Geodesic
On the identification of solutions to Riccati equation and the other polynomial systems of differential equations
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 14 (2022) no. 3, pp. 23-27 Cet article a éte moissonné depuis la source Math-Net.Ru

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The authors previously proposed a general method for finding particular solutions for overdetermined PDE systems, where the number of equations is greater than the number of unknown functions. The essence of the method is to reduce the PDE to systems of PDE of a lower dimension, in particular, to ODEs by overdetermining them by additional constraint equations. Reduction of some PDE systems generates overdetermined systems of polynomial ODEs, which are studied in this paper. A method for transforming polynomial ODE systems to linear ODE systems is proposed. The result is interesting from a theoretical point of view if these systems of polynomial ODEs are with constant coefficients. The solution of such nonlinear systems using our method can be represented as a sum of a very large but finite number of oscillations. The amplitudes of these oscillations depend on the initial data nonlinearly. The Navier-Stokes equations and unified PDE systems obtained by the authors earlier can be transformed to such systems. The Riccati equation is also investigated. New special cases are indicated when it is possible to find its solution. Numerical estimates of the complexity of this method for practical implementation are presented.
Keywords: overdetermined systems of differential equations, reduction, polynomial ODE systems, dimension of differential equations, Cauchy problem, linear ODE systems, Navier-Stokes equations, unification of PDE systems, symbolic calculations.
Mots-clés : Riccati equation 
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M. L. Zaytsev; V. B. Akkerman. On the identification of solutions to Riccati equation and the other polynomial systems of differential equations. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 14 (2022) no. 3, pp. 23-27. http://geodesic.mathdoc.fr/item/VYURM_2022_14_3_a2/

[1] A.N. Tikhonov, A.A. Samarskii, Uravneniya matematicheskoi fiziki, Nauka, M., 1966, 724 pp. | MR

[2] R. Kurant, Uravneniya s chastnymi proizvodnymi, 1964, 830 pp.

[3] M.L. Zaitsev, V.B. Akkerman, “Preobrazovanie sistem uravnenii v chastnykh proizvodnykh k sistemam kvazilineinykh i lineinykh differentsialnykh uravnenii. Ikh reduktsiya i unifikatsiya”, Matematicheskaya fizika i kompyuternoe modelirovanie, 21:1 (2018), 18–33 | MR

[4] M.L. Zaytsev, V.B. Akkerman, Unification of Solution of the Cauchy Problem for Overdetermined Systems of Differential Equations. Version 3, Research Gate, March 2019 | Zbl

[5] V.B. Akkerman, M.L. Zaitsev, “Snizhenie razmernosti v uravneniyakh gidrodinamiki”, Zhurnal vychislitelnoi matematiki i matematicheskoi fiziki, 51:8 (2011), 1518–1530 | MR | Zbl

[6] M.L. Zaitsev, V.B. Akkerman, “Gipoteza ob uproschenii pereopredelennykh sistem differentsialnykh uravnenii i ee primenenie k uravneniyam gidrodinamiki”, Vestnik VGU. Seriya: Fizika. Matematika, 2015, no. 2, 5–27 | Zbl

[7] M.L. Zaitsev, V.B. Akkerman, “Esche odin sposob nakhozhdeniya chastnykh reshenii uravnenii matematicheskoi fiziki”, Vestnik VolGU. Seriya 1, Matematika. Fizika, 2016, no. 6 (37), 119–127

[8] M.L. Zaitsev, V.B. Akkerman, “Reduktsiya pereopredelennykh sistem differentsialnykh uravnenii matematicheskoi fiziki”, Matematicheskaya fizika i kompyuternoe modelirovanie, 20:4 (2017), 43–67 | MR

[9] M.L. Zaitsev, V.B. Akkerman., “Zadacha obtekaniya i sokraschenie razmernosti v uravneniyakh Nave-Stoksa”, Trudy MFTI, 7:3 (2015), 18–30

[10] M.L. Zaytsev, V.B. Akkerman, Algoritm nakhozhdeniya reshenii pereopredelennykh sistem differentsialnykh uravnenii v yavnom vide, Research Gate, July 2020

[11] M.V. Fedoryuk, Obyknovennye differentsialnye uravneniya, Lan, SPb., 2003, 447 pp.

[12] A.I. Egorov, Uravneniya Rikkati, Fizmatlit, M., 2001, 318 pp.

[13] M.L. Zaitsev, V.B. Akkerman, “Preobrazovanie uravneniya Rikkati i drugikh polinomialnykh ODU k sistemam lineinykh ODU v yavnom vide”, Vestnik TGU. Matematika i mekhanika, 2021, no. 72, 5–14