Advanced Mathematical Models & Applications

This paper presents a sequential polynomial expansion method for solving inverse boundary condition problems in two-dimensional parabolic heat conduction. The approach combines temporal discretization via backward Euler scheme with spatial approximation using polynomial expansions, transforming the original time-dependent problem into a sequence of modified Helmholtz equations. The method effectively handles the ill-posed nature of inverse Cauchy problems through careful selection of optimal polynomial degrees and demonstrates robust performance across diverse solution profiles including oscillatory, hyperbolic-type, and exponentially decaying functions. Numerical experiments reveal that the method achieves optimal accuracy at specific polynomial degrees (m=7 or m=9 depending on solution characteristics) while maintaining computational efficiency with stable iteration counts. Despite challenges posed by ill-conditioned matrices at higher polynomial degrees, the approach provides accurate boundary condition reconstruction with controlled error propagation, making it a valuable tool for practical inverse problem applications in heat transfer and related fields.