arXiv subjects

Ravshan Ashurov

Publications and source records attributed to Ravshan Ashurov.

At least 19 recordsLinked to original sources

arXiv2026

Uniqueness for an inverse problem of determining order and temporal factor of the source for time-fractional evolution equations

This paper addresses the inverse problem of simultaneously recovering the fractional order $α\in (0,1)\cup (1,2)$ and the time-dependent source factor $p(t)$ in the Cauchy problem for an evolution equation with a general self-adjoint operator $A$ in a Hilbert space $X$. The overdetermination condition is given by the scalar product $( u(t), ψ)_X$ for $0 < t < T$, where $ψ\in D(A)$ is an arbitrary fixed element. Uniqueness of the fractional order $α$ is established independently of the specific form of the elliptic operator $A$ and the source function $p(t)$. Furthermore, uniqueness of the factor $p(t)$ is proved not only under the trivial overdetermination $( u(t), ψ)_X = 0$ for all $t \in (0,T)$, but also when the function $t \mapsto ( u(t), ψ)_X$ possesses sufficient smoothness. The proof relies on a decomposition of the solution near $t=0$ into a least smooth component and a smoother remainder.

math.AP↗

arXiv2026

Forward and inverse problems for a time-fractional pseudo-parabolic equation with variable coefficients

In this work, forward and inverse problems for a time-fractional pseudo-parabolic equation $D_t^ρ [u(t) + μAu(t)] + σ(t) Au(t) = r(t)g$ are investigated in a Hilbert space, where $A$ is an unbounded, positive, self-adjoint operator. According to the known papers, the forward problem has been studied only in the case $σ(t) = const$. The main novelty of the forward problem in this work is that the model is further generalized and investigated for a time-dependent coefficient $σ(t)$. To determine the solution of the forward problem, the Fourier method is employed, and the global existence and uniqueness of the solution are proved. Moreover, when the operator $A$ is a second-order differential operator, a numerical scheme and an efficient computational algorithm are developed. The inverse problem of determining a time-dependent source function is considered under the overdetermination condition of the form $F[u(t)] = Φ(t)$. The functional $F$ is taken in a general form, and such an inverse problem has not been considered before. The global existence of the solution to the inverse problem is proved by applying Schauder's fixed point theorem, and its uniqueness is established. Furthermore, several examples related to the operator $A$ and the functional $F$ are provided.

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arXiv2026

The backward problem for a multi-term time-fractional diffusion equation

This paper is devoted to the investigation of the backward problem for a multi-term time-fractional diffusion equation. Backward problems for fractional diffusion equations are typically studied using regularization methods due to their ill-posedness in the sense of Hadamard; that is, a small change in u(T) may lead to large changes in the initial data. Nevertheless, we show that if sufficiently smooth current data are considered, then the solution exists, is unique, and is stable. A principal difficulty in the analysis of the backward problem stems from the structure of the solution, in which the multinomial Mittag-Leffler function appears in the denominator. Accordingly, a precise characterization of the asymptotic behavior of this function is required. Such asymptotic properties are nontrivial and have been rigorously established in the authors' recent work, which serves as a fundamental basis for the present study. In addition, we investigate the conditional stability of the backward problem. It is shown that, although the problem is ill-posed in general, stability can be restored under an appropriate a priori bound imposed on the initial data. The main novelty of the paper lies in proving the best smoothing property of the solution, showing that it belongs to the domain of the operator A for any positive time.

math.AP↗

arXiv2026

Non-local time problem for the Rayleigh--Stokes type fractional equations

Despite the growing interest in fractional generalizations of classical fluid dynamics equations, the fractional Rayleigh--Stokes problem has previously been studied almost exclusively using the Riemann--Liouville fractional derivative. To the authors' knowledge, an explicit analytical form of the solution for the Caputo derivative case has not been established in the literature, and before this work, no systematic study of the existence, uniqueness, or regularity properties of this formulation has been conducted. In this paper, we fill this gap by considering the Rayleigh--Stokes equation with the Caputo fractional time derivative of order $ρ\in (0, \, 1)$. Using the Laplace transform and Fourier methods, as well as special functions, we perform a rigorous well-posedness analysis of the corresponding initial boundary-value, non-local, and backward problems.

math.AP↗

arXiv2026

Inverse problem for a multi-term time-fractional diffusion equation with the Caputo derivatives

This paper investigates an inverse source problem for a multi-term time-fractional diffusion equation with Caputo derivatives. The source term is separable as $f(x)g(t)$, with the unknown spatial component $f(x)$ reconstructed from an overdetermination condition at interior time $t_0 \in (0, T]$. The elliptic part is governed by a self-adjoint positive differential operator $A(x, D)$ of order $m \ge 2$. The solution features a spectral representation using the multinomial Mittag-Leffler function, for which we derive novel precise asymptotic expansions. These asymptotics provide a uniform lower bound for the solution's characteristic denominator, enabling sufficient conditions for the existence of a classical solution. Uniqueness of the reconstructed source holds under natural assumptions on the data and $g(t)$. Despite the problem's ill-posedness, high-regularity classical solutions are achievable under suitable structural conditions.

