TARU PUBLICATIONS
Journal of Interdisciplinary Mathematics cover
Open Access ·Peer-reviewed·ISSN (Online): 2169-012X·ISSN (Print): 0972-0502

Freq.: MONTHLY - Publishes the methodological and theoretical role of mathematics and mathematical applications underpinning scientific research.

Issues up to 2022 co-published with and available at:Taylor & Francis Online
submissions@tarupublications.com
Open Access Research Article

Solvability and factorization of mixed problem for the heat conduction equation

* , , ,

* Corresponding author · click or hover a name for details

pp. 1595–1605Vol. 28Issue 4June 2025DOI: 10.47974/JIM-2154XML
Received:
11 Jun 2024
Published Online:
10 Jun 2025
Article type:
Research Article
Language:
EN
Article no.:
JIM-2154
Pages:
1595–1605

Abstract

The study aims to analyse the solvability of mixed problems for the heat conduction equation. To achieve the objectives, analytical methods, methods of mathematical analysis of mixed problems and heat conduction equations were applied. Functional analysis and differential equation theory methods were also used to study the solvability. The study determined that the equation under consideration has a wide range of applications and plays an important role in describing heat transfer processes in various fields of science and technology. The study revealed that analysing mixed problems involving a variety of initial and boundary conditions provides a key tool for gaining a deeper understanding of heat transfer processes. Furthermore, properly chosen initial and boundary conditions have a significant impact on the long-term system operation. The study results have important implications for understanding and modelling heat conduction processes in a variety of physical systems. The results provide a deeper understanding of heat conduction processes and their mathematical description in a variety of systems, which opens new perspectives for improving the prediction and control of heat conduction phenomena. This study represents a significant contribution to the field of thermal conduction, not only expanding the theoretical understanding of this physical phenomenon but also providing a basis for practical applications.

