Construction of approximate solutions for a class of first-order nonlinear differential equations in the analyticity region
Authors: Pchelova A.Z. | Published: 15.06.2016 |
Published in issue: #3(66)/2016 | |
DOI: 10.18698/1812-3368-2016-3-3-15 | |
Category: Mathematics and Mechanics | Chapter: Differential Equations and Mathematical Physics | |
Keywords: nonlinear ordinary differential equation, movable singular point, Cauchy problem, analytical approximate solution, error of the approximate solution, perturbation of the initial condition, analyticity region |
Nonlinear ordinary differential equations are mathematical models of various processes and phenomena of the real world; they belong to one of the most complicated categories of differential equations due to the presence of integrals of movable singular points. The study tested a class of first-order nonlinear ordinary differential equations with polynomial right part of not lower than the third degree. The solutions of these equations have movable singular points. The equations are not integratable in quadratures in a common case. For solving nonlinear differential equations with movable singular points of the algebraic type, we use the approximate method proposed by V.N. Orlov. We prove the existence and uniqueness of Cauchy problem solution for the class of differential equations in the analyticity region. In proving this theorem, we use the method of majorants for solving the examined nonlinear differential equations, rather than for solving the right part of differential equations, as it is done in classic literature. We offer the structure of approximate analytical solutions to the equations under study with the exact and perturbed values of the initial conditions, and we estimate the errors for these approximate solutions. The findings of the research are illustrated with the examples of calculations which are compared with similar results performed by other researchers.
References
[1] Lukashevich N.A., Orlov V.N. Studies of the approximate solution of the second Painleve. Differ. Uravn. [Differential Equations], 1989, vol. 25, no. 10, pp. 1829-1832 (in Russ.).
[2] Orlov V.N. About the approximate solution of the first Painleve equation. Vestn. Kazan. Gos. Tekh. Univ. im. A.N. Tupoleva [Herald of the A. Tupolev Kazan State Technical University], 2008, no. 2, pp. 42-46 (in Russ.).
[3] Orlov V.N. The method for the approximate solution of Riccati differential equation. Nauch.-tekh. vedomosti Sankt-Peterb. Politekh. Univ. [Scientific and technical statements of the St. Petersburg State Polytechnical University], 2008, no. 63, pp. 102-108 (in Russ.).
[4] Orlov V.N. About a teachnique to solve approximately matrix differential Riccati equations. Vestn. Moskovskogo aviatsionnogo inst. [Bull. of Moscow Aviation. Inst.], 2008, vol. 15, no. 5, pp. 128-135 (in Russ.). Available at: http://www.mai.ru/science/vestnik/eng/publications.php?ID=7837&eng=Y
[5] Orlov V.N. The exact application area borders of Abel differential equation approximate solution in the area of the movable special point approximate meaning. Vesto. Vorcmezh. Gos. Tekh. Umv. [Herald of the Voronezh State Technical University], 2009, vol. 5, no. 10, рр. 192-195 (in Russ.).
[6] Orlov V.N. Metod priblizhennogo resheniya pervogo, vtorogo differentsial’nykh uravneniy Penleve i Abelya [Method of approximate solution of the first, second differential Painleve’s and Abel’s equations]. Moscow, MPGU Publ., 2013. 174 р.
[7] Orlov V.N., Guz M.P. The approximate solution of nonlinear differential equation in the domain of analyticity. Vestn. Mordovsk. Gos. Univ. im. N.P. Ogareva, Fiz.-Matem. Nauki [Bull. of the Ogarev Mordovia State University, Phys.-Math. Sci.], 2012, no. 2, pp. 187-191 (in Russ.).
[8] Orlov V.N., Leontieva T.Yu. Construction of approximate solution of one nonlinear differential second-order equation in the neighborhood of movable singular point. Vestn. Mosk. Gos. Tekh. Urnv. im N.E. Baumam, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2015, no. 2, pp. 26-37 (in Russ.). DOI: 10.18698/1812-3368-2015-2-26-37
[9] Orlov V.N., Ivanov S.A. Numerical Solution of Three-Dimensional Problems of Seismic Stability of Large Structures. Vestn. Chuvash Gos. Ped. Univ. im. I.Ya. Yakovleva, Mekh. predel. sost. [I. Yakovlev Chuvash State Pedagogical University Bulletin, Mechanics of Limit State], 2014, no. 4, pp. 204-214 (in Russ.).
[10] Orlov V.N., Redkozubov S.A., Pchelova A.Z. The research of the approximate solution of the Cauchy problem of a nonlinear differential equation in the neighborhood of movable singular point. Izvestiya Inst. inzhener. fiziki [Proceedings of the Institute of Engineering Physics], 2013, no. 2, pp. 21-27 (in Russ.).
[11] Golubev V.V. Lektsii po analiticheskoi teorii differentsial’nykh uravnenii [Lectures on the analytic theory of differential equations]. Moscow-Leningrad, Gostechizdat Publ., 1950. 436 p.