. Numerical experiments confirm the theoretical predictions, with absolute errors decreasing from \(10^{-2}\) to \(10^{-15}\) as the number of basis functions increases. The results demonstrate that the proposed framework provides reliable solutions for heat transfer problems with complex boundary regimes in circular geometries.