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arXiv2025

Inverse problem of determining a time-dependent coefficient in the time-fractional subdiffusion equation

This paper explores the forward and inverse problems for a fractional subdiffusion equation characterized by time-dependent diffusion and reaction coefficients. Initially, the forward problem is examined, and its unique solvability is established. Subsequently, the inverse problem of identifying an unknown time-dependent reaction coefficient is addressed, with rigorous proofs of the existence and uniqueness of its solution. Both problems' existence and uniqueness are demonstrated using Banach's contraction mapping theorem. Notably, this work is the first to investigate direct and inverse problems for such equations with time-dependent coefficients.

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arXiv2025

Forward and inverse problems for a mixed-type equation with the Caputo fractional derivative and Dezin-type non-local condition

This work is dedicated to the study of a mixed-type partial differential equation involving a Caputo fractional derivative in the time domain $t > 0$ and a classical parabolic equation in the domain $t < 0$, along with Dezin-type non-local boundary and gluing conditions. The forward and inverse problems are studied in detail. For the forward problem, the existence and uniqueness of solutions are established using the Fourier method, under appropriate assumptions on the initial data and the right-hand side. We also analyze the dependency of solvability on the parameter $λ$, from the Dezin-type condition. For the inverse problem, where the right-hand side is separable as $F(x,t) = f(x)g(t)$ (the unknown function is $f(x)$), the existence and uniqueness of a solution are proven under a certain condition on the function $g(t)$ (a constant sign is sufficient).

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arXiv2025

A Nonlinear Nonlocal Problem for the Caputo Fractional Subdiffusion Equation

In this paper, we study a time-fractional subdiffusion equation with a nonlinear nonlocal initial condition involving the unknown solution at the final time. The considered problem is formulated using the Caputo fractional derivative of order $0 < α< 1$, along with homogeneous Dirichlet boundary conditions. The nonlocal initial condition is of the form $ u(x,0) = g(x, u(x,T)) $, where $g$ is a nonlinear function satisfying a Lipschitz condition. The main challenge arises from the implicit dependence on the unknown final state. Using an explicit representation of the solution in terms of the Green function and applying the Banach fixed point theorem, we establish the existence and uniqueness of a regular solution. We also provide uniform estimates for the Green function and analyze the influence of the Lipschitz constant on solvability.

math.AP↗

arXiv2025

Monotonicity in the parameter of the Mittag-Leffler function and determining the fractional exponent of the subdiffusion equation

In this paper, we prove the strict monotonicity in the parameter $ρ$ of the Mittag-Leffler functions $E_ρ(-t^ρ)$ and $t^{ρ-1}E_{ρ,ρ}(-t^ρ)$. Then, these results are applied to solve the inverse problem of determining the order of the fractional derivative in subdiffusion equations, where the available measurement is given at one point in space-time. In particular, we find the missing conditions in the previously known work in this area. Moreover, the obtained results are valid for a wider class of subdiffusion equations than those considered previously. An example of an initial boundary value problem constructed by Sh.A. Alimov is given, for which the inverse problem under consideration has a unique solution. We also point out the application of the monotonicity of the Mittag-Leffler functions to solving some other inverse problems of determining the order of a fractional derivative.

math.AP↗

arXiv2024

On determining the fractional exponent of the subdiffusion equation

Determining the unknown order of the fractional derivative in differential equations simulating various processes is an important task of modern applied mathematics. In the last decade, this problem has been actively studied by specialists. A number of interesting results with a certain applied significance were obtained. This paper provides a short overview of the most interesting works in this direction. Next, we consider the problem of determining the order of the fractional derivative in the subdiffusion equation, provided that the elliptic operator included in this equation has at least one negative eigenvalue. An asymptotic formula is obtained according to which, knowing the solution at least at one point of the domain under consideration, the required order can be calculated.

math.AP↗

arXiv2024

On fractional parabolic systems of vector order

The paper considers the Cauchy problem for the system of partial differential equations of fractional order $D_t^{\mathcal{B}} {U}(t,x) + \mathbb{A}(D) {U} (t,x)=H(t,x) $. Here $U$ and $H$ are vector-functions, the $m\times m$ matrix of differential operators $\mathbb{A}(D)$ is triangular (elements above or below the diagonal are zero). Operators located on the diagonal are elliptic. The main distinctive feature of this system is that the vector-order $\mathcal{B}$ has different components $β_j\in (0,1]$, and $β_j$ are not necessarily rational. Sufficient conditions (in some cases they are necessary) on the initial function and the right-hand side of the equation are found to ensure the existence of a classical solution. Note that the existence of a classical solution to systems of fractional differential equations was studied by various authors, but in all these works the fractional order had the same components for each equation: $β_j=β$, $j=1,...,m$.

