{"title":"On initial value problem for elliptic equation on the plane under Caputo derivative","authors":"Tran Thanh Binh, Bui Dinh Thang, Nguyen Duc Phuong","doi":"10.1515/dema-2022-0257","DOIUrl":null,"url":null,"abstract":"Abstract In this article, we are interested to study the elliptic equation under the Caputo derivative. We obtain several regularity results for the mild solution based on various assumptions of the input data. In addition, we derive the lower bound of the mild solution in the appropriate space. The main tool of the analysis estimation for the mild solution is based on the bound of the Mittag-Leffler functions, combined with analysis in Hilbert scales space. Moreover, we provide a regularized solution for our problem using the Fourier truncation method. We also obtain the error estimate between the regularized solution and the mild solution. Our current article seems to be the first study to deal with elliptic equations with Caputo derivatives on the unbounded domain.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Demonstratio Mathematica","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/dema-2022-0257","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Abstract In this article, we are interested to study the elliptic equation under the Caputo derivative. We obtain several regularity results for the mild solution based on various assumptions of the input data. In addition, we derive the lower bound of the mild solution in the appropriate space. The main tool of the analysis estimation for the mild solution is based on the bound of the Mittag-Leffler functions, combined with analysis in Hilbert scales space. Moreover, we provide a regularized solution for our problem using the Fourier truncation method. We also obtain the error estimate between the regularized solution and the mild solution. Our current article seems to be the first study to deal with elliptic equations with Caputo derivatives on the unbounded domain.
期刊介绍：
Demonstratio Mathematica publishes original and significant research on topics related to functional analysis and approximation theory. Please note that submissions related to other areas of mathematical research will no longer be accepted by the journal. The potential topics include (but are not limited to): -Approximation theory and iteration methods- Fixed point theory and methods of computing fixed points- Functional, ordinary and partial differential equations- Nonsmooth analysis, variational analysis and convex analysis- Optimization theory, variational inequalities and complementarity problems- For more detailed list of the potential topics please refer to Instruction for Authors. The journal considers submissions of different types of articles. "Research Articles" are focused on fundamental theoretical aspects, as well as on significant applications in science, engineering etc. “Rapid Communications” are intended to present information of exceptional novelty and exciting results of significant interest to the readers. “Review articles” and “Commentaries”, which present the existing literature on the specific topic from new perspectives, are welcome as well.
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