{"title":"Group Analysis, Reductions, and Exact Solutions of the Monge–Ampère Equation in Magnetic Hydrodynamics","authors":"A. V. Aksenov, A. D. Polyanin","doi":"10.1134/s001226612406003x","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the Monge–Ampère equation with three independent variables, which\noccurs in electron magnetohydrodynamics. A group analysis of this strongly nonlinear partial\ndifferential equation is carried out. An eleven-parameter transformation preserving the form of the\nequation is found. A formula is obtained that permits one to construct multiparameter families of\nsolutions based on simpler solutions. Two-dimensional reductions leading to simpler partial\ndifferential equations with two independent variables are considered. One-dimensional reductions\nare described that permit one to obtain self-similar and other invariant solutions that satisfy\nordinary differential equations. Exact solutions with additive, multiplicative, and generalized\nseparation of variables are constructed, many of which admit representation in elementary\nfunctions. The obtained results and exact solutions can be used to evaluate the accuracy and\nanalyze the adequacy of numerical methods for solving initial–boundary value problems described\nby strongly nonlinear partial differential equations.\n</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s001226612406003x","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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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