Analytical solutions to generalized problems of locally nonequilibrium heat transfer: Operational method

Cover Page

Cite item

Full Text

Abstract

This study develops an analytical framework for mathematical modeling of locally nonequilibrium heat transfer in the context of boundary value problems for hyperbolic-type equations with generalized boundary conditions. Nonstandard operational relations based on the Laplace transform and their corresponding originals, which are absent from known handbooks on operational calculus, are presented. The obtained image–original relations are characteristic of operational solutions to a broad class of generalized boundary value problems arising in various branches of mathematical physics (heat conduction, diffusion, hydrodynamics, oscillation theory, electrodynamics, thermomechanics). The lack of a developed mathematical apparatus, including complex operational relations, has previously precluded the existence of functional constructs serving as exact analytical solutions for this class of heat transfer problems. The present study proposes an approach to solving this problem and significantly expands the analytical capabilities in the field of generalized locally nonequilibrium heat transfer problems. Solutions for partially bounded and finite domains of canonical shape are provided as illustrations.

About the authors

Eduard M. Kartashov

Moscow Aviation Institute (National Research University); MIREA — Russian Technological University

Email: professor.kartashov@gmail.com
ORCID iD: 0000-0002-7808-4246
Scopus Author ID: 7004134344
https://www.mathnet.ru/rus/person27518

Dr. Phys. & Math. Sci., Professor; Professor; Dept. of Computational Mathematics and Programming; Professor; Dept. of Higher and Applied Mathematics

Russian Federation, 125993, Moscow, Volokolamskoe Shosse, 4; 119454, Moscow, prosp. Vernadskogo, 78

Sergey S. Krylov

Moscow Aviation Institute (National Research University)

Email: compgra@yandex.ru
ORCID iD: 0000-0003-3267-6411
SPIN-code: 8634-7203
Scopus Author ID: 55453093500
https://www.mathnet.ru/eng/person232488

Cand. Phys. & Math. Sci., Associate Professor; Head of Department; Dept. of Computational Mathematics and Programming

Russian Federation, 125993, Moscow, Volokolamskoe Shosse, 4

Evgenii V. Nenakhov

Moscow Aviation Institute (National Research University)

Author for correspondence.
Email: newnew94@mail.ru
ORCID iD: 0000-0003-4165-0520
SPIN-code: 8305-3922
Scopus Author ID: 57221920841
https://www.mathnet.ru/rus/person169372

Cand. Phys. & Math. Sci.; Associate Professor; Dept. of Computational Mathematics and Programming