Keywords

Subject Classifications

35K0535Pxx

References

[1] N. Acharya, “Spectral quasi linearization simulation of radiative nanofluidic transport over a bended surface considering the effects of multiple convective conditions,” European Journal of Mechanics - B/Fluids, vol. 84, pp. 139–154 (2020).
[2] T. K. Yuldashev and B. J. Kadirkulov, “Boundary value problem for weak nonlinear partial differential equations of mixed type with fractional Hilfer operator,” Axioms, vol. 9, no. 2, p. 68 (2020).
[3] Y. Gouari and Z. Dahmani, “Solvability for a class of nonlocal singular fractional differential equations of Lane-Emden type,” Journal of Interdisciplinary Mathematics, vol. 24, no. 5, pp. 1221–1240 (2021).
[4] S. Kumar, S. Ghosh, B. Samet, and E. F. D. Goufo, “An analysis for heat equations arises in diffusion process using new Yang-Abdel-Aty-Cattani fractional operator,” Mathematical Methods in the Applied Sciences, vol. 43, no. 9, pp. 6062–6080 (2020).
[5] T. Sh. Kalmenov, A. Sh. Shaldanbayev, M. I. Akylbayev, and A. N. Urmatova, “On completeness of root vectors of the Cauchy problem of the first order equation with deviating argument,” News of the National Academy of Sciences of the Republic of Kazakhstan: Physico-Mathematical Series, no. 2, pp. 50–57 (2020).
[6] E. Cheremnikh, H. Ivasyk, V. Alieksieiev, M. Kuchma, and O. Brodyak, “Construction of spectral decomposition for nonself-adjoint Friedrichs model operator,” Eastern-European Journal of Enterprise Technologies, vol. 4, no. 4-94, pp. 6–18 (2018).
[7] A. Sh. Shaldanbayev and M. T. Shomanbayeva, “About the existence and uniqueness of strong solution of the antiperiodic problem for the heat equation with deviating argument,” Bulletin of Karaganda University. Mathematics Series, vol. 81, no. 1, pp. 83–91 (2016).
[8] K. Rustemova, G. Takibaeva, A. Sabalakhova, and B. Tursynbek, “Spectral properties of some differentiation operator with deviated argument,” Scientific Heritage, no. 90, pp. 108–116 (2022).
[9] S. Carpentier, A. De Sole, and V. G. Kac, “Some algebraic properties of differential operators,” Journal of Mathematical Physics, vol. 53, no. 6, p. 063501 (2012).
[10] P. Koiran, “Hilbert’s Nullstellensatz is in the polynomial hierarchy,” Journal of Complexity, vol. 12, no. 4, pp. 273–286 (1996).
[11] T. K. Yuldashev and E. T. Karimov, “Inverse problem for a mixed type integro-differential equation with fractional order Caputo operators and spectral parameters,” Axioms, vol. 9, no. 4, pp. 121 (2020).
[12] J. Y. Salah, “Properties of the modified Caputo’s derivative operator for certain analytic functions,” International Journal of Pure and Applied Mathematics, vol. 109, no. 3, pp. 665–671 (2016).
[13] J. Salah, “Note on the modified Caputo’s fractional calculus derivative operator,” Far East Journal of Mathematical Sciences, vol. 100, no. 4, pp. 609–615 (2016).
[14] X. R. Rasulov, “On the solvability of a boundary value problem for a quasilinear equation of mixed type with two degeneration lines,” Journal of Physics: Conference Series, vol. 2070, p. 012002 (2021).
[15] Z. Suranchiyeva, B. Bostanov, S. Kenesbayev, S. Idrissov, and K. Turganbay, “Unveiling the digital equation through innovative approaches for teaching discrete mathematics to future computer science educators,” Journal of Information Technology Education: Innovations in Practice, vol. 22, pp. 215–234 (2023).
[16] A. P. Rotshtein, M. Posner, and H. B. Rakytyanska, “Cause and effect analysis by fuzzy relational equations and a genetic algorithm,” Reliability Engineering & System Safety, vol. 91, no. 9, pp. 1095–1101 (2006).
[17] Y. Mysak, I. Galyanchuk, and M. Kuznetsova, “Development of mathematical models and the calculations of elements of convective heat transfer systems,” Eastern-European Journal of Enterprise Technologies, vol. 4, no. 8-82, pp. 33–41 (2016).
[18] A. R. Seadawy, M. Iqbal, and D. Lu, “Nonlinear wave solutions of the Kudryashov-Sinelshchikov dynamical equation in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity,” Journal of Taibah University for Science, vol. 13, no. 1, pp. 1060–1072 (2019).
[19] S. Lyubchik, E. Lygina, A. Lyubchyk, S. Lyubchik, J. M. Loureiro, I. M. Fonseca, A. B. Ribeiro, M. M. Pinto, and A. M. S. Figueiredo, “The kinetic parameters evaluation for the adsorption processes at ‘liquid-solid’ interface,” in Electrokinetics Across Disciplines: Contemporary Approaches from Chemistry, Physics, Biology and Engineering Towards Sustainable Development, pp. 81–109 (2015).
[20] M. Simoncelli, N. Marzari, and A. Cepellotti, “Generalization of Fourier’s law into viscous heat equations,” Physical Review X, vol. 10, no. 1, pp. 011019 (2020).
[21] E. Schiassi, R. Furfaro, C. Leake, M. De Florio, H. Johnston, and D. Mortari, “Extreme theory of functional connections: A fast physics-informed neural network method for solving ordinary and partial differential equations,” Neurocomputing, vol. 457, pp. 334–356 (2021).
[22] M. Simoncelli, N. Marzari, and A. Cepellotti, “Generalization of Fourier’s law into viscous heat equations,” Physical Review X, vol. 10, no. 1, art. no. 011019 (2020).
[23] O. V. Makhnei, “Mixed problem for the differential equation of parabolic type with measures,” Mathematical Methods of Physics and Mechanics of Fields, vol. 61, no. 4, pp. 49–55, 2020.
[24] I. Boussaada, S. I. Niculescu, A. El-Ati, R. Pérez-Ramos, and K. Trabelsi, “Multiplicity-induced-dominancy in parametric second-order delay differential equations: Analysis and application in control design,” ESAIM: Control, Optimisation and Calculus of Variations, vol. 26, art. no. 57 (2020).
[25] J. Ciula, A. Generowicz, A. Oleksy-Gebczyk, A. Gronba-Chyla, I. Wiewiorska, P. Kwasnicki, P. Herbut, and V. Koval, “Technical and economic aspects of environmentally sustainable investment in terms of the EU taxonomy,” Energies, vol. 17, no. 10, pp. 2239 (2024).

Views: 109Downloads: 8Citations: 0