Application of the Fourier Transform to the Solution of a One-Dimensional Non-Stationary Heat Conduction Problem

Article Sidebar

PDF
Published: 16.09.2026

Main Article Content

Akhmetova Dilnaz Saidovna
Abai Kazakh National Pedagogical University
Papyshev A. A.
Abai Kazakh National Pedagogical University

Abstract

This article examines the application of the Fourier transform to the analytical solution of a one-dimensional unsteady heat conduction problem in an unbounded homogeneous rod. The original parabolic heat conduction equation is transformed with respect to the spatial coordinate into an ordinary differential equation with respect to time, after which an inverse Fourier transform is performed, yielding an integral representation of the solution in terms of the heat conduction kernel. An explicit analytical formula is derived for a localised Gaussian initial temperature distribution. Based on this, a parametric numerical experiment is carried out, allowing one to track the change in the maximum temperature and the expansion of the heated region over time, as well as to assess the influence of the thermal conductivity coefficient on the rate at which the temperature field equilibrates.

Downloads

Download data is not yet available.

Article Details

Data Availability Statement

All data supporting the findings of this study are presented in the text of the scientific work.

Issue

No. 309 (2026): 14th ISPC «Global and Regional Aspects of Sustainable Development» (September 16-18, 2026; Copenhagen, Denmark)

Section

Physics and Maths
Creative Commons License

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

Author Biographies

Akhmetova Dilnaz Saidovna, Abai Kazakh National Pedagogical University

Master’s student, 2nd year, 7M05401 – Mathematics Program, Faculty of Mathematics, Physics and Informatics

Papyshev A. A., Abai Kazakh National Pedagogical University

Ph.D.

How to Cite

Akhmetova, D., & Papyshev, A. A. (2026). Application of the Fourier Transform to the Solution of a One-Dimensional Non-Stationary Heat Conduction Problem. Scientific Collection «InterConf», 309, 88–96. https://interconf.openpubarchive.com/index.php/proceeding/article/view/97

References

Carslaw H.S., Jaeger J.C. Conduction of Heat in Solids. 2nd ed. Oxford: Clarendon Press, 1959. 510 p.

Özisik M.N. Heat Conduction. 2nd ed. New York: John Wiley & Sons, 1993. 692 pp.

Evans L.C. Partial Differential Equations. 2nd ed. Providence, RI: American Mathematical Society, 2010. 749 pp.

Li X., Zhao S., Qu W. Localised Fourier collocation method for 2D transient heat conduction problems // Applied Mathematics Letters. 2025. Vol. 160. Article 109331. DOI: 10.1016/j.aml.2024.109331.

Yu J., An C., He Y., Su J. Transient nonlinear heat conduction in a slab with temperature -dependent thermal conductivity: Integral transform and lumped model solutions // International Communications in Heat and Mass Transfer. 2024. Vol. 158. Article 107886. DOI: 10.1016/j.icheatmasstransfer.2024.107886.

Delouei A.A., Emamian A., Ghorbani S., He F. A general unsteady Fourier solution for orthotropic heat transfer in 2D functionally graded cylinders // Mathematical Methods in the Applied Sciences. 2024. Vol. 47, No. 6. pp. 3942–3959. DOI: 10.1002/mma.9796.

Wu D., Wang Y., Tao Y., Li X., Ren H., Yang Q. A Fast Fourier Transform Solution for a 1D Unsteady Heat Transfer Model Bounded by a Varying Temperature // Archives of Thermodynamics. 2025. Vol. 46, No. 2. pp. 167– 172. DOI: 10.24425/ather.2025.154915.

Koilyshov U.K., Sadybekov M.A., Beisenbayeva K.A. Solution of an initial - boundary value problem for the heat equation with a discontinuous coefficient and a general conjugation condition // Filomat. 2025. Vol. 39, No. 23. pp. 7997–8005. DOI: 10.2298/FIL2523997K.

Crank J. The Mathematics of Diffusion. 2nd ed. Oxford: Clarendon Press, 1975. 414 pp.

Beck J.V., Cole K.D., Haji -Sheikh A., Litkouhi B. Heat Conduction Using Green’s Functions. 2nd ed. Boca Raton: CRC Press, 2011. 694 pp.