Research into a Class of Third-Order Nonlinear Differential Equations in the Domain of Analyticity
Authors: Orlov V.N., Kovalchuk O.A., Linnik E.P., Linnik I.I. | Published: 01.08.2018 |
Published in issue: #4(79)/2018 | |
DOI: 10.18698/1812-3368-2018-4-24-35 | |
Category: Mathematics and Mechanics | Chapter: Differential Equations and Mathematical Physics | |
Keywords: nonlinear differential equation, existence theorem, analytical approximate solution, movable singular point, a posteriori error estimation |
We consider the class of ordinary third-order nonlinear differential equations with a polynomial right-hand side of the second degree, which has movable singular points of algebraic type and is in general unsolvable in quadratures. The existing classical theory, in particular the Cauchy theorem of the existence of a solution to a differential equation, in this case is practically inapplicable. To solve this category of equations, one of the authors has developed an analytical approximate method consisting of six mathematical problems. The paper presents a study of the analytical approximate solution in the domain of analyticity, including the solution existence theorem proof, the making of an analytical approximate solution, and the investigation of the effect of initial data perturbation on the analytical approximate solution. The existence theorem proof is based on the majorant method in a new version, which makes it possible to carry out the planned investigations. A computational experiment with the use of a posteriori error estimation is presented, which makes it possible to significantly improve the a priori error estimation obtained
References
[1] Kalman R. Contribution to the theory of optimal control. Boletin de la Sociedad Matematica Mexicana, 1960, vol. 5, no. 1, pp. 102–119.
[2] Gorin V.A., Konakov A.P., Popov N.S. Research on work of feed meter. Mekhanizatsiya i elektrifikatsiya selskogo khozyaystva, 1981, no. 1, pp. 24–26 (in Russ.).
[3] Airault H. Rational solutions of Painlevé equations. Studies in Applied Mathematics, 1979, vol. 61, iss. 1, pp. 31–53. DOI: 10.1002/sapm197961131
[4] Samodurov A.A., Chudnovskiy V.M. Simple method for determination time delay of superradiant boson avalanche. Doklady AN BSSR, 1985, vol. 29, no. 1, pp. 9–10 (in Russ.).
[5] Hill J.M. Radial deflections of thin precompressed cylindrical rubber bush mountings. International Journal of Solids and Structures, 1977, vol. 13, iss. 2, pp. 93–104. DOI: 10.1016/0020-7683(77)90125-1
[6] Ockendon J.R. Numerical and analytical solutions of moving boundary problems. Proc. Symp. Moving Boundary Problems. New York, 1978, pp. 129–145.
[7] Axford R.A. The exact solution of singular arc problems in rector core optimization. Proc. Nuclear Utilities Planning Methods Symp. Tennessee, 1974, pp. 1–14.
[8] Hill J.M. Abels differential equation. J. Math. Scientist., 1982, vol. 7, no. 2, pp. 115–125.
[9] Kovalchuk O.A. Simulation of the state of the rod elements of the building construction. Procedia Engineering, 2016, vol. 153, pp. 304–309. DOI: 10.1016/j.proeng.2016.08.120
[10] Kovalchuk O.A. Stability of rod elements of building structures. PGS [Industrial and Civil Engineering], 2014, no. 11, pp. 60–62 (in Russ.).
[11] Orlov V.N. Metod priblizhennogo resheniya pervogo, vtorogo differentsialnykh uravneniy Penleve i Abelya [Approximate method for solving first and second Abel and Painlevé differential equations]. Moscow, MPGU Publ., 2013. 174 p.
[12] Orlov V.N. Study of approximate solution of Abels differential equation in the vicinity of movable singularity. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2009, no. 4, pp. 23–32 (in Russ.).
[13] Orlov V.N. The exact application area borders of Abel differential equation approximate solution in the area of the movable special point approximate meaning. Vestnik Voronezhskogo gos. tekhn. un-ta, 2009, vol. 5, no. 10, pp. 192–195 (in Russ.).
[14] Redkozubov S.A., Orlov V.N. Exact criteria of movable singularity existence for Abel differential equation. Izvestiya instituta inzhenernoy fiziki, 2009, vol. 4, no. 14, pp. 12–14 (in Russ.).
[15] Orlov V.N. Exact boundaries of approximate solution of Abel differential equation in the vicinity of movable singularity approximate value in сomplex domain. Vestnik ChGPU im. I.N. Yakovleva. Ser. Mekhanika predelnogo sostoyaniya [Bulletin of the Yakovlev Chuvash State Pedagogical University. Series: Mechanics of Limit State], 2010, no. 2 (8), pp. 399–405 (in Russ.).
[16] Orlov V.N., Guz M.P. [Relation between nonlinear differential equation and existence and properties of movable singularities]. Fundamentalnye i prikladnye problemy mekhaniki deformiruemogo tverdogo tela, matematicheskogo modelirovaniya i informatsionnykh tekhnologiy. Sb. statey po mat. mezhdunar. nauch.-prakt. konf. Ch. 2 [Fundamental and applied problems of deformable solid mechanics, mathematical simulation and information technologies. Proc. Int. Sci.-Pract. Conf. P. 2]. Cheboksary, Yakovlev Chuvash State Pedagogical University Publ., 2013. Pp. 30–35 (in Russ.).
[17] Pchelova A.Z., Kolle K.V. [Theorem of the solution existence for one nonlinear differential third order equation with polynomial second order right-hand member in the vicinity of movable singularity]. Mat. Vseros. nauch. shkoly-konf. "Mekhanika predelnogo sostoyaniya i smezhnye voprosy". Ch. 2 [Proc. Russ. Sci.-Pract. School-Conf. Limit-state mechanics and related issues]. Cheboksary, Yakovlev Chuvash State Pedagogical University Publ., 2015. Pp. 221–226 (in Russ.).