A.L. Kazakov, P.A. Kuznetsov, L.F. Spevak. The problem of diffusion wave initiation for a nonlinear second-order parabolic system ... P. 67-86

The study of nonlinear singular parabolic equations occupies a key place in the scientific school of A. F. Sidorov. In particular, the problem on initiating a heat wave has been studied since the 1980s. The present study aims to extend the results of Sidorov and his followers, including the authors, to the case of systems of the corresponding type. We find that the heat (diffusion) wave for the system considered has a more complex (three-part) structure, which follows from the fact that the zero fronts are different for the unknown functions. A theorem on the existence and uniqueness of a piecewise analytical solution, which has the form of special series, is proved. We find an exact solution of the desired type, the construction of which is reduced to the integration of ordinary differential equations (ODEs). We managed to integrate the ODEs by quadratures. In addition, we propose an algorithm based on the collocation method, which allows us to effectively construct an approximate solution on a given time interval. Illustrative numerical calculations are performed. Since we have not managed to prove the convergence in this case (this is far from always possible for nonlinear singular equations and systems), exact solutions, both obtained in this paper and previously known, have been used to verify the calculation results.

Keywords: nonlinear parabolic system, singularity, existence theorem, special series, exact solution, collocation method, computational experiment

Received February 15, 2023

Revised March 16, 2023

Accepted March 20, 2023

Funding Agency: The research by A.L. Kazakov and P.A. Kuznetsov was funded by Ministry of Science and Higher Education of the Russian Federation within the framework of the project “Analytical and numerical methods of mathematical physics in problems of tomography, quantum field theory, and fluid mechanics” (No. of state registration: 121041300058-1).

Alexandr Leonidovich Kazakov, Dr. Phys.-Math. Sci., Matrosov Institute for System Dynamics and Control Theory SB RAS, 134, Lermontov st., Irkutsk, 664033 Russia, e-mail: kazakov@icc.ru

Pavel Alexandrovich Kuznetsov, Cand. Sci. (Phys. Math.), Matrosov Institute for System Dynamics and Control Theory SB RAS, Irkutsk, 664033 Russia, e-mail: kuznetsov@icc.ru

Lev Fridrikhovich Spevak, Cand. Sci. (Engineering Sciences), Institute of Engineering Science of UB RAS, Yekaterinburg, 620049 Russia, e-mail: lfs@imach.uran.ru

REFERENCES

1.   Sidorov A.F. Izbrannyye trudy: Matematika. Mekhanika [Selected works: Mathematics. Mechanics]. Moscow: Fizmatlit, 2001, 576 p.

2.   Vasin V.V., Sidorov A.F. Some methods of the approximate solution for differential and integral equations. Iz. VUZ. Math., 1983, vol. 27, no. 7, pp. 14–33.

3.   Sidorov A.F. On some classes of solutions of the nonsteady filtration equation. In: Computational Methods of Continuum Mechanics, 1984, vol. 15, no. 2, pp. 121–133 (in Russian).

4.   Sidorov A.F. Analytic representations of solutions of nonlinear parabolic equations of time-dependent filtration Dokl. Academy of Sciences of the USSR, 1985, vol. 31, pp. 40–44.

5.    Osipov Yu.S., Berdyshev V.I., Il’in A.M., Korotkiy A.I., Samofalov V.V., Titov S.S., Ul’yanov O.N., Khayrullina O.B. Anatoliy Fyodorovich Sidorov (1933–1999). Proc. Steklov Inst. Math. (Suppl.), 2003, suppl. 2, pp. S1–S7.

6.   Sidorov A.F. On some analytical representations of solutions of the nonlinear equation of nonsteady filtration. In: Computational Methods of Continuum Mechanics, 1987, pp. 247–257 (in Russian).

7.   Bautin S.P. Analiticheskaya teplovaya volna [Analytical heat wave]. Moscow: Fizmatlit Publ., 2003, 88 p. (in Russian). ISBN: 978-5-9221-0443-2.

8.   Kovrizhnykh O.O. On construction of an asymptotic solution to the degenerate nonlinear parabolic equation. Comput. Math. and Math. Phys., 2003, vol. 43, no. 10, pp. 1430–1436.

