Article 8120

Title of the article

ON APPLYING THE CONTINUOUS OPERATOR METHOD TO SOLVE THE DIRECT PROBLEM FOR NONLINEAR PARABOLIC EQUATIONS 

Authors

Boykov Il'ya Vladimirovich, Doctor of physical and mathematical sciences, professor, head of the sub-department of higher and applied mathematics, Penza State University (40, Krasnaya street, Penza, Russia), E-mail: boikov@pnzgu.ru
Ryazantsev Vladimir Andreevich, Candidate of engineering sciences, associate professor, sub-department of higher and applied mathematics, Penza State University (40, Krasnaya street, Penza, Russia), E-mail: math@pnzgu.ru 

Index UDK

519.633 

DOI

10.21685/2072-3040-2020-1-8 

Abstract

Background. Parabolic differential equations of mathematical physics play very important role in mathematical modeling of the wide range of phenomena in physical and technical sciences. In particular, parabolic equations are widely used for modeling diffusion processes, processes of fluid dynamics as well as biological and ecological phenomena. These equations are also occur in the problems of heat and mass transfer, combustion theory, filter theory etc. Besides, in spite of sufficiently wide amount of known results in the field of approximate solution of parabolic equations, there is important need for developing effective numerical methods for solving nonlinear parabolic equations. Such methods must be quite simple and in the same time be resistant to initial data disturbances as well as be applicable to a wide range of equations.
Materials and methods. The main subject of this paper is Cauchy problem for one-dimensional parabolic equations that is nonlinear in unknown function. We consider the problem of constructing the numerical method for solving the mentioned equation. In order to do that we change from the Cauchy problem for parabolic differential equation to a nonlinear integral equation. The integral equation is then solved by means of continuous operator method for nonlinear equations: an auxiliary system of integro-differential equations of special type is constructed and then it solved with one of the numerical methods for solving differential equations. The result of the method is a set of approximate values of unknown function in the nodes of the uniform mesh, which is constructed in a finite domain.
Results. A numerical method for solving the Cauchy problem for nonlinear onedimensional parabolic differential equation is proposed in the paper. A high potential of this method is primarily due to its simplicity and also its universality that allows to apply the same algorithm for very wide range of nonlinearities.
Conclusions. An effective iterative method for solving the Cauchy problem for nonlinear one-dimensional parabolic differential equation is proposed. Extending the method to boundary problems as well as to multidimensional equations is of considerable theoretical and practical interest. 

Key words

nonlinear parabolic equations, Cauchy problem, continuous operator method, nonlinear integral equations, logarithmic norm 

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References

1. Lions Zh.-L. Nekotorye metody resheniya nelineynykh kraevykh zadach [Some methods for solving nonlinear boundary value problems]. Moscow: Mir, 1972, 588 p. [In Russian]
2. Gaevskiy Kh., Greger K., Zakharias K. Nelineynye operatornye uravneniya i operatornye differentsial'nye uravneniya [Nonlinear operator equations and operator differential equations]. Moscow: Mir, 1978, 336 p. [In Russian]
3. Ablovits M., Sigur Kh. Solitony i metod obratnoy zadachi [Solitons and the inverse problem method]. Moscow: Mir, 1987, 479 p. [In Russian]
4. Mors F. M., Feshbakh G. Metody teoreticheskoy fiziki [Methods of theoretical physics]. Moscow: Media, 2012, vol. 2. [In Russian]
5. Kalitkin N. N. Chislennye metody [Numerical method]. Moscow: Nauka, 1978, 512 p. [In Russian]
6. Samarskiy A. A., Vabishchevich P. N. Vychislitel'naya teploperedacha [Computational heat transfer]. Moscow: LIBROKOM, 2009, 784 p. [In Russian]
7. Vabishchevich P. N. Vychislitel'nye metody matematicheskoy fiziki. Nestatsionarnye zadachi [Computational methods of mathematical physics. Non-stationary tasks]. Moscow: Vuzovskaya kniga, 2008, 228 p. [In Russian]
8. Kalodzhero F., Degasperis A. Spektral'nye preobrazovaniya i solitony. Metody resheniya i issledovaniya nelineynykh evolyutsionnykh uravneniy [Spectral transforms and solitons. Methods for solving and studying nonlinear evolution equations]. Moscow: Mir, 1985, 472 p. [In Russian]
9. Polyanin A. D., Zaytsev V. F., Zhurov A. I. Metody resheniya nelineynykh uravneniy matematicheskoy fiziki i mekhaniki [Methods for solving nonlinear equations of mathematical physics and mechanics]. Moscow: FIZMATLIT, 2009, 256 p. [In Russian]
10. Polyanin A. D. Spravochnik po lineynym uravneniyam matematicheskoy fiziki [Reference book of linear equations of mathematical physics]. Moscow: FIZMATLIT, 2001, 576 p. [In Russian]
11. Polyanin A. D., Zaytsev V. F. Spravochnik po nelineynym uravneniyam matematicheskoy fiziki: tochnye resheniya [Reference book of nonlinear equations of mathematical physics: exact solutions]. Moscow: FIZMATLIT, 2002, 432 p. [In Russian]
12. Boykov I. V. Differentsial'nye uravneniya [Differential equations]. 2012, vol. 48, no. 9, pp. 1308–1314. [In Russian]
13. Boykov I. V., Ryazantsev V. A. Zhurnal Srednevolzhskogo matematicheskogo obshchestva [Middle Volga Mathematical Society Journal]. 2019, vol. 21, no. 2, pp. 149–163. [In Russian]
14. Boykov I. V., Ryazantsev V. A. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fiziko-matematicheskie nauki [University proceedings. Volga region. Physical and mathematical sciences]. 2019, no. 3 (51), pp. 47–62. [In Russian]

 

Дата создания: 06.05.2020 16:25
Дата обновления: 06.05.2020 17:18