Heat Transfer in <i>n</i> ‐Dimensional Parallelepipeds Under Zero Dirichlet Conditions

ABSTRACT This article presents a comprehensive analytical study of heat propagation in an n ‐dimensional bounded domain subject to zero Dirichlet boundary conditions, corresponding to a thermally insulated parallelepiped with internal heat sources. While classical heat conduction problems are well understood in one‐, two‐, and three‐dimensional geometries, their extension to higher‐dimensional domains remains both mathematically challenging and practically significant. The present research addresses this gap by employing affine transformations to generalize the standard unit cube into an arbitrary n ‐dimensional parallelepiped, thereby enabling a more versatile description of multidimensional thermal systems. The governing heat equation is solved using the method of separation of variables, leading to Fourier‐type series solutions constructed from the eigenfunctions of a multidimensional Sturm–Liouville problem. The convergence, smoothness, and stability of these series are rigorously established, ensuring that the solutions satisfy both the mathematical requirements of well‐posedness and the physical principles of heat transfer. A Green's function is further developed for the multidimensional setting, providing a fundamental tool for characterizing the system's response to localized heat sources and enabling broader applicability to engineering and computational models. The results of this study extend the classical theory of mathematical physics into higher dimensions while maintaining analytical transparency and physical interpretability. The findings hold theoretical importance for advancing the understanding of multidimensional parabolic partial differential equations and offer practical value for engineering disciplines that require precise thermal modeling in complex geometries, such as aerospace, materials science, and mechanical design.

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