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Abstrakty
In this paper we have investigated the existence, uniqueness and possibility of constructing of two-sided approximations to the positive solution of a heat conduction problem with two sources. The investigation is based on methods in operator equations theory in half-ordered spaces. In this case we have considered a nonlinear operator equation that corresponds to the initial boundary value problem in a cone of non-negative continous functions. The properties of the corresponding operator define conditions which provide the existence and uniqueness of the solution. The conditions link the parameters of the problem implicitly meaning that they don’t provide the range of allowed values but need to be verified for each specific parameters value set separately. During the investigation we have provided the scheme of a two-sided iteration process which must satisfy the conditions in order to converge to the positive solution from both sides. The computational experiment have been conducted in two domains – unit disk and unit half disk. We have applied both two-sided approximations method and Green’s quasifunction method for the problem solving. The obtained results are presented as a surface and level lines plots and also as a table. The results in corresponding domains obtained by different methods have been compared with each other.
Czasopismo
Rocznik
Tom
Strony
11--19
Opis fizyczny
Bibliogr. 21 poz., rys., tab., wz.
Twórcy
autor
- Kharkiv National University of Radio Electronics
autor
- Kharkiv National University of Radio Electronics
Bibliografia
- 1. Bellman R., Kalaba R., 1968. Quasilinearization and Nonlinear Boundary Value Problems. Moscow:Mir, 186. (In Russian).
- 2. Frank-Kamenetskii D., 2008. Basics of Macrokinetics. Diffusion and Heat Transfer in Chemical Kinetics. Dolgoprudny:Intellekt Press, 408. (In Russian).
- 3. Dong Ye, Feng Zhou., 2001. A generalized two dimensional Emden-Fowler equation with exponential nonlinearity. Calculus of Variations and Partial Differential Equations. Volume 13. Issue 2, 141-158.
- 4. Bozhkov Y., 2005. Noether Symmetries and Critical Exponents. Symmetry, Integrability and Geometry: Methods and Applications. Volume 1. Paper 022, 1-12.
- 5. Rvachev V., Slesarenko A., Safonov N., 1992. Mathematical modelling of thermal self-ignition under stationary conditions by the R-functions method. Doklady Akademii Nauk Ukrainskoi SSR. Seriya A. n. 12, 24-27. (In Russian).
- 6. Galakhov E., 2005. Positive Solutions of Quasilinear Elliptic Equations. Mat. Zametki. Volume 78. Issue 2, 202-211. (in Russian).
- 7. Matinfar M., Nemati K., 2008. A numerical extension on a convex nonlinear elliptic problem. International Mathematical Forum. Volume 3. No. 17, 811-816.
- 8. Ambrosetti A., Brezis H., Cerami G., 1994. Combined effects of concave and convex nonlinearities in some elliptic problems. Journal of Functional Analysis. Volume 122. Issue 2, 519-543.
- 9. Baraket S. and Dong Ye, 2001. Singular limit solutions for two-dimensional elliptic problems with exponentially dominated nonlinearity. Chinese Annals of Mathematics. Series B. Volume 22. Issue 03, 287-296.
- 10. Junping Shi, Miaoxin Yao, 2005. Positive solutions for elliptic equations with singular nonlinearity. Electronic Journal of Differential Equations. Volume 04, 1-11.
- 11. Kolosova S., Lukhanin V., Sidorov M., 2013. About construction iterative methods of boundary value problems for nonlinear elliptic equations. Visnyk of Zaporizhzhya National University. Physical and mathematical Sciences. № 1, 35-42. (In Russian).
- 12. Kolosova S., Lukhanin V., Sidorov M., 2015. On the construction of two-sided approximations to the positive solution of the Lane-Emden equation. Visnyk of Zaporizhzhya National University. Physical and mathematical Sciences. № 3, 107-120. (In Russian).
- 13. Lukhanin V., 2015. On the construction of twosided approximations to the positive solution of the elliptic boundary value problem with exponential dominant. Radioelektronika i informatika. № 2 (69), 16- 18. (In Ukrainian).
- 14. Rvachev V., 1982. Theory of R-functions and Some Applications. Kiev: Naukova Dumka, 552. (In Russian).
- 15. Artyukh A., Sidorov M., 2014. Mathematical Modeling and numerical analysis of nonstationary planeparallel flows of viscous incompressible fluid by Rfunctions and Galerkin Method. Econtechmod. An International Quarterly Journal. Vol. 3. № 3, 3-11.
- 16. Lamtyugova S., Sidorov M., 2014. Numerical analysis of the problem of flow past a cylindrical body applying the R-functions method and the Galerkin method. Econtechmod. An International Quarterly Journal. Vol. 3. № 3, 43-50.
- 17. Krasnosel'skii M., 1962. Positive Solutions of Operator Equations. Moscow:Fizmatgiz, 394. (In Russian).
- 18. Opojtsev V., 1978. A generalization of the theory of monotone and concave operators. Trans. Mosc. Math. Soc. Volume 36, 237-273. (In Russian).
- 19. Opojtsev V., Khurodze T., 1984. Nonlinear operators in spaces with a cone. Tbilisi: Tbilis. Gos. Univ., 269. (In Russian).
- 20. Svirsky I., 1968. Methods of the Bubnov- Galerkin Type and a Sequence of Approximation. Мoscow: Nauka, 287. (In Russian).
- 21. Mikhlin S., 1970. Variational methods in mathematical physics. Мoscow: Nauka, 512. (In Russian).
Uwagi
Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).
Typ dokumentu
Bibliografia
Identyfikator YADDA
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