COMPARISON OF SOLVING A STIFF EQUATION ON A SPHERE BY THE MULTI-LAYER METHOD AND METHOD OF CONTINUING AT THE BEST PARAMETER
Abstract
A stiff equation, linked with the solution of singularly perturbed differential equations with the use of standard methods of numeral solutions of simple differential equations often lead to major difficulties. First difficulty is the loss of stability of the counting process, when small errors on separate steps lead to an increase in the systematic errors in general. Another difficulty is, directly linked with the first one, consists of the need of decreasing the integrating step by a lot, which leads to a major decrease in the time taken for the calculations. On an example of a boundary value problem for a differential equation of second power on a sphere, comparison of our two approaches of constructing approximate values are held. The first approach is connected with the construction of an approximate multi-layer solution of the problem and is based on the use of recurrent equalities, which come out from classical numeral methods to the interval of a non-constant length. As a result, a numeral, approximated solution is replaced with an approximate solution in form of a function, which is easier to use for adaptation, building a graph and other needs. The second approach is linked with the continuation of the solutions by the best parameter. This method allows us to decrease majorly the number of steps and increase the stability of the computing process compared to standard methods.
References
[2] Tikhonov A.N. Dependence of solutions of differential equations on a small parameter. Sbornik: Mathematics. 1948; 22(64)-2:193-204. Available at: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=6075&option_lang=rus (accessed 21.06.2018). (In Russian)
[3] Tikhonov A.N. On systems of differential equations containing parameters. Sbornik: Mathematics. 1950; 27(69)-1:147-156. Available at: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=5907&option_lang=rus (accessed 21.06.2018). (In Russian)
[4] Tikhonov A.N. Systems of differential equations containing small parameters for derivatives. Sbornik: Mathematics. 1952; 31(73)-3:575-586. Available at: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=5548&option_lang=rus (accessed 21.06.2018). (In Russian)
[5] Vasil'eva A.B., Butuzov V.F. Asymptotic methods in the theory of singular perturbations. M.: Higher School, 1990. 208 p. (In Russian)
[6] Vasilyeva A.B., Plotnikov A.A. Asymptotic theory of singularly perturbed problems. M.: Physics Faculty of Moscow State University, 2008. 398 p. (In Russian)
[7] Vasil'eva A.B., Butuzov V.F., Nefedov N.N. Contrast structures in singularly perturbed problems. Fundamentalnaya i prikladnaya matematika = Fundamental and Applied Mathematics. 1998; 4(3):799-851. Available at: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=fpm&paperid=344&option_lang=rus (accessed 21.06.2018). (In Russian)
[8] Butuzov V.F., Vasilyeva A.B., Nefedov N.N. Asymptotic theory of contrast structures (review). Automation and Remote Control. 1997; 58(7):1068–1091. (In Russian)
[9] Chang K., Howes F. Nonlinear Singularly Perturbed Boundary Value Problems. Theory and applications. M.: Mir, 1988. 247 p. (In Russian)
[10] Nagumo M. Über das Verhalten der Integrals von λy''+f(x,y,y',λ)=0 für λ→0. Proc. Phys. Math. Soc. Japan. 1939; 21:529-534.