9.   Vaganova N.A. Constructing of new classes of solutions of a nonlinear filtration equation by special consistent series. Proc. Steklov Inst. Math. (Suppl.), 2003, suppl. 2, pp. S182–S193.

10.   Filimonov M.Yu. Representation of solutions of boundary value problems for nonlinear evolution equations by special series with recurrently calculated coefficients. Journal of Physics: Conference Series, 2019, vol. 1268, article no. 012071. doi: 10.1088/1742-6596/1268/1/012071

11.   Kazakov A.L., Lempert A.A. Existence and uniqueness of the solution of the boundary-value problem for a parabolic equation of unsteady filtration. J. Appl. Mech. and Tech. Phys., 2013, vol. 54, no. 2, pp. 251–258. doi: 10.1134/S0021894413020107

12.   Kazakov A.L., Kuznetsov P.A. On the analytic solutions of a special boundary value problem for a nonlinear heat equation in polar coordinates. J. Appl. Industr. Math., 2018, vol. 12, pp. 255–263. doi: 10.1134/S1990478918020060

13.   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. Comput. Math. and Math. Phys., 2019, vol. 59, no. 6, pp. 1015–1029. doi: 10.1134/S0965542519060083

14.   Kazakov A.L. Application of characteristic series for constructing solutions of nonlinear parabolic equations and systems with degeneracy. Tr. Inst. Math. Mekh. UrO RAN, 2012, vol. 18, no. 2, pp. 114–122 (in Russian).

15.   Kazakov A.L., Kuznetsov P.A., Lempert A.A. Analytical solutions to the singular problem for a system of nonlinear parabolic equations of the reaction-diffusion type. Symmetry, 2020, vol. 12, no. 6, pp. 999–1013. doi: 10.3390/SYM12060921

16.   Kazakov A.L., Spevak L.F. Exact and approximate solutions of a degenerate reaction-diffusion system. J. Appl. Mech. and Tech. Phys., 2021, vol. 62, no. 4, pp. 673–683. doi: 10.1134/S0021894421040179

17.   Kazakov A.L., Kuznetsov P.A., Spevak L.F. Construction of solutions to a boundary value problem with a singularity for a nonlinear parabolic system. J. Appl. and Industr. Math., 2021, vol. 15, no. 4, pp. 1–13. doi: 10.1134/S1990478921040050

18.   Kazakov A.L., Kuznetsov P.A. Diffusion-wave type solutions with two fronts to a nonlinear degenerate reaction-diffusion system. J. Appl. Mech. and Tech. Phys., 2022, vol. 63, no. 6, pp. 1–10. doi: 10.1134/S0021894422060128

19.   Kosov A.A., Semenov E.I. Distributed model of space exploration by two types of interacting robots and its exact solutions Journal of Physics: Conference Series, 2021, vol. 1847, article no. 012007. doi: 10.1088/1742-6596/1847/1/012007

20.   Ladyzenskaja O.A., Solonnikov V.A., Ural’ceva N.N. Linear and quasi-linear equations of parabolic type. Ser. Translations of Mathematical monographs, vol. 23, Providence: Amer. Math. Soc., 1968. ISBN: 978-0-8218-1573-1 Original Russian text was published in Ladyzenskaja O.A., Solonnikov V.A., Ural’ceva N.N. Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Moscow, Nauka Publ., 1967, 738 p.

21.   Grindrod P. Patterns and waves: theory and applications of reaction-diffusion equations. NY, Clarendon Press, 1991, 256 p.

22.   Samarskii A.A., Galaktionov V.A., Kurdyumov S.P., Mikhailov A.P. Blow-up in quasilinear parabolic equations. Berlin, Walter de Gruyte, 1995, 554 p. doi: 10.1515/9783110889864. Original Russian text was published in Samarskii A.A., Galaktionov V.A., Kurdyumov S.P., Mikhailov A.P. Rezhimy s obostreniem v zadachakh dlya kvazilineinykh parabolicheskikh uravnenii, Moscow, Nauka Publ., 1987, 480 p.

23.   Vazquez J. The porous medium equation: mathematical theory. Oxford, Clarendon Press, 2007, 624 p. doi: 10.1093/acprof:oso/9780198569039.001.0001.

