Related papers: L1 scheme for solving an inverse problem subject t…
An initial-boundary value problem with a Caputo time derivative of fractional order $\alpha\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. An L2-type discrete fractional-derivative…
In this paper, we consider a diffusion equation with fractional-time derivative with nonsingular Mittag-Leffler kernel in Hilbert spaces. Existence and uniqueness of solution are proved by means of a spectral argument. The existence of…
In this paper, we investigate the direct and linear inverse problems of identifying time-dependent and time-independent source terms in a time-fractional diffusion-wave equation, using measured data at an interior point of the time…
The numerical approximation of an inverse problem subject to the convection--diffusion equation when diffusion dominates is studied. We derive Carleman estimates that are on a form suitable for use in numerical analysis and with explicit…
We study a linear-quadratic optimal control problem involving a parabolic equation with fractional diffusion and Caputo fractional time derivative of orders $s \in (0,1)$ and $\gamma \in (0,1]$, respectively. The spatial fractional…
Fractional Gaussian noise models the time series with long-range dependence; when the Hurst index $H>1/2$, it has positive correlation reflecting a persistent autocorrelation structure. This paper studies the numerical method for solving…
In this work, we consider a FDE (fractional diffusion equation) $${}^C D_t^\alpha u(x,t)-a(t)\mathcal{L} u(x,t)=F(x,t)$$ with a time-dependent diffusion coefficient $a(t)$. For the direct problem, given an $a(t),$ we establish the…
In this work, we explore a time-fractional diffusion equation of order $\alpha \in (0,1)$ with a stochastic diffusivity parameter. We focus on efficient estimation of the expected values (considered as an infinite dimensional integral on…
In this paper, we present an inverse problem of identifying the reaction coefficient for time fractional diffusion equations in two dimensional spaces by using boundary Neumann data. It is proved that the forward operator is continuous with…
In this paper a fractional differential equation of the Euler-Lagrange / Sturm-Liouville type is considered. The fractional equation with derivatives of order $\alpha \in \left( 0,1 \right]$ in the finite time interval is transformed to the…
An inverse problem to determine a space-dependent factor in a semilinear time-fractional diffusion equation is considered. Additional data are given in the form of an integral with the Borel measure over the time. Uniqueness of the solution…
This paper analyzes the well-known L1 scheme for fractional wave equations with nonsmooth data. A new stability estimate is obtained, and the temporal accuracy $ \mathcal O(\tau^{3-\alpha}) $ is derived for the nonsmooth initial data. In…
This paper develops a two-level fourth-order scheme for solving time-fractional convection-diffusion-reaction equation with variable coefficients subjected to suitable initial and boundary conditions. The basis properties of the new…
In this paper, we are concerned with the numerical solution for the two-dimensional time fractional Fokker-Planck equation with tempered fractional derivative of order $\alpha$. Although some of its variants are considered in many recent…
In this paper, we obtain the sharp uniqueness for an inverse $x$-source problem for a one-dimensional time-fractional diffusion equation with a zeroth-order term by the minimum possible lateral Cauchy data. The key ingredient is the unique…
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…
In this paper, we consider an inverse problem for a time-fractional diffusion equation with a nonlinear source. We prove that the considered problem is ill-posed, i.e. the solution does not depend continuously on the data. The problem is…
Fractional Dzherbashian-Nersesian operator is considered and three famous fractional order derivatives namely Riemann-Liouville, Caputo and Hilfer derivatives are shown to be special cases of the earlier one. The expression for Laplace…
We consider the inverse source problem of determining a source term depending on both time and space variable for fractional and classical diffusion equations in a cylindrical domain from boundary measurements. With suitable boundary…
The backwards diffusion equation is one of the classical ill-posed inverse problems, related to a wide range of applications, and has been extensively studied over the last 50 years. One of the first methods was that of {\it…