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ORIGINAL RESEARCH article

Front. Appl. Math. Stat.
Sec. Dynamical Systems
Volume 10 - 2024 | doi: 10.3389/fams.2024.1466569
This article is part of the Research Topic Approximation Methods and Analytical Modeling Using Partial Differential Equations View all 20 articles

Some class of nonlinear partial differential equations in the ring of copolynomials over a commutative ring

Provisionally accepted
Aleksey Piven' Aleksey Piven' *Sergiy Gefter Sergiy Gefter
  • V. N. Karazin Kharkiv National University, Kharkiv, Ukraine

The final, formatted version of the article will be published soon.

    We study the copolynomials, i.e. K-linear mappings from the ring of polynomials K[x] into the commutative ring K. With the help of the Cauchy-Stieltjes transform of a copolynomial we introduce and study a multiplication of copolynomials. We investigate a Cauchy problem for the nonlinear partial differential equationring of copolynomials. To find a solution we use the series in powers of the δ-function. As examples we consider a Cauchy problem for the Euler-Hopf equation ∂u ∂t + u ∂u ∂x = 0, for a Hamilton-Jacobi type equation ∂u ∂t = ( ∂u ∂x ) 2 and for the Harry Dym equation ∂u ∂t = u 3 ∂ 3 u ∂x 3 .

    Keywords: copolynomial, δ-function, Partial differential equation, Cauchy problem, Cauchy-Stieltjes transform, multiplication of copolynomials

    Received: 18 Jul 2024; Accepted: 01 Nov 2024.

    Copyright: © 2024 Piven' and Gefter. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

    * Correspondence: Aleksey Piven', V. N. Karazin Kharkiv National University, Kharkiv, Ukraine

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