Russian Federation, 125993, Moscow, Volokolamskoe Shosse, 4

References

  1. Kartashov E. M., Kudinov V. A. Analiticheskaya teoriya teploprovodnosti i prikladnoy termouprugosti [Analytical Theory of Heat Conduction and Applied Thermoelasticity]. Moscow, URSS, 2012, 651 pp. (In Russian). EDN: QJZZGX.
  2. Kartashov E. M., Kudinov V. A. Matematicheskie modeli teploprovodnosti i termouprugosti [Mathematical Models of Heat Conduction and Thermoelasticity]. Moscow, MIREA, 2013, 1200 pp. (In Russian)
  3. Zarubin V. S., Kuvyrkin G. N. Matematicheskie modeli mekhaniki i elektrodinamiki sploshnoi sredy [Mathematical Models of Continuum Mechanics and Electrodynamics], Mathematical Modeling in Engineering and Technology. Moscow, Bauman Moscow State Technical Univ., 2008, 511 pp. (In Russian). EDN: QJUOAF.
  4. Zarubin V. S., Kuvyrkin G. N. Matematicheskie modeli termomekhaniki [Mathematical Models of Thermomechanics]. Moscow, Fizmatlit, 2002, 168 pp. (In Russian). EDN: UGLEJJ.
  5. Kudinov I. V., Kudinov V. A. Mathematical simulation of the locally nonequilibrium heat transfer in a body with account for its nonlocality in space and time, J. Eng. Phys. Thermophys., 2015, vol. 88, no. 2, pp. 406–422. EDN: SZQXMU. DOI: https://doi.org/10.1007/s10891-015-1206-6.
  6. Kudinov V. A., Eremin A. V., Kudinov I. V. The development and investigation of a strongly non-equilibrium model of heat transfer in fluid with allowance for the spatial and temporal non-locality and energy dissipation, Thermophys. Aeromech., 2017, vol. 24, no. 6, pp. 901–907. EDN: XXHYWT. DOI: https://doi.org/10.1134/S0869864317060087.
  7. Kirsanov Yu. A., Kirsanov A. Yu. On measuring the thermal relaxation time of a solid body, Izv. RAN. Energetika, 2015, no. 1, pp. 113–122 (In Russian). EDN: TLUFJZ.
  8. Kirsanov Yu. A. Tsiklicheskie teplovye protsessy i teoriya teploprovodnosti v regenerativnykh vozdukhopodogrevatelyakh [Cyclic Thermal Processes and Heat Conduction Theory in Re- generative Air Heaters]. Moscow, Fizmatlit, 2007, 240 pp. (In Russian). EDN: RXGTPJ.
  9. Formalev V. F. Uravneniya matematicheskoi fiziki [Equations of Mathematical Physics]. Moscow, URSS, 2020, 646 pp. (In Russian)
  10. Baumeister K., Hamill T. Hyperbolic heat conduction equation. Solution of the semi-infinite body problem, Teploperedacha, 1969, no. 4, pp. 112–119 (In Russian).
  11. Demirchan K. S., Butyrin P. A. Modelirovanie i mashinnyi raschet elektricheskikh polei [Modeling and Computer Calculation of Electric Fields]. Moscow, Vyssh. shk., 1983, 335 pp. (In Russian)
  12. Kartashov E. M. Analytical solutions of hyperbolic heat-conduction models, J. Eng. Phys. Thermophys., 2014, vol. 87, no. 5, pp. 1116–1125. EDN: UFUQDP. DOI: https://doi.org/10.1007/s10891-014-1113-2.
  13. Fok I. A. Solution of the theory of diffusion problem by the finite difference method and its application to light diffusion, Trudy GOI [Proc. State Opt. Inst.], 1926, no. 4, pp. 1–31 (In Russian).
  14. Davydov B. I. Diffusion Equation Taking into Account Molecular Velocity, Dokl. Akad. Nauk SSSR, 1935, no. 2, pp. 474–475 (In Russian).
  15. Predvoditelev A. S. The theory on heat and Riemannian manifolds, In: Problems of Heat and Mass Transfer. Moscow, Energiya, 1970, pp. 151–192 (In Russian).
  16. Kartashov E. M., Krylov S. S. New analytical solutions of mathematical models of thermal shock in local non-equilibrium heat exchange, Izv. RAN. Energetika, 2023, no. 6, pp. 44–60 (In Russian). EDN: PWWXZS. DOI: https://doi.org/10.31857/S0002331023060031.
  17. Kartashov E. M., Krylov S. S. New functional relations in analytical thermal physics for locally non-equilibrium heat exchange processes, Teplovye protsessy v tekhnike, 2024, vol. 16, no. 6, pp. 243–256 (In Russian). EDN: ERMOQJ.
  18. Carslaw H. S., Jaeger J. C. Operational Methods in Applied Mathematics. New York, Oxford Univ., 1941, xvi+264 pp.
  19. Kartashov E. M. Developing generalized model representations of thermal shock for local non-equilibrium heat transfer processes, Russ. Technol. J., 2023, vol. 11, no. 3, pp. 70–85 (In Russian). EDN: YTYOXG. DOI: https://doi.org/10.32362/2500-316X-2023-11-3-70-85.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2025 Authors; Samara State Technical University (Compilation, Design, and Layout)

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

Согласие на обработку персональных данных

 

Используя сайт https://journals.rcsi.science, я (далее – «Пользователь» или «Субъект персональных данных») даю согласие на обработку персональных данных на этом сайте (текст Согласия) и на обработку персональных данных с помощью сервиса «Яндекс.Метрика» (текст Согласия).