[11] Lomov S.A. Lomov I.S. Fundamentals of the mathematical theory of the boundary layer. M.: Publishing House of Moscow University, 2011. 456 p. (In Russian)
[12] Lomov S.A. Introduction to the general theory of singular perturbations. M.: Nauka, 1981. 400 p. (In Russian)
[13] Butuzov V.F., Levashova N.T., Mel'nikova A.A. A steplike contrast structure in a singularly perturbed system of elliptic equations. Computational Mathematics and Mathematical Physics. 2013. 53; 9:1239–1259. DOI: 10.1134/S0965542513090054
[14] Butuzov V.F., Denisov I.V. Corner Boundary Layer in Nonlinear Elliptic Problems Containing Derivatives of First Order. Modeling and Analysis of Information Systems. 2014; 21(1):7-31. Available at: https://elibrary.ru/item.asp?id=21351116 (accessed 21.06.2018). (In Russian)
[15] Butuzov V.F., Beloshapko V.A. Singularly perturbed elliptic Dirichlet problem with multiple root of the degenerate equation. Modeling and Analysis of Information Systems. 2016; 23(5):515-528. (In Russian) DOI: 10.18255/1818-1015-2016-5-515-528
[16] Butuzov V.F., Bychkov A.I. Asymptotics of the solution of an initial-boundary value problem for a singularly perturbed parabolic equation in the case of a triple root of a degenerate equation. Computational Mathematics and Mathematical Physics. 2016; 56(4):593-611. (In Russian) DOI: 10.1134/S0965542516040060
[17] Butuzov V.F. On contrast structures with a multi-zone inner layer. Modeling and Analysis of Information Systems. 2017; 24(3):288-308. (In Russian) DOI: 10.18255/1818-1015-2017-3-288-308
[18] Lazovskaya T., Tarkhov D. Multilayer neural network models, based on grid methods. IOP Conference Series: Materials Science and Engineering. 2016; 158(1):012061. DOI: 10.1088/1757-899X/158/1/012061
[19] Tarkhov D., Shershneva E. Approximate analytical solutions of Mathieu's equations based on classical numerical methods. V. Sukhomlin, E. Zubareva, M. Shneps-Shneppe (Eds.) Proceedings of the XI International Scientific-Practical Conference “Modern Information Technologies and IT-Education” (SITITO 2016). Moscow, Russia, November 25-26. 2016. CEUR Workshop Proceedings. Vol. 1761. Pр. 356-362. Available at: http://ceur-ws.org/Vol-1761/paper46.pdf (accessed 21.06.2018). (In Russian)
[20] Vasilyev A., Tarkhov D., Shemyakina T. Approximate analytical solutions of ordinary differential equations. V. Sukhomlin, E. Zubareva, M. Shneps-Shneppe (Eds.) Proceedings of the XI International Scientific-Practical Conference “Modern Information Technologies and IT-Education” (SITITO 2016). Moscow, Russia, November 25-26. 2016. CEUR Workshop Proceedings. Vol. 1761. Pр. 393-400. Available at: http://ceur-ws.org/Vol-1761/paper50.pdf (accessed 21.06.2018). (In Russian)
[21] Lazovskaya T., Tarkhov D., Vasilyev А. Multi-Layer Solution of Heat Equation. B. Kryzhanovsky, W. Dunin-Barkowski, V. Redko (Eds.) Advances in Neural Computation, Machine Learning, and Cognitive Research. Studies in Computational Intelligence. Vol. 736. Springer International Publishing, 2018. Pp. 17–22. DOI: 10.1007/978-3-319-66604-4_3
[22] Borovskaya O.D., Lazovskaya T.V., Skolis X.V., Tarkhov D.A., Vasilyev A.N. Multilayer parametric models of processes in a porous catalyst pellet. Modern Information Technologies and IT-Education. 2018; 14(1):27-37. DOI: 10.25559/SITITO.14.201801.027-037
[23] Kartavchenko A.E., Tarhov D.A. Comparison of methods for construction of approximate analytical solutions of differential equations on the example of elementary functions. Modern Information Technologies and IT-Education. 2017; 13(3):16-23. (In Russian) DOI: 10.25559/SITITO.2017.3.440
[24] Vasilyev A.N., Svintsov M.V., Tarkhov D.A. Development of analysis of multilayer methods for solving wave equation with special initial conditions. V. Sukhomlin, E. Zubareva, M. Shneps-Shneppe (Eds.) Proceedings of the 2nd International scientific conference "Convergent cognitive information technologies" (Convergent’2017). Moscow, Russia: November 24–26, 2017. CEUR Workshop Proceedings. Vol. 2064. Pp. 393-400. Available at: http://ceur-ws.org/Vol-2064/paper16.pdf (accessed 21.06.2018). (In Russian)
[25] Tarkhov D.A., Kaverzneva T.T., TereshinV.A., Vinokhodov T.V., Kapitsin D.R., Zulkarnay I.U. New methods of multilayer semiempirical models in nonlinear bending of the cantilever. V. Sukhomlin, E. Zubareva, M. Shneps-Shneppe (Eds.) Proceedings of the 2nd International scientific conference "Convergent cognitive information technologies" (Convergent’2017). Moscow, Russia: November 24–26, 2017. CEUR Workshop Proceedings. Vol. 2064. Pp. 143-149. Available at: http://ceur-ws.org/Vol-2064/paper17.pdf (accessed 21.06.2018).
