Geodesic
Integral representation of solutions of an ordinary differential equation and the Loewner–Kufarev equation
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 67 (2020), pp. 28-39 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The article presents a method of integral representation of solutions of ordinary differential equations and partial differential equations with a polynomial right-hand side part, which is an alternative to the construction of solutions of differential equations in the form of different series. The method is based on the introduction of additional analytical functions establishing the equation of connection between the introduced functions and the constituent components of the original differential equation. The implementation of the coupling equations contributes to the representation of solutions of the differential equation in the integral form, which allows solving some problems of mathematics and mathematical physics. The first part of the article describes the coupling equation for an ordinary differential equation of the first order with a special polynomial part of a higher order. Here, the integral representation of the solution of a differential equation with a second-order polynomial part is indicated in detail. In the second part of the paper, we consider the integral representation of the solution of a partial differential equation with the polynomial second-order part of the Loewner-Kufarev equation, which is an equation for univalent functions.
Keywords: differential equations, integral representations of solutions, univalent functions, conformal mappings. 
@article{VTGU_2020_67_a2,
     author = {O. V. Zadorozhnaya and V. K. Kochetkov},
     title = {Integral representation of solutions of an ordinary differential equation and the {Loewner{\textendash}Kufarev} equation},
     journal = {Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika},
     pages = {28--39},
     year = {2020},
     number = {67},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTGU_2020_67_a2/}
}
TY  - JOUR
AU  - O. V. Zadorozhnaya
AU  - V. K. Kochetkov
TI  - Integral representation of solutions of an ordinary differential equation and the Loewner–Kufarev equation
JO  - Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika
PY  - 2020
SP  - 28
EP  - 39
IS  - 67
UR  - http://geodesic.mathdoc.fr/item/VTGU_2020_67_a2/
LA  - ru
ID  - VTGU_2020_67_a2
ER  - 
%0 Journal Article
%A O. V. Zadorozhnaya
%A V. K. Kochetkov
%T Integral representation of solutions of an ordinary differential equation and the Loewner–Kufarev equation
%J Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika
%D 2020
%P 28-39
%N 67
%U http://geodesic.mathdoc.fr/item/VTGU_2020_67_a2/
%G ru
%F VTGU_2020_67_a2
O. V. Zadorozhnaya; V. K. Kochetkov. Integral representation of solutions of an ordinary differential equation and the Loewner–Kufarev equation. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 67 (2020), pp. 28-39. http://geodesic.mathdoc.fr/item/VTGU_2020_67_a2/

[1] I. A. Aleksandrov, Methods of geometrical theory of analytical functions, Tomsk State University, Tomsk, 2001

[2] V. I. Arnold, A. N. Varchenko, S. M. Gusein-Zade, Singularities of Differentiable Maps, Birkhäuser, Basel, 2012 | MR

[3] I. E. Bazilevich, “On a case of integrability in quadratures of the Loewner-Kufarev equation”, Mathematicheskiy sbornik, 37:3 (1995), 471–476

[4] V. P. Derevenskii, “First-order polynomial differential equations over matrix skew series”, Russian Mathematics, 58:9 (2014), 1–12 | DOI | MR | Zbl

[5] O. V. Zadorozhnaya, V. K. Kochetkov, “The structure of integrals of the second Loewner-Kufarev differential equation in a particular case”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika – Tomsk State University Journal of Mathematics and Mechanics, 2018, no. 55, 12–21 | DOI | MR

[6] O. V. Zadorozhnaya, V. K. Kochetkov, “Some study methods for ordinary differential equation integrability of the second order of a certain type”, Matematika i matematicheskoye modelirovaniye, 2019, no. 2, 48–62 | DOI

[7] O. V. Zadorozhnaya, V. K. Kochetkov, “Alternative methods of integrability of an ordinary nonlinear differential equation of the first order with a polynomial part”, Ekologicheskiy vestnik nauchnykh tsentrov Chernomorskogo ekonomicheskogo sotrudnichestva, 16:2 (2019), 6–14 | DOI

[8] O. V. Zadorozhnaya, V. K. Kochetkov, “A practical method for integral representation of solutions to a homogeneous second-order differential equation”, Innovatsionnyye issledovaniya kak lokomotiv razvitiya sovremennoy nauki: ot teoreticheskikh paradigm k praktike – Innovative investigations as a locomotive of the development of present-day science, 2019, 23–28

[9] Yu. S. Il'yashenko, S. Yu. Yakovenko, Analytical theory of differential equations, MCCME, M., 2013

[10] P. N. Matveev, Lectures on analytical theory of differential equations, LAN, St. Petersburg, 2009