Related papers: Numerical reconstructions of a source