analysis of elliptic inverse heat conduction problems using a pascal polynomial numerical approach

This study presents a pascal polynomial-based numerical framework for solving inverse heat conduction problems governed by the steady-state poisson equation. the proposed methodology employs the pascal polynomial collocation method (ppcm) and its regularized variant (ppcm-t) to reconstruct unknown boundary conditions and source terms in two-dimensional bounded domains. two complementary strategies are adopted: one directly approximates both the temperature field and the unknown source term using pascal polynomials, while the other reformulates the inverse problem as a direct fourth-order system by assuming that the unknown source satisfies laplace's equation. several benchmark examples defined over rectangular and annular geometries are investigated to assess the method's accuracy and stability. the results demonstrate that both ppcm and ppcm-t achieve excellent agreement with analytical or reference solutions under noise-free data, with systematic error reduction as the number of collocation nodes increases. under noisy boundary data, ppcm-t exhibits superior robustness due to its built-in regularization, maintaining acceptable accuracy and numerical stability. overall, the pascal polynomial-based framework provides a flexible, mesh-free, and computationally efficient approach for addressing inverse elliptic and heat conduction problems, offering strong potential for broader applications in computational mechanics and thermal analysis.