{"title":"The Dirichlet problem in an infinite layer for a system of differential equations with shifts","authors":"Zinovii Nytrebych, Roman Shevchuk, Ivan Savka","doi":"10.1515/gmj-2023-2104","DOIUrl":null,"url":null,"abstract":"In this paper, we study the problem with data on the boundary of the infinite layer <jats:disp-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>x</m:mi> <m:mo rspace=\"0.278em\" stretchy=\"false\">)</m:mo> </m:mrow> <m:mo rspace=\"0.278em\">:</m:mo> <m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>h</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo rspace=\"0.337em\">,</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi mathvariant=\"double-struck\">R</m:mi> <m:mi>s</m:mi> </m:msup> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">}</m:mo> </m:mrow> <m:mo rspace=\"1.167em\">,</m:mo> <m:mi>h</m:mi> </m:mrow> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> <m:mo rspace=\"0.337em\">,</m:mo> <m:mrow> <m:mi>s</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant=\"double-struck\">N</m:mi> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2104_eq_9999.png\" /> <jats:tex-math>\\{(t,x):t\\in(0,h),\\,x\\in\\mathbb{R}^{s}\\},\\quad h>0,\\,s\\in\\mathbb{N},</jats:tex-math> </jats:alternatives> </jats:disp-formula> for the system of two differential equations of the second order in the time variable 𝑡 with shifts in the spatial variables <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>x</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>x</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>x</m:mi> <m:mi>s</m:mi> </m:msub> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2104_ineq_0001.png\" /> <jats:tex-math>x_{1},x_{2},\\ldots,x_{s}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We propose a differential-symbol method of constructing a solution of the problem and identify a class of vector functions in which the obtained solution is unique. The method of solving the Dirichlet problem in the layer is illustrated by examples.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"14 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Georgian Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/gmj-2023-2104","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we study the problem with data on the boundary of the infinite layer {(t,x):t∈(0,h),x∈Rs},h>0,s∈N,\{(t,x):t\in(0,h),\,x\in\mathbb{R}^{s}\},\quad h>0,\,s\in\mathbb{N}, for the system of two differential equations of the second order in the time variable 𝑡 with shifts in the spatial variables x1,x2,…,xsx_{1},x_{2},\ldots,x_{s}. We propose a differential-symbol method of constructing a solution of the problem and identify a class of vector functions in which the obtained solution is unique. The method of solving the Dirichlet problem in the layer is illustrated by examples.
在本文中,我们研究了无限层 { ( t , x ) : t∈ ( 0 , h ) , x∈ R s } 边界上的数据问题。 , h > 0 , s ∈ N , \{(t,x):t\in(0,h),\,x\in\mathbb{R}^{s}\},\quad h>0,\,s\in\mathbb{N}, 为时间变量 x 1 , x 2 , ... , x s x_{1},x_{2},\ldots,x_{s} 的二阶微分方程系统。我们提出了一种构建问题解的微分符号法,并确定了一类向量函数,在这类向量函数中,得到的解是唯一的。我们通过实例来说明层中 Dirichlet 问题的求解方法。
期刊介绍:
The Georgian Mathematical Journal was founded by the Georgian National Academy of Sciences and A. Razmadze Mathematical Institute, and is jointly produced with De Gruyter. The concern of this international journal is the publication of research articles of best scientific standard in pure and applied mathematics. Special emphasis is put on the presentation of results obtained by Georgian mathematicians.