Diabetes mellitusDiabetes mellitus2072-03512072-0378Endocrinology research centre1064410.17816/RFD10644Original ArticleNumerical determination of the singularity order of a system of differential equationsBaddourAliPhD student of Department of Applied Probability and Informaticsalibddour@gmail.comMalykhMikhail D.Doctor of Physical and Mathematical Sciences, assistant professor of Department of Applied Probability and Informaticsmalykhmd-md@rudn.ruPaninAlexander A.Candidate of Physical and Mathematical Sciences, assistant professor of Faculty of Physicsa-panin@yandex.ruSevastianovLeonid A.Doctor of Physical and Mathematical Sciences, professor of Department of Applied Probability and Informaticssevastianov-la@rudn.ruPeoples’ Friendship University of Russia (RUDN University)M. V. Lomonosov Moscow State University15122020281173421052020Copyright © 2020, Baddour A., Malykh M.D., Panin A.A., Sevastianov L.A.2020We consider moving singular points of systems of ordinary differential equations. A review of Painlevé’s results on the algebraicity of these points and their relation to the Marchuk problem of determining the position and order of moving singularities by means of finite difference method is carried out. We present an implementation of a numerical method for solving this problem, proposed by N. N. Kalitkin and A. Al’shina (2005) based on the Rosenbrock complex scheme in the Sage computer algebra system, the package CROS for Sage. The main functions of this package are described and numerical examples of usage are presented for each of them. To verify the method, computer experiments are executed (1) with equations possessing the Painlevé property, for which the orders are expected to be integer; (2) dynamic Calogero system. This system, well-known as a nontrivial example of a completely integrable Hamiltonian system, in the present context is interesting due to the fact that coordinates and momenta are algebraic functions of time, and the orders of moving branching points can be calculated explicitly. Numerical experiments revealed that the applicability conditions of the method require additional stipulations related to the elimination of superconvergence points.CROSFinite-difference methodssageCalogero systemPainlevé propertyCROSSageметод конечных разностейсистема Калоджеросвойство ПенлевеW. A. Stein, Sage Mathematics Software (Version 6.7), The Sage Development Team, 2015.W. W. Golubev, Vorlesungen über Differentialgleichungen im Komplexen. Berlin: VEB Deutscher Verlag der Wissenschaften, 1958.P. Painlevé, “Leçons sur la theorie analytique des equations differentielles,” in Œuvres de Paul Painlevé. 1973, vol. 1.C. L. Siegel and J. Moser, Lectures on Celestial Mechanics. Springer, 1995.E. A. Al’shina, N. N. Kalitkin, and P. V. Koryakin, “Diagnostics of singularities of exact solutions in computations with error control,” Computational Mathematics and Mathematical Physics, vol. 45, no. 10, pp. 1769-1779, 2005.A. A. Belov, “Numerical detection and study of singularities in solutions of differential equations,” Doklady Mathematics, vol. 93, no. 3, pp. 334- 338, 2016. DOI: 10.1134/S1064562416020010.A. A. Belov, “Numerical diagnostics of solution blowup in differential equations,” Computational Mathematics and Mathematical Physics, vol. 57, no. 1, pp. 122-132, 2017. DOI: 10.31857/S004446690002533-7.M. O. Korpusov, D. V. Lukyanenko, A. A. Panin, and E. V. Yushkov, “Blow-up for one Sobolev problem: theoretical approach and numerical analysis,” Journal of Mathematical Analysis and Applications, vol. 442, no. 2, pp. 451-468, 2016. DOI: 10.26089/NumMet.v20r328.M. O. Korpusov, D. V. Lukyanenko, A. A. Panin, and E. V. Yushkov, “Blow-up phenomena in the model of a space charge stratification in semiconductors: analytical and numerical analysis,” Mathematical Methods in the Applied Sciences, vol. 40, no. 7, pp. 2336-2346, 2017. DOI: 10.1002/mma.4142.M. O. Korpusov and D. V. Lukyanenko, “Instantaneous blow-up versus local solvability for one problem of propagation of nonlinear waves in semiconductors,” Journal of Mathematical Analysis and Applications, vol. 459, no. 1, pp. 159-181, 2018. DOI: 10.1016/j.jmaa.2017.10.062.M. O. Korpusov, D. V. Lukyanenko, A. A. Panin, and G. I. Shlyapugin, “On the blow-up phenomena for a one-dimensional equation of ion-sound waves in a plasma: analytical and numerical investigation,” Mathematical Methods in the Applied Sciences, vol. 41, no. 8, pp. 2906-2929, 2018. DOI: 10.1002/mma.4142.A. Baddour and M. D. Malykh. (2019). Cros for sage, RUDN, [Online]. Available: https://malykhmd.neocities.org.H. Airault, “Rational solutions of Painlevé equations,” Stud. Appl. Math., vol. 61, no. 1, pp. 31-53, 1979. DOI: 10.1002/sapm197961131.(2019). Nist digital library of mathematical functions. version 1.0.25, The National Institute of Standards and Technology, [Online]. Available: http://dlmf.nist.gov.J. Moser, Integrable Hamiltonian Systems and Spectral Theory. Edizioni della Normale, 1983.