Group Analysis, Reductions, and Exact Solutions of the Monge–Ampère Equation in Magnetic Hydrodynamics

IF 0.8 4区数学 Q2 MATHEMATICS

Differential Equations Pub Date : 2024-09-19 DOI:10.1134/s001226612406003x

A. V. Aksenov, A. D. Polyanin

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引用次数: 0

Abstract

We study the Monge–Ampère equation with three independent variables, which occurs in electron magnetohydrodynamics. A group analysis of this strongly nonlinear partial differential equation is carried out. An eleven-parameter transformation preserving the form of the equation is found. A formula is obtained that permits one to construct multiparameter families of solutions based on simpler solutions. Two-dimensional reductions leading to simpler partial differential equations with two independent variables are considered. One-dimensional reductions are described that permit one to obtain self-similar and other invariant solutions that satisfy ordinary differential equations. Exact solutions with additive, multiplicative, and generalized separation of variables are constructed, many of which admit representation in elementary functions. The obtained results and exact solutions can be used to evaluate the accuracy and analyze the adequacy of numerical methods for solving initial–boundary value problems described by strongly nonlinear partial differential equations.

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磁流体力学中蒙日-安培方程的组分析、还原和精确解

摘要我们研究了电子磁流体动力学中出现的具有三个独立变量的 Monge-Ampère 方程。我们对这个强非线性偏微分方程进行了群分析。找到了保留方程形式的十一参数变换。得到的公式允许人们在较简单解的基础上构建多参数解族。此外，还考虑了二维还原法，从而简化了具有两个独立变量的偏微分方程。描述了一维还原，从而获得满足常微分方程的自相似解和其他不变解。还构建了具有加法、乘法和广义变量分离的精确解，其中许多解可以用基本函数表示。所获得的结果和精确解可用来评估求解强非线性偏微分方程所描述的初始边界值问题的数值方法的准确性和分析其适当性。

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来源期刊

Differential Equations 数学-数学

CiteScore

1.30

自引率

33.30%

发文量

审稿时长

3-8 weeks

期刊介绍： Differential Equations is a journal devoted to differential equations and the associated integral equations. The journal publishes original articles by authors from all countries and accepts manuscripts in English and Russian. The topics of the journal cover ordinary differential equations, partial differential equations, spectral theory of differential operators, integral and integral–differential equations, difference equations and their applications in control theory, mathematical modeling, shell theory, informatics, and oscillation theory. The journal is published in collaboration with the Department of Mathematics and the Division of Nanotechnologies and Information Technologies of the Russian Academy of Sciences and the Institute of Mathematics of the National Academy of Sciences of Belarus.