On the Nonlocal Problem for the Equation with the Hilfer Fractional Derivative
R. R. AshurovDepartment of Mathematics, New Uzbekistan University, 100000, Tashkent, UzbekistanYusuf FayzievNational University of Uzbekistan, 100174, Tashkent, UzbekistanNazokat TukhtaevaRomanovskii Institute of Mathematics, Academy of Science of Uzbekistan, 100174, Tashkent, Uzbekistan
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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.
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