Comparative Analysis of the Finite-Difference Method of Finite Elements in Heat Conduction Modeling

Received: 2026-10-14

Published: 2026-09-30

Abstract

In this paper, heat transfer equations describing heat propagation processes in solids, liquids, and gases are considered. The physical nature of heat conduction, the fundamental laws governing heat transfer, and their mathematical description are investigated. Particular attention is paid to the derivation of the heat conduction equation, the formulation of initial and boundary conditions, and the methods for solving the corresponding problems. Practical applications of heat transfer equations in physics, engineering, and technical calculations are also discussed. The obtained results can be used for the modeling of thermal processes and the analysis of temperature distribution in various media.

List of references

  1. Fletcher K. Vychislitel’nye metody v dinamike zhidkostey [Computational methods in fluid dynamics]. vol. 1–2. Mir; 1991. (In

  2. Russian).

  3. Malikov Z, Nazarov F, Khaydarov S. Numerical study of subsonic hot and near-sonic submerged jets based on a two-fluid model. J Appl Comput Mech. 2025;11(4):871–886.

  4. Nazarov FK, et al. Calculation of axisymmetric transonic jet with various turbulence models using COMSOL Multiphysics package. AIP Conf Proc. 2025;3256(1):040040.

  5. Malikov Z, Nazarov F. Numerical study of flow on a flat plate with zero pressure gradient based on a two-fluid model. AIP Conf Proc. 2025;3265(1):060003.

  6. Nazarov F, et al. Numerical calculation of turbulent flow in a two-dimensional flat diffuser based on the COMSOL Multiphysics software. In: Proceedings of the 7th International Conference on Future Networks and Distributed Systems; 2023:372–376.

  7. Parneix S, Durbin PA, Behnia M. Computation of 3-D turbulent boundary layers using the V2F model. Flow Turbul Combust. 1998;60(1):19–46.

  8. Mazumder MHM. Comparative study of turbulence models in CFD for transonic flow over the ONERA M6 wing. Discov Mech Eng. 2024;3:46. https://doi.org/10.1007/s44245-024-00080-5.

  9. Kuznetsov GV, Sheremet MA. Raznostnye metody resheniya zadach teploprovodnosti [Finite-difference methods for solving heat conduction problems]. Tomsk Polytechnic University; 2007. (In Russian).

  10. Samarskii AA, Vabishchevich PN. Vychislitel’naya teploperedacha [Computational heat transfer]. Editorial URSS; 2003. (In Russian).

  11. Mayeur J, Dumont A, Destarac D, Gleize V. RANS simulations on TMR test cases and M6 wing with the Onera elsA flow solver. In: 54th AIAA Aerospace Sciences Meeting; 2016. January 4–8, 2016. https://doi.org/10.2514/6.2015-1745.

  12. Zhukov MY, Shiryaeva EV. Ispol’zovanie paketa konechnykh elementov FreeFem++ dlya zadach gidrodinamiki, elektroforeza i biologii [Use of the FreeFem++ finite element package for problems of hydrodynamics, electrophoresis, and biology]. Southern Federal University; 2008. (In Russian).

  13. NASA Langley Research Center. 3D ONERA M6 Wing Validation Case. NASA Langley Research Center; 2026. Accessed October 6, 2026. Turbulence Modeling Resource. Available from: https://turbmodels.larc.nasa.gov/onerawingnumerics_val.html.

About the Authors

Farrukh Kh.Nazarov
Institute of Mechanics and Seismic Stability of Structures named after M.T. Urazbaev, Academy of Sciences of the Republic of Uzbekistan
Dilnoza A.Amirova
Institute of Mechanics and Seismic Stability of Structures named after M.T. Urazbaev, Academy of Sciences of the Republic of Uzbekistan

License

Copyright (c) 2026 Uzbek Journal “Problems of Mechanics”

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Comparative Analysis of the Finite-Difference Method of Finite Elements in Heat Conduction Modeling. (2026). Uzbek Journal “Problems of Mechanics”, 37(3), 84-91. https://pmjournal.uz/jour/article/view/27

Similar Articles

You may also start an advanced similarity search for this article.

ISSN 2010-7250 (Print)