About new nonlinear properties of the problem of nonlinear thermal conductivity
Keywords:
degenerate nonlinear, estimate, exact solution, localization, parabolic equationAbstract
In this paper we consider a problem of nonlinear heat conduction with double nonlinearity under the action of strong absorption. For which an exact analytical solution is found, analysis of which makes it possible to reveal several characteristic features of thermal processes in nonlinear media. The following nonlinear effects are established: an inertial effect of a finite velocity of propagation of thermal disturbances, spatial heat localization, and finite time effect i.e. existence of a thermal structure in a medium with strong absorption. The following nonlinear effects are observed in the problem under consideration: the inertial effect of an ultimate speed of propagation of thermal perturbations, the effect of spatial localization of heat, and the effect of ultimate time of the thermal structure in an absorption medium.
Downloads
References
Abbasov, I. B. (2019). A Research and Modeling of Wave Processes at the Scattering of Nonlinear Acoustic Waves on Cylindrical Bodies. International Research Journal of Engineering, IT and Scientific Research, 5(5), 32-44.
Aripov, M., & Sayfullayeva, M. (2020). On the new nonlinear properties of the nonlinear heat conductivity problem in nondivergence form. Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences, 3(2), 200-208.
Bernis, F., & Friedman, A. (1990). Higher order nonlinear degenerate parabolic equations. Journal of differential equations, 83(1), 179-206. https://doi.org/10.1016/0022-0396(90)90074-Y
Bhagavannarayana, G., Riscob, B., & Shakir, M. (2011). Growth and characterization of l-leucine l-leucinium picrate single crystal: a new nonlinear optical material. Materials Chemistry and Physics, 126(1-2), 20-23. https://doi.org/10.1016/j.matchemphys.2010.12.040
Bhat, M. N., & Dharmaprakash, S. M. (2002). New nonlinear optical material: glycine sodium nitrate. Journal of crystal growth, 235(1-4), 511-516. https://doi.org/10.1016/S0022-0248(01)01799-7
Dehghan, M. (2001). An inverse problem of finding a source parameter in a semilinear parabolic equation. Applied Mathematical Modelling, 25(9), 743-754. https://doi.org/10.1016/S0307-904X(01)00010-5
Isaenko, L., Vasilyeva, I., Merkulov, A., Yelisseyev, A., & Lobanov, S. (2005). Growth of new nonlinear crystals LiMX2 (M= Al, In, Ga; X= S, Se, Te) for the mid-IR optics. Journal of crystal growth, 275(1-2), 217-223. https://doi.org/10.1016/j.jcrysgro.2004.10.089
Jüngel, A. (2017). Cross-diffusion systems with entropy structure. arXiv preprint arXiv:1710.01623.
Kurdyumov, S. P. (1990). Evolution and self-organization laws in complex systems. Advances in Theoretical Physics, 134.
Li, C. L., & Cui, M. G. (2003). The exact solution for solving a class nonlinear operator equations in the reproducing kernel space. Applied Mathematics and Computation, 143(2-3), 393-399. https://doi.org/10.1016/S0096-3003(02)00370-3
Martinson, L. K., & Pavlov, K. B. (1972). The problem of the three-dimensional localization of thermal perturbations in the theory of non-linear heat conduction. USSR Computational Mathematics and Mathematical Physics, 12(4), 261-268.
Mercaldo, A., Peral, I., & Primo, A. (2011). Results for degenerate nonlinear elliptic equations involving a Hardy potential. Journal of Differential Equations, 251(11), 3114-3142. https://doi.org/10.1016/j.jde.2011.07.024
Mersaid, A. (2013). To properties of solutions to reaction-di ? usion equation with double nonlinearity with distributed parameters. Journal of the Siberian Federal University. Mathematics and Physics , 6 (2).
Shkir, M., & Abbas, H. (2014). Physico chemical properties of l-asparagine l-tartaric acid single crystals: A new nonlinear optical material. Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy, 118, 172-176. https://doi.org/10.1016/j.saa.2013.08.086
Wei, Z., Li, G., & Qi, L. (2006). New nonlinear conjugate gradient formulas for large-scale unconstrained optimization problems. Applied Mathematics and computation, 179(2), 407-430. https://doi.org/10.1016/j.amc.2005.11.150
Yang, L., Yu, J. N., & Deng, Z. C. (2008). An inverse problem of identifying the coefficient of parabolic equation. Applied Mathematical Modelling, 32(10), 1984-1995. https://doi.org/10.1016/j.apm.2007.06.025
Zel'Dovich, Y. B., & Raizer, Y. P. (2002). Physics of shock waves and high-temperature hydrodynamic phenomena. Courier Corporation.
Zmitrenko, N. V., & Kurdyumov, S. P. (1992). Time Reversion of Processes in Dissipative Systems. Modern Physics Letters B, 6(01), 49-54.
Published
How to Cite
Issue
Section
Copyright (c) 2021 International journal of physics & mathematics
![Creative Commons License](http://i.creativecommons.org/l/by-nc-nd/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Articles published in the International Journal of Physics & Mathematics (IJPM) are available under Creative Commons Attribution Non-Commercial No Derivatives Licence (CC BY-NC-ND 4.0). Authors retain copyright in their work and grant IJPM right of first publication under CC BY-NC-ND 4.0. Users have the right to read, download, copy, distribute, print, search, or link to the full texts of articles in this journal, and to use them for any other lawful purpose.
Articles published in IJPM can be copied, communicated and shared in their published form for non-commercial purposes provided full attribution is given to the author and the journal. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.