Nonlinear differential equation with first order partial derivatives

  • T. М. Aldibekov al-Farabi Kazakh National University
  • M. M. Aldazharova Scientific Research Institute of the al-Farabi Kazakh National University

Abstract

The asymptotic behavior of solutions of a nonlinear differential equation with first-order partialderivatives solved with respect to one of the derivatives is investigated. Each first-order partialdifferential equation under certain conditions has a fundamental system of integrals or an integralbasis. We note that for a general linear partial differential equation of the first order there canbe no nontrivial integral. For a linear homogeneous first-order partial differential equation, wherethe coefficients of the equation are given on an unbounded set and have continuous first-orderpartial derivatives, with the first coefficient equal to one, an integral basis exists. In this paper,a nonlinear partial differential equation of the first order, which is solved with respect to oneof the derivatives, is estimated from two sides by first-order partial differential equations. Usingdifferential inequalities it is proved that a nonlinear differential equation with first-order partialderivatives solved with respect to one of the derivatives has a solution that tends to zero as onetends to infinity to one of the independent variables. At present, the theory of partial differentialequations finds its application in various fields of natural science.

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Published
2018-12-21
How to Cite
ALDIBEKOV, T. М.; ALDAZHAROVA, M. M.. Nonlinear differential equation with first order partial derivatives. Journal of Mathematics, Mechanics and Computer Science, [S.l.], v. 99, n. 3, p. 3-11, dec. 2018. ISSN 1563-0277. Available at: <http://bm.kaznu.kz/index.php/kaznu/article/view/508>. Date accessed: 20 jan. 2019.
Keywords equation, first order partial derivatives