Related papers: Carleman estimate and its application for anomalou…
We construct a novel estimator for the diffusion coefficient of the limiting homogenized equation, when observing the slow dynamics of a multiscale model, in the case when the slow dynamics are of bounded variation. Previous research…
We derive backward and forward fractional Schr\"odinger type of equations for the distribution of functionals of the path of a particle undergoing anomalous diffusion. Fractional substantial derivatives introduced by Friedrich and…
In the development of controllability and inverse problem results for semi-discrete systems, by using Carleman estimates, it is required to estimate of the discrete operators applied to Carleman weight functions. This work aims to establish…
We present an analysis of existence, uniqueness, and smoothness of the solution to a class of fractional ordinary differential equations posed on the whole real line that models a steady state behavior of a certain anomalous diffusion,…
The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order $\beta \in (0,1)$. The fundamental solution for the Cauchy problem is…
In this paper, two kinds of high-order compact finite difference schemes for second-order derivative are developed. Then a second-order numerical scheme for Riemann-Liouvile derivative is established based on fractional center difference…
The fractional Sturm-Liouville eigenvalue problem appears in many situations, e.g., while solving anomalous diffusion equations coming from physical and engineering applications. Therefore to obtain solutions or approximation of solutions…
This paper intends on obtaining the explicit solution of $n$-dimensional anomalous diffusion equation in the infinite domain with non-zero initial condition and vanishing condition at infinity. It is shown that this equation can be derived…
In mathematical modeling of the non-squared frequency-dependent diffusions, also known as the anomalous diffusions, it is desirable to have a positive real Fourier transform for the time derivative of arbitrary fractional or odd integer…
In this article we study the numerical approximation of a variable coefficient fractional diffusion equation. Using a change of variable, the variable coefficient fractional diffusion equation is transformed into a constant coefficient…
The first part of this paper introduces sufficient conditions to determine conservation laws of diffusion equations of arbitrary fractional order in time. Numerical methods that satisfy a discrete analogue of these conditions have…
In the paper, we show a global Carleman estimate for the non-local heat equation. To be more precise, let $\Omega\subset\RR^d$ be a bounded domain and $\CO\subset\Omega$ an open subdomain, $s\in(0,1)$. We show that there exist constants…
We present a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional derivative of order $\alpha\in(0,1)$ in time. The framework relies on three…
Weighted averaged finite difference methods for solving fractional diffusion equations are discussed and different formulae of the discretization of the Riemann-Liouville derivative are considered. The stability analysis of the different…
In the present paper, we address a class of the fractional derivatives of constant and variable orders for the first time. Fractional-order relaxation equations of constants and variable orders in the sense of Caputo type are modeled from…
In this paper, we study discrete Carleman estimates for space semi-discrete approximations of one-dimensional stochastic parabolic equation. As applications of these discrete Carleman estimates, we apply them to study two inverse problems…
We examine the short and long-time behaviors of time-fractional diffusion equations with variable space-dependent order. More precisely, we describe the time-evolution of the solution to these equations as the time parameter goes either to…
The nonparametric estimation of the volatility and the drift coefficient of a scalar diffusion is studied when the process is observed at random time points. The constructed estimator generalizes the spectral method by Gobet, Hoffmann and…
Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations, but is better described by fractional diffusion models. The nonlocal nature of the fractional diffusion operators makes substantially…
The time-fractional convection-diffusion equation is performed by Lie symmetry analysis method which involves the Riemann-Liouville time-fractional derivative of the order $\alpha\in(0,2)$. In eight cases, the symmetries are obtained and…