arXiv subjects

Yusuf Fayziev

Publications and source records attributed to Yusuf Fayziev.

4 recordsLinked to original sources

arXiv2025

On the Cauchy problem for the Langevin-type fractional equation

In this article, the Cauchy problem for the Langevin-type time-fractional equation $D_t^β(D_t^αu(t))+D_t^β(Au(t))=f(t),(0<t\leq T)$ is studied. Here $α,β\in(0,1)$, $D_t^α, D_t^β$ is the Caputo derivative and $A$ is an unbounded self-adjoint operator in a separable Hilbert space. Under certain conditions, we establish the existence and uniqueness of the solution and provide an explicit representation of it using eigenfunction expansions.

math.AP↗

arXiv2024

On the non-local problem for Boussinesq type fractional equation

In recent years, the Boussinesq type fractional partial differential equation has attracted much attentions of researchers for its practical importance. In this paper we study a non-local problem for the Boussinesq type equation $D_t^αu(t)+A D_t^αu(t)+ν^2A u(t)=0,\,\, 0< t< T,\,\, 1<α<2,$ where $D_t^α$ is the Caputo fractional derivative and $A$ is abstract operator. In the classical case, i.e. at $α=2$, this problem was studied earlier and an interesting effect was discovered: the well-posedness of the problem significantly depends on the length of the time interval and the parameter $ν$. This note shows that for the case of a fractional equation there is no such effect: the problem is well-posed for any $T$ and $ν$.

math.AP↗

arXiv2023

Forward and Inverse Problems for Subdiffusion Equation with Time-Dependent Coefficients

In this paper, we consider forward and inverse problems for subdiffusion equations with time-dependent coefficients. The fractional derivative is taken in the sense of Riemann-Liouville. Using the classical Fourier method, the theorem of the uniqueness and existence of forward and inverse problems for determining the right-hand side of the equation are proved.

math.AP↗

arXiv2022

On the uniqueness of solutions of two inverse problems for the subdiffusion equation

Let $A$ be an arbitrary positive selfadjoint operator, defined in a separable Hilbert space $H$. The inverse problems of determining the right-hand side of the equation and the function $ϕ$ in the non-local boundary value problem $D_t^ρ u(t) + Au(t) = f(t)$ ($0 < ρ< 1, 0 < t \leq T$), $u(ξ) = αu(0) + ϕ$, ($α$ is a constant and $0 < ξ\leq T)$, is considered. Operator $D_t$ on the left-hand side of the equation expresses the Caputo derivative. For both inverse problems $u(ξ_1) = V$ is taken as the over-determination condition. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant $α$ on the existence and uniqueness of a solution to problems is investigated. An interesting effect was discovered: when solving the forward problem, the uniqueness of the solution $u(t)$ was violated, while when solving the inverse problem for the same values of $α$, the solution $u(t)$ became unique.

math.AP↗