非线性方程模型的精确解

IF 0.8 Q2 MATHEMATICS Lobachevskii Journal of Mathematics Pub Date : 2024-08-28 DOI:10.1134/s1995080224602571
A. I. Aristov
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引用次数: 0

摘要

摘要自二十世纪下半叶以来,对索伯列夫型方程进行了广泛的研究。这些方程包含的项目是未知函数相对于空间变量的二阶导数的时间导数。它们可以描述半导体、等离子体中的非稳态过程、流体力学现象和其他现象。我们注意到,对索伯列夫型方程的解的定性性质存在广泛的研究。也就是说,关于解的存在性和唯一性、其渐近性和炸毁的结果都是已知的。但是关于索波列夫方程的精确解的结果却很少。虽然也有关于偏方程精确解的书籍和论文,但它们主要针对的是经典方程,即未知函数相对于时间的一阶或二阶导数或相对于空间变量的一阶导数相对于时间的导数等于静止表达式。因此,研究 Sobolev 型方程的精确解很有意义。本文研究了一个三阶模型非线性偏方程。建立了六类精确解。它们用基本函数和特殊函数(某些常微分方程的解)表示。
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Exact Solutions of a Model Nonlinear Equation

Abstract

Since the second half of the twentieth century, wide studies of Sobolev-type equations are undertaken. These equations contain items that are derivatives with respect to time of the second order derivatives of the unknown function with respect to space variables. They can describe nonstationary processes in semiconductors, in plasm, phenomena in hydrodinamics and other ones. Notice that wide studies of qualitative properties of solutions of Sobolev-type equations exist. Namely, results about existence and uniqueness of solutions, their asymptotics and blow-up are known. But there are few results about exact solutions of Sobolev-type equations. There are books and papers about exact solutions of partial equations, but they are devoted mainly to classical equations, where the first or second order derivative with respect to time or the derivative with respect to time of the first order derivative of the unknown function with respect to the space variable is equal to a stationary expression. Therefore it is interesting to study exact solutions of Sobolev-type equations. In the present paper, a third order model nonlinear partial equation is studied. Six classes of its exact solutions are built. They are expressed in terms of elementary and special functions (solutions of some ordinary differential equations).

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来源期刊
CiteScore
1.50
自引率
42.90%
发文量
127
期刊介绍: Lobachevskii Journal of Mathematics is an international peer reviewed journal published in collaboration with the Russian Academy of Sciences and Kazan Federal University. The journal covers mathematical topics associated with the name of famous Russian mathematician Nikolai Lobachevsky (Lobachevskii). The journal publishes research articles on geometry and topology, algebra, complex analysis, functional analysis, differential equations and mathematical physics, probability theory and stochastic processes, computational mathematics, mathematical modeling, numerical methods and program complexes, computer science, optimal control, and theory of algorithms as well as applied mathematics. The journal welcomes manuscripts from all countries in the English language.
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