Electronic Scientific Journal
 
Diagnostics, Resource and Mechanics 
         of materials and structures
Рус/Eng  

 

advanced search

IssuesAbout the JournalAuthorContactsNewsRegistration

2021 Issue 6

All Issues
 
2024 Issue 2
 
2024 Issue 1
 
2023 Issue 6
 
2023 Issue 5
 
2023 Issue 4
 
2023 Issue 3
 
2023 Issue 2
 
2023 Issue 1
 
2022 Issue 6
 
2022 Issue 5
 
2022 Issue 4
 
2022 Issue 3
 
2022 Issue 2
 
2022 Issue 1
 
2021 Issue 6
 
2021 Issue 5
 
2021 Issue 4
 
2021 Issue 3
 
2021 Issue 2
 
2021 Issue 1
 
2020 Issue 6
 
2020 Issue 5
 
2020 Issue 4
 
2020 Issue 3
 
2020 Issue 2
 
2020 Issue 1
 
2019 Issue 6
 
2019 Issue 5
 
2019 Issue 4
 
2019 Issue 3
 
2019 Issue 2
 
2019 Issue 1
 
2018 Issue 6
 
2018 Issue 5
 
2018 Issue 4
 
2018 Issue 3
 
2018 Issue 2
 
2018 Issue 1
 
2017 Issue 6
 
2017 Issue 5
 
2017 Issue 4
 
2017 Issue 3
 
2017 Issue 2
 
2017 Issue 1
 
2016 Issue 6
 
2016 Issue 5
 
2016 Issue 4
 
2016 Issue 3
 
2016 Issue 2
 
2016 Issue 1
 
2015 Issue 6
 
2015 Issue 5
 
2015 Issue 4
 
2015 Issue 3
 
2015 Issue 2
 
2015 Issue 1

 

 

 

 

 

A. L. Kazakov, L. F. Spevak

ON THE CONSTRUCTION OF A HEAT WAVE GENERATED BY A BOUNDARY CONDITION ON A MOVING BORDER

DOI: 10.17804/2410-9908.2021.6.054-067

The paper deals with the construction of solutions to a nonlinear heat equation, which have the type of heat waves propagating over a cold (zero) background with a finite velocity. Such solutions are atypical for parabolic equations. They appear due to the degeneration of the parabolic type of equation on a manifold where the desired function becomes zero. Various kinds of boundary conditions provide the existence of solutions with the desired properties. The most complicated of them, specifying nonzero values of the desired function on a moving manifold, is considered in this paper. A new theorem of the existence and uniqueness of the solution to the heat wave initiation problem under the considered boundary condition is proved. A method for constructing an approximate solution based on expansion in radial basis functions and the collocation method is proposed. The solution is constructed in two steps. At the first step, we construct a solution in the domain situated between the specified moving manifold and the zero front, which is determined in the process of solving. A special variable change similar to hodograph transformation is used. At the second step, we complete the solution in the domain situated between the initial and actual position of the moving manifold. Calculations are made showing that the new approach gives good results and more stable convergence as compared with the boundary element method used by the authors earlier.

Acknowledgments: The work was supported by the RFBR, project No. 20-07-00407.

Keywords: nonlinear heat equation, heat wave, power series, boundary element method, radial basis functions

References:

