{"title":"Weakly nonlocal boundary value problems with application to geology","authors":"D. Maroncelli, E. Collins","doi":"10.7153/DEA-2021-13-12","DOIUrl":null,"url":null,"abstract":"In many cases, groundwater flow in an unconfined aquifer can be simplified to a one-dimensional Sturm-Liouville model of the form: \\begin{equation*} x''(t)+\\lambda x(t)=h(t)+\\varepsilon f(x(t)),\\hspace{.1in}t\\in(0,\\pi) \\end{equation*} subject to non-local boundary conditions \\begin{equation*} x(0)=h_1+\\varepsilon\\eta_1(x)\\text{ and } x(\\pi)=h_2+\\varepsilon\\eta_2(x). \\end{equation*} In this paper, we study the existence of solutions to the above Sturm-Liouville problem under the assumption that $\\varepsilon$ is a small parameter. Our method will be analytical, utilizing the implicit function theorem and its generalizations.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2021-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Differential Equations & Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.7153/DEA-2021-13-12","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 1
Abstract
In many cases, groundwater flow in an unconfined aquifer can be simplified to a one-dimensional Sturm-Liouville model of the form: \begin{equation*} x''(t)+\lambda x(t)=h(t)+\varepsilon f(x(t)),\hspace{.1in}t\in(0,\pi) \end{equation*} subject to non-local boundary conditions \begin{equation*} x(0)=h_1+\varepsilon\eta_1(x)\text{ and } x(\pi)=h_2+\varepsilon\eta_2(x). \end{equation*} In this paper, we study the existence of solutions to the above Sturm-Liouville problem under the assumption that $\varepsilon$ is a small parameter. Our method will be analytical, utilizing the implicit function theorem and its generalizations.