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Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2014, Volume 14, Issue 4(1), Pages 374–381 (Mi isu524)  

This article is cited in 5 scientific papers (total in 5 papers)

Mathematics

On the Solutions of Multi-dimensional Clairaut Equation with Multi-homogeneous Function of the Derivatives

I. V. Rakhmelevich

Nizhny Novgorod State University, 23, Gagarin ave., Nizhny Novgorod, 603950, Russia

Abstract: The analysis of the solutions of Clairaut equation with an arbitrary number of independent variables is completed. It is assumed that the function of the derivatives, which is part of the equation is multi-homogeneous. This means that the set of function arguments can be represented as the union of subsets, and the function is homogeneous on each of these subsets. We consider solutions of equations depending on linear combinations of the original variables, each of which contains only a certain subset of variables. Original equation is transformed to a reduced one, which can be solved by separation of variables. It is shown that the reduced equation has solutions in the form of arbitrary homogeneous functions with index of homogeneity 1 and solutions in the form of some generalized polynomials.

Key words: Clairaut equation, reduced equation, multi-homogeneous function, variables separation method.

DOI: https://doi.org/10.18500/1816-9791-2014-14-4-374-381

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UDC: 517.952

Citation: I. V. Rakhmelevich, “On the Solutions of Multi-dimensional Clairaut Equation with Multi-homogeneous Function of the Derivatives”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 14:4(1) (2014), 374–381

Citation in format AMSBIB
\Bibitem{Rak14}
\by I.~V.~Rakhmelevich
\paper On the Solutions of Multi-dimensional Clairaut Equation with Multi-homogeneous Function of the Derivatives
\jour Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform.
\yr 2014
\vol 14
\issue 4(1)
\pages 374--381
\mathnet{http://mi.mathnet.ru/isu524}
\crossref{https://doi.org/10.18500/1816-9791-2014-14-4-374-381}
\elib{https://elibrary.ru/item.asp?id=22575444}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. I. V. Rakhmelevich, “O nekotorykh novykh resheniyakh mnogomernogo uravneniya v chastnykh proizvodnykh pervogo poryadka so stepennymi nelineinostyami”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2015, no. 3(35), 18–25  mathnet  crossref  elib
    2. I. V. Rakhmelevich, “O resheniyakh dvumernogo uravneniya Monzha–Ampera so stepennoi nelineinostyu po pervym proizvodnym”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2016, no. 4(42), 33–43  mathnet  crossref  elib
    3. I. V. Rakhmelevich, “On multi-dimensional partial differential equations with power nonlinearities in first derivatives”, Ufa Math. J., 9:1 (2017), 98–108  mathnet  crossref  isi  elib
    4. L. L. Ryskina, “O singulyarnykh resheniyakh mnogomernogo differentsialnogo uravneniya tipa Klero so stepennoi i pokazatelnoi funktsiyami”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 23:2 (2019), 394–401  mathnet  crossref  elib
    5. L. L. Ryskina, “Nakhozhdenie osobykh reshenii mnogomernykh differentsialnykh uravnenii tipa Klero v chastnykh proizvodnykh s trigonometricheskimi funktsiyami”, Kompyuternye issledovaniya i modelirovanie, 12:1 (2020), 33–42  mathnet  crossref
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