[26] Vasilyev A.N., Tarkhov D.A., Tereshin V.A., Berminova M.S., Galyautdinova A.R. Semi-empirical Neural Network Model of Real Thread Sagging. B. Kryzhanovsky, W. Dunin-Barkowski, V. Redko (Eds.) Advances in Neural Computation, Machine Learning, and Cognitive Research. Studies in Computational Intelligence. Vol. 736. Springer International Publishing, 2018. Pp. 138–146. DOI: 10.1007/978-3-319-66604-4_21
[27] Zulkarnay I.U., Kaverzneva T.T., Tarkhov D.A., Tereshin V.A., Vinokhodov T.V., Kapitsin D.R. A Two-layer Semi-Empirical Model of Nonlinear Bending of the Cantilevered Beam. IOP Conference Series: Journal of Physics: Conference Series. 2018; 1044(conf. 1):012005. DOI: 10.1088/1742-6596/1044/1/012005
[28] Bortkovskaya M.R., Vasilyev P.I., Zulkarnay I.U., Semenova D.A., Tarkhov D.A., Udalov P.P., Shishkina I.A. Modeling of the membrane bending with multilayer semi-empirical models based on experimental data. V. Sukhomlin, E. Zubareva, M. Shneps-Shneppe (Eds.) Proceedings of the 2nd International scientific conference "Convergent cognitive information technologies" (Convergent’2017). Moscow, Russia: November 24–26, 2017. CEUR Workshop Proceedings. Vol. 2064. Pp. 150-156. Available at: http://ceur-ws.org/Vol-2064/paper18.pdf (accessed 21.06.2018).
[29] Hairer E., Norsett S.P., Wanner G. Solving Ordinary Differential Equations I. Nonstiff Problem. Springer-Verlag Berlin Heidelberg, 1987. 482 p. DOI: 10.1007/978-3-662-12607-3
[30] Shalashilin V.I., Kuznetsov E. Parametric Continuation and Optimal Parametrization in Applied Mathematics and Mechanics. Springer Netherlands, 2003. DOI: 10.1007/978-94-017-2537-8
[31] Samoylenko A.M., Ronto N.I. Numerically-analytical methods for investigating solutions of boundary-value problems [Chislenno-analiticheskie metody issledovaniya reshenij kraevyh zadach]. Kiev: Naukova Dumka, 1986. 224 p. (In Russian)
[32] Budkina E.M., Kuznetsov E.B. Solution of boundary value problems for differential algebraic equations. Proceedings of the 11th International Conference on Computational Mechanics and Contemporary Applied Software Systems. M.: Izd-vo MAI, 2015. Pp. 44-46. Available at: https://elibrary.ru/item.asp?id=24658093 (accessed 21.06.2018). (In Russian)
This work is licensed under a Creative Commons Attribution 4.0 International License.
Publication policy of the journal is based on traditional ethical principles of the Russian scientific periodicals and is built in terms of ethical norms of editors and publishers work stated in Code of Conduct and Best Practice Guidelines for Journal Editors and Code of Conduct for Journal Publishers, developed by the Committee on Publication Ethics (COPE). In the course of publishing editorial board of the journal is led by international rules for copyright protection, statutory regulations of the Russian Federation as well as international standards of publishing.
Authors publishing articles in this journal agree to the following: They retain copyright and grant the journal right of first publication of the work, which is automatically licensed under the Creative Commons Attribution License (CC BY license). Users can use, reuse and build upon the material published in this journal provided that such uses are fully attributed.