  1. Vazquez J.L. The Porous Medium Equation: Mathematical Theory, Oxford, Clarendon Press, 2007, 648 р. ISBN-10: 0198569033, ISBN-13: 978-019856903.
  2. Samarskii A.A., Galaktionov V.A., Kurdyumov S.P., Mikhailov A.P. Blow-up in Quasilinear Parabolic Equations, NY, Berlin, Walter de Gruyte, 1995, 534 p. ISBN 3-11-012754-7.
  3. Zeldovich Ya.B., Kompaneets A.S. On the theory of heat propagation with temperature-dependent thermal conductivity. In: Sbornik, posvyashchennyi 70-letiyu akademika A.F. Ioffe [Collection dedicated to the 70th anniversary of Academician A.F. Ioffe]. Moscow, Izd-vo AN SSSR Publ., 1950, pp. 61–71. (In Russian).
  4. Barenblatt G.I., Vishik I.M. On the Final Velocity of Propagation in Problems of Non-stationary Filtration of Liquid and Gas. Prikladnaya matematika i mekhanika, 1956, vol. 20, no. 3, pp. 411–417. (In Russian).
  5. Oleynik O.A., Kalashnikov A.S., Chzhou Yuy-Lin. The Cauchy Problem and Boundary Value Problems for Equations of the type of Non-stationary Filtration. Izvestiya AN SSSR. Seriya matematicheskaya, 1958, vol. 22, no. 5, pp. 667–704. (In Russian).
  6. Sidorov A.F. In: Izbrannye Trudy: Matematika. Mekhanika [Selected Works: Mathematics. Mechanics]. Moscow, Fizmatlit Publ., 2001, 576 p. (In Russian). ISBN 5-9221-0103-Х.
  7. Kazakov A.L., Spevak L.F. Boundary Elements Method and Power Series Method for One-dimensional Nonlinear Filtration Problems. Izvestiya IGU, Seriya Matematika, 2012, vol. 5, no. 2, pp. 2–17. (In Russian).
  8. Kazakov A.L., Spevak L.F. Numerical and analytical studies of a nonlinear parabolic equation with boundary conditions of a special form. Applied Mathematical Modelling, 2013, vol. 37, iss. 10–11, pp. 6918–6928. DOI: 10.1016/j.apm.2013.02.026.
  9. Kazakov A.L., Kuznetsov P.A., Spevak L.F. On a Degenerate Boundary Value Problem for the Porous Medium Equation in Spherical Coordinates. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2014, vol. 20, no. 1, pp. 119–129. (In Russian).
  10. Kazakov A.L., Nefedova O.A., Spevak L.F. Solution of the Problem of Initiating the Heat Wave for a Nonlinear Heat Conduction Equation Using the Boundary Element Method. Computational Mathematics and Mathematical Physics, 2019, vol. 59, No. 6, pp. 1015–1029. DOI: 10.1134/S0965542519060083.
  11. Banerjee P.К., Butterheld R. Boundary element methods in engineering science, US, McGraw-Hill Inc., 1981, 452 р. ISBN-10: 0070841209, ISBN-13: 978-0070841208.
  12. Brebbia C.A., Telles J.F.C., Wrobel L.C. Boundary Element Techniques, Berlin, Neidelberg, New-York, Tokyo, Springer-Verlag, 1984, 466 р. ISBN 978-3-642-48862-7. DOI: 10.1007/978-3-642-48860-3.
  13. Nardini D., Brebbia C.A. A New Approach to Free Vibration Analysis using Boundary Elements. Applied Mathematical Modelling, 1983, vol. 7, No. 3, pp. 157–162. DOI: 10.1016/0307-904X(83)90003-3.
  14. Wrobel L.C., Brebbia C.A., Nardini D. The dual reciprocity boundary element formulation for transient heat conduction. In: Finite elements in water resources VI, Springer-Verlag, Berlin, Germany, 1986, pp. 801–811.
  15. Tanaka M., Matsumoto T., Yang Q.F. Time-stepping boundary element method applied to 2-D transient heat conduction problems. Appl. Math. Model, 1994, vol. 18, pp. 569–576. DOI: 10.1016/0307-904X(94)90142-2.
  16. Powell M.J.D. The Theory of Radial Basis Function Approximation. In: Light W., ed. Advances in Numerical Analysis, Oxford Science Publications, Oxford, UK, 1992, vol. 2.
  17. Golberg M.A., Chen C.S., Bowman H. Some recent results and proposals for the use of radial basis functions in the BEM. Engineering Analysis with Boundary Elements, 1999, vol. 23, pp. 285–296. DOI: 10.1016/S0955-7997(98)00087-3.
  18. Spevak L.F., Nefedova O.A. Solving a two-dimensional nonlinear heat conduction equation with degeneration by the boundary element method with the application of the dual reciprocity method. AIP Conference Proceedings, 2016, vol. 1785, iss. 1, pp. 040077. – DOI: 10.1063/1.4967134.
  19. Kazakov A.L. On exact solutions to a heat wave propagation boundary-value problem for a nonlinear heat equation. Sibirskie Elektronnye Matematicheskiye Izvestiya, 2019, vol. 16, pp. 1057–1068. (In Russian). Available at: http://semr.math.nsc.ru/v16
  20. Kazakov A.L., Spevak L.F., Spevak E.L. On Numerical Methods for Constructing Benchmark Solutions to a Nonlinear Heat Equation with a Singularity. Diagnostics, Resource and Mechanics of materials and structures, 2020, iss. 5, pp. 26–44. DOI: 10.17804/2410-9908.2020.5.026-044. Available at: http://dream-journal.org/issu2020-5/2020-5_306.html (accessed: 14.12.2021).
  21. Kazakov A.L., Spevak L.F., Nefedova O.A. On the Numerical-Analytical Approaches to Solving a Nonlinear Heat Conduction Equation with a Singularity. Diagnostics, Resource and Mechanics of materials and structures, 2018, iss. 6, pp. 100–116. DOI: 10.17804/2410-9908.2018.6.100-116. Available at: http://dream-journal.org/issues/2018-6/2018-6_232.html (accessed: 14.12.2021).
  22. Kazakov A.L., Spevak L.F. Approximate and Exact Solutions to the Singular Nonlinear Heat Equation with a Common Type of Nonlinearity. Izvestiya IGU. Seriya Matematika, 2020, vol. 34, pp. 18–34. DOI: 10.26516/1997-7670.2020.34.18. (In Russian).
  23. Bautin S.P., Kazakov A.L. Obobshchennaya zadacha koshi i ee prilozheniya [The Generalized Cauchy problem and its applications]. Novosibirsk, Nauka Publ., 2006, 397 p. ISBN: 5-02-032540-6. (In Russian).
  24. Courant R., Hilbert D. Methods of Mathematical Physics. Vol. II: Partial Differential Equations, New York, Interscience Publishers, 2008.
  25. Golberg M.A. Numerical evaluation of particular solutions in the BEM–a review. Boundary Element Comm., 1995, vol. 6, pp. 99–106.


PDF      

Article reference

Kazakov A. L., Spevak L. F. On the Construction of a Heat Wave Generated by a Boundary Condition on a Moving Border // Diagnostics, Resource and Mechanics of materials and structures. - 2021. - Iss. 6. - P. 54-67. -
DOI: 10.17804/2410-9908.2021.6.054-067. -
URL: http://eng.dream-journal.org/issues/2021-6/2021-6_350.html
(accessed: 06/22/2024).

 

impact factor
RSCI 0.42

 

MRDMS 2024
Google Scholar


NLR

 

Founder:  Institute of Engineering Science, Russian Academy of Sciences (Ural Branch)
Chief Editor:  S.V. Smirnov
When citing, it is obligatory that you refer to the Journal. Reproduction in electronic or other periodicals without permission of the Editorial Board is prohibited. The materials published in the Journal may be used only for non-profit purposes.
Contacts  
 
Home E-mail 0+
 

ISSN 2410-9908 Registration SMI Эл № ФС77-57355 dated March 24, 2014 © IMACH of RAS (UB) 2014-2024, www.imach.uran.ru