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↗

arXiv2024

Determining the order of time and spatial fractional derivatives

The paper considers the initial-boundary value problem for equation $D^ρ_t u(x,t)+ (-Δ)^σu(x,t)=0$, $ρ\in (0,1)$, $σ>0$, in an N-dimensional domain $Ω$ with a homogeneous Dirichlet condition. The fractional derivative is taken in the sense of Caputo. The main goal of the work is to solve the inverse problem of simultaneously determining two parameters: the order of the fractional derivative $ρ$ and the degree of the Laplace operator $σ$. A new formulation and solution method for this inverse problem are proposed. It is proved that in the new formulation the solution to the inverse problem exists and is unique for an arbitrary initial function from the class $L_2(Ω)$. Note that in previously known works, only the uniqueness of the solution to the inverse problem was proved and the initial function was required to be sufficiently smooth and non-negative.

math.AP↗

arXiv2024

On systems of fractional nonlinear partial differential equations

The work considers a system of fractional order partial differential equations. The existence and uniqueness theorems for the classical solution of initial-boundary value problems are proved in two cases: 1) the right-hand side of the equation does not depend on the solution of the problem and 2) it depends on the solution, but at the same time satisfies the classical Lipschitz condition with respect to this variable and an additional condition which guarantees a global existence of the solution. Sufficient conditions are found (in some cases they are necessary) on the initial function and on the right-hand side of the equation, which ensure the existence of a classical solution. In previously known works, linear but more general systems of fractional pseudodifferential equations were considered and the existence of a weak solution was proven in the special classes of distributions.

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arXiv2023

Inverse problem for the subdiffusion equation with non-local in time condition

In the Hilbert space $H$, the inverse problem of determining the right-hand side of the abstract subdiffusion equation with the fractional Caputo derivative is considered. For the forward problem, a non-local in time condition $u(0)=u(T)$ is taken. The right-hand side of the equation has the form $fg(t)$, and the unknown element is $f\in H$. If function $g(t)$ does not change sign, then under a over-determination condition $ u (t_0)= ψ$, $t_0\in (0, T)$, it is proved that the solution of the inverse problem exists and is unique. An example is given showing the violation of the uniqueness of the solution for some sign-changing functions $g(t)$. For such functions $g(t)$, under certain conditions on this function, one can achieve well-posedness of the problem by choosing $t_0$. And for some $g(t)$, for the existence of a solution to the inverse problem, certain orthogonality conditions must be satisfied and in this case there is no uniqueness. All the results obtained are also new for the classical diffusion equations.

math.AP↗

arXiv2023

Time-dependent identification problem for a fractional Telegraph equation with the Caputo derivative

This study investigates the inverse problem of determining the right-hand side of a telegraph equation given in a Hilbert space. The main equation under consideration has the form $(D_{t}^ρ)^{2}u(t)+2αD_{t}^ρu(t)+Au(t)=p( t)q+f(t)$, where $0<t\leq T$, $0<ρ<1$ and $D_{t}^ρ$ is the Caputo derivative. The equation contains a self-adjoint positive operator $A$ and a time-varying multiplier $p(t)$ in the source function, which, like the solution of the equation, is unknown. To solve the inverse problem, an additional condition $B[u(t)] = ψ(t)$ is imposed, where $B$ is an arbitrary bounded linear functional. The existence and uniqueness of a solution to the problem are established and stability inequalities are derived. It should be noted that, as far as we know, such an inverse problem for the telegraph equation is considered for the first time. Examples of the operator $A$ and the functional $B$ are discussed.

math.AP↗

arXiv2023

Fractional Telegraph equation with the Caputo derivative

The Cauchy problem for the telegraph equation $(D_{t}^{ρ})^{2}u(t)+2αD_{t}^{ρ}u(t)+Au(t)=f(t)$ ($0<t\leq T, \, 0<ρ<1$), with the Caputo derivative is considered. Here $A$ is a selfadjoint positive operator, acting in a Hilbert space $H$, $D_t$ is the Caputo fractional derivative. Existence and uniqueness theorems for the solution to the problem under consideration is proved. Inequalities of stability are obtained.

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.

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