{"title":"Analysis of a class of completely non-local elliptic diffusion operators","authors":"","doi":"10.1007/s13540-024-00254-8","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>This work explores the possibility of developing the analog of some classic results from elliptic PDEs for a class of fractional ODEs involving the composition of both left- and right-sided Riemann-Liouville (R-L) fractional derivatives, <span> <span>\\({D^\\alpha _{a+}}{D^\\beta _{b-}}\\)</span> </span>, <span> <span>\\(1<\\alpha +\\beta <2\\)</span> </span>. Compared to one-sided non-local R-L derivatives, these composite operators are completely non-local, which means that the evaluation of <span> <span>\\({D^\\alpha _{a+}}{D^\\beta _{b-}}u(x)\\)</span> </span> at a point <em>x</em> will have to retrieve the information not only to the left of <em>x</em> all the way to the left boundary but also to the right up to the right boundary, simultaneously. Therefore, only limited tools can be applied to such a situation, which is the most challenging part of the work. To overcome this, we do the analysis from a non-traditional perspective and eventually establish elliptic-type results, including Hopf’s Lemma and maximum principles. As <span> <span>\\(\\alpha \\rightarrow 1^-\\)</span> </span> or <span> <span>\\(\\alpha ,\\beta \\rightarrow 1^-\\)</span> </span>, those operators reduce to the one-sided fractional diffusion operator and the classic diffusion operator, respectively. For these reasons, we still refer to them as “elliptic diffusion operators", however, without any physical interpretation.</p>","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s13540-024-00254-8","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
引用次数: 0
Abstract
This work explores the possibility of developing the analog of some classic results from elliptic PDEs for a class of fractional ODEs involving the composition of both left- and right-sided Riemann-Liouville (R-L) fractional derivatives, \({D^\alpha _{a+}}{D^\beta _{b-}}\), \(1<\alpha +\beta <2\). Compared to one-sided non-local R-L derivatives, these composite operators are completely non-local, which means that the evaluation of \({D^\alpha _{a+}}{D^\beta _{b-}}u(x)\) at a point x will have to retrieve the information not only to the left of x all the way to the left boundary but also to the right up to the right boundary, simultaneously. Therefore, only limited tools can be applied to such a situation, which is the most challenging part of the work. To overcome this, we do the analysis from a non-traditional perspective and eventually establish elliptic-type results, including Hopf’s Lemma and maximum principles. As \(\alpha \rightarrow 1^-\) or \(\alpha ,\beta \rightarrow 1^-\), those operators reduce to the one-sided fractional diffusion operator and the classic diffusion operator, respectively. For these reasons, we still refer to them as “elliptic diffusion operators", however, without any physical interpretation.
Abstract This work explores the possibility of developing the analog of some classic results from elliptic PDEs for a class of fractional ODEs involving the composition of both left-side and right-sided Riemann-Liouville (R-L) fractional derivatives, \({D^\alpha _{a+}}{D^\beta _{b-}}) , \(1<\alpha +\beta <2\) .与单边非局部 R-L 导数相比，这些复合算子是完全非局部的，这意味着在对 x 点的\({D^\alpha _{a+}}{D^\beta _{b-}}u(x)\) 求值时，不仅要检索 x 左侧一直到左边界的信息，还要同时检索右侧一直到右边界的信息。因此，在这种情况下只能使用有限的工具，这也是这项工作最具挑战性的部分。为了克服这个问题，我们从非传统的角度进行分析，最终建立了椭圆型结果，包括霍普夫定理和最大原则。作为 \(\alpha \rightarrow 1^-\) 或 \(\alpha ,\beta \rightarrow 1^-\) ，这些算子分别简化为单边分数扩散算子和经典扩散算子。由于这些原因，我们仍然称它们为 "椭圆扩散算子"，但没有任何物理解释。
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