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.

中文翻译：

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希尔弗分数阶导数方程的非局部问题

 抽象的

在本文中，我们研究了带有 Hilfer 导数的分数阶偏微分方程的非局部问题。非局部边值问题，\(D^{\alpha,\beta}u(t)+Au(t)=f(t)\) (\(0<\alpha<1\), \(0 \leq\beta\leq 1\) 和 \(0<t\leq T\)), \(I^{\delta}u(t)=\gamma I^{\delta}u(+0)+\考虑具有强正自伴算子 \(A\) 的任意可分离希尔伯特空间 H 中的 varphi\) （\(\gamma\) 是常数）。除了正问题之外，文章还探讨了确定方程右侧的逆问题。存在唯一性定理被证明可以解决正问题和反问题。