Boykov Il'ya Vladimirovich, Doctor of physical and mathematical sciences, head of sub-department of higher and applied mathematics, Penza State University (40 Krasnaya street, Penza, Russia), firstname.lastname@example.org
Baulina Ol'ga Aleksandrovna, Postgraduate student, Penza State University (40 Krasnaya street, Penza, Russia), email@example.com
Background. Solution of mathematical physics’ problems on artificial neural networks is an actively developing concept combining methods of calculus mathematics and computer science. Application of neural networks is especially effective in solution of reverse and incorrect problems and equations with inaccurately set parameters. At the present time the main method of solution of mathematical physics’ problems on artificial neural networks is minimization of functional error. The study is aimed at development of a stable and quick method of solving mathematical physics’ problems on artificial neural networks, based on the theory of differential equation solution stability.
Materials and methods. The article describes an approximate method of elliptic equations’ solution on Hopfield neural networks. The method consists in approximation of the source boundary problem of difference scheme and formation of a system of regular differential equations, the solution of which is reduced to solution of the difference scheme.
Results. The authors suggest a method of boundary problem solution for linear and non-linear elliptic equations, based on the methods of the stability theory. Effectivenes of the method is demonstrated by the model examples.
Conclusions. The results of the study may be used for solution of a wide class of boundary problems for linear and non-linear elliptic equations, determined in sec-tionally smooth areas.
1. Hopfield J. J. Proc. Natl. Acad. Sci. USA. 1982, April, vol. 79, pp. 2554–2558.
2. Hopfield J. J. Proc. Natl. Acad. Sci. USA. 1984, May, vol. 81, pp. 3088–3092.
3. Gorbachenko V. I. Neyrokomp'yutery v reshenii kraevykh zadach teorii polya [Neuro-computers in solution of boundary problems of the field theory]. Moscow: Radio-tekhnika, 2003, 336 p.
4. Boykov I. V., Rudnev V. A., Boykova A. I. Neyrokomp'yutery, razrabotka, primenenie [Neurocomputers, development, application]. 2013, no. 10, pp. 13–22.
5. Boykov I. V., Baulina O. A. Zhurnal Srednevolzhskogo matematicheskogo obshchestva [Journal of Middle Volga mathematical society]. 2013, vol. 15, no. 1, pp. 41–51.
6. Daletskiy Yu. L., Kreyn M. G. Ustoychivost' resheniy differentsial'nykh uravneniy v ba-nakhovom prostranstve [Stability of differential equations solution in Banach space]. Moscow: Nauka, 1970, 534 p.
7. Boykov I. V. DAN SSSR [Reports of the USSR Academy of Sciences]. 1990, vol. 314, no. 6, pp. 1298–1300.
8. Boykov I. V. Ustoychivost' resheniy differentsial'nykh uravneniy [Stability of differential equations solutions]. Penza: Izd-vo PGU, 2008, 244 p.
9. Boykov I. V. Differentsial'nye uravneniya [Differential equations]. 2012, vol. 48, no. 9, pp. 1308–1314.
10. Gyunter N. M. Teoriya potentsiala i ee primenenie k osnovnym zadacham matematicheskoy fiziki [Theory of potentials and application thereof to main problems of mathematical physics]. Moscow: GITTL, 1953, 415 p.