{"title":"On the Nonlocal Problem for the Equation with the Hilfer Fractional Derivative","authors":"R. R. Ashurov, Yu. E. Fayziev, N. M. Tukhtaeva","doi":"10.1134/s1995080224600729","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In the paper, we study the nonlocal problem for a fractional partial differential equation with the Hilfer derivative. The non-local boundary value problem, <span>\\(D^{\\alpha,\\beta}u(t)+Au(t)=f(t)\\)</span> (<span>\\(0<\\alpha<1\\)</span>, <span>\\(0\\leq\\beta\\leq 1\\)</span> and <span>\\(0<t\\leq T\\)</span>), <span>\\(I^{\\delta}u(t)=\\gamma I^{\\delta}u(+0)+\\varphi\\)</span> (<span>\\(\\gamma\\)</span> is a constant), in an arbitrary separable Hilbert space H with the strongly positive self-adjoint operator <span>\\(A\\)</span>, is considered. In addition to the forward problem, the article also explores the inverse problem of determining the right-hand side of the equation. Existence and uniqueness theorems are proved to solve the forward and inverse problems.</p>","PeriodicalId":46135,"journal":{"name":"Lobachevskii Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Lobachevskii Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s1995080224600729","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In the paper, we study the nonlocal problem for a fractional partial differential equation with the Hilfer derivative. The non-local boundary value problem, \(D^{\alpha,\beta}u(t)+Au(t)=f(t)\) (\(0<\alpha<1\), \(0\leq\beta\leq 1\) and \(0<t\leq T\)), \(I^{\delta}u(t)=\gamma I^{\delta}u(+0)+\varphi\) (\(\gamma\) is a constant), in an arbitrary separable Hilbert space H with the strongly positive self-adjoint operator \(A\), is considered. In addition to the forward problem, the article also explores the inverse problem of determining the right-hand side of the equation. Existence and uniqueness theorems are proved to solve the forward and inverse problems.
期刊介绍:
Lobachevskii Journal of Mathematics is an international peer reviewed journal published in collaboration with the Russian Academy of Sciences and Kazan Federal University. The journal covers mathematical topics associated with the name of famous Russian mathematician Nikolai Lobachevsky (Lobachevskii). The journal publishes research articles on geometry and topology, algebra, complex analysis, functional analysis, differential equations and mathematical physics, probability theory and stochastic processes, computational mathematics, mathematical modeling, numerical methods and program complexes, computer science, optimal control, and theory of algorithms as well as applied mathematics. The journal welcomes manuscripts from all countries in the English language.