24.   Bekezhanova V.B., Stepanova I.V. Evaporation convection in two-layers binary mixtures: equations, structure of solution, study of gravity and thermal diffusion effects on the motion. Appl. Math. Comput., 2022, vol. 414, article no. 126424. doi: 10.1016/j.amc.2021.126424

25.   Cantrell R.S., Cosner C. Spatial ecology via reaction-diffusion equations, Chichester, Wiley, 2003, 432 p.

26.   DiBenedetto E. Degenerate parabolic equations. NY, Springer-Verlag, 1993, 388 p. doi: 10.1007/978-1-4612-0895-2.

27.   Oleinik O.A., Kalashnikov A.S., Czou Juj-lin. The Cauchy problem and boundary problems for equations of the type of non-stationary filtration. Izv. Akad. Nauk SSSR Ser. Mat., 1958, vol. 22, no. 5, pp. 667–704 (in Russian).

28.   Stepanova K.V., Shishkov A.E. Initial evolution of supports of solutions of quasilinear parabolic equations with degenerate absorption potential. Sbornik: Math., 2013, vol. 204, no. 3, pp. 383–410. doi: 10.1070/SM2013v204n03ABEH004305

29.   Antontsev S.N., Shmarev S.I. Evolution PDEs with nonstandard growth conditions: Existence, uniqueness, localization, blow-up. Paris: Atlantis Press, 2015, 409 p. doi: 10.2991/978-94-6239-112-3

30.   Bautin S.P., Kazakov A.L. Obobshchennaya zadacha Koshi i ee prilozheniya [Generalized Cauchy Problem with Applications]. Novosibirsk, Nauka Publ., 2006, 399 p.

31.   Godunov S.K., Ryaben’kiy V.S. Raznostnye skhemy (vvedenie v teoriyu) [Difference schemes (An introduction to the theory)]. Moscow, Nauka Publ., 1977, 440 p.

32.   Rubina L.I. On the characteristics and solutions of the one-dimensional non-stationary seepage equation. J. Appl. Math. Mech., 2005, vol. 69, no. 5, pp. 743–750. doi: 10.1016/j.jappmathmech.2005.09.009

33.   Rubina L.I., Ul’yanov O.N. On some method for solving a nonlinear heat equation. Sib. Math. J., 2012, vol. 53, no. 5, pp. 872–881. doi: 10.1134/S0037446612050126

34.   Polyanin A.D., Zaitsev V.F., Zhurov A.I. Metody resheniya nelineynykh uravneniy matematicheskoy fiziki i mekhaniki [Solution methods for nonlinear equations of mathematical physics and mechanics]. Moscow: Fizmatlit, 2005, 256 p.

35.   Sidorov A.F. Some new analytical methods of exploring of nonlinear wave processes in gas dynamics. In: Fundamental investigations of reliability and quality of machines, 1990, pp. 37–48 (in Russian).

36.   Brebbia C.A., Telles J.C.F., Wrobel L.C. Boundary element techniques. Berlin, Springer-Verlag, 1984, 464 p.

37.   Banerjee P.K., Butterfield R. Boundary element methods in engineering science. London, McGraw-Hill Book Company, 1981, 452 p.

38.   Nardini N., Brebbia C.A. A new approach to free vibration analysis using boundary elements. Appl. Math. Modelling, 1983, vol. 7, pp. 157–162.

39.   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, Berlin, Springer-Verlag, 1986, pp. 801–811.

40.   Chen C.S., Chen W., Fu Z.J. Recent advances in radial basis function collocation methods, Berlin, Heidelberg, Springer, 2013, 165 p. doi: 10.1007/978-3-642-39572-7

41.   Buhmann M.D. Radial basis functions, Cambridge: Cambridge University Press, 2003, 259 p. doi: 10.1017/CBO9780511543241

42.   Fornberg B., Flyer N. Solving PDEs with radial basis functions. Acta Numerica, 2015, vol. 24, pp. 215–258. doi: 10.1017/S0962492914000130

43.   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

Cite this article as: A.L. Kazakov, P.A. Kuznetsov, L.F. Spevak, The problem of diffusion wave initiation for a nonlinear second-order parabolic system. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 2, pp. 67–86. Proceedings of the Steklov Institute of Mathematics (Suppl.), 2023, Vol. 321, Suppl. 1, pp. S109–S126.