Analysis of Elliptic Inverse Heat Conduction Problems Using a Pascal Polynomial Numerical Approach

Document Type : Research Paper

Authors
Muhammad Nawaz Khan 1
Imtiaz Ahmad 2
Hijaz Ahmad 3 , 4 , 5 , 6 , 7
Osama Oqilat 8
1 Jadara University Research Center, Jadara University, 21110, Jordan
2 Department of Mathematics, Saveetha School of Engineering, SIMATS, Saveetha University, Chennai 602105, Tamil Nadu, India
3 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah, 42351, Saudi Arabia
4 Irfan Suat Gunsel Operational Research Institute, Near East University, Nicosia/TRNC, 99138 Mersin 10, Turkey
5 Sustainability Competence Centre, Széchenyi István University, Egyetem tér 1, H-9026 Győr, Hungary
6 VIZJA University, Okopowa 59, 01-043 Warsaw, Poland
7 Department of Mathematics, College of Science, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul 02841, South Korea
8 Department of Basic Sciences, Faculty of Arts and Science, Hourani Center for Applied Scientific Research, Al-Ahliyya Amman University, Amman, Jordan
10.22055/jacm.2026.48835.5535
Abstract
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.
Keywords
Inverse heat conduction
Poisson equation
Pascal polynomials
Numerical stability
Mesh-free numerical methods
Subjects

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Journal of Applied and Computational Mechanics
Volume 12, Issue 3
Summer 2026
Pages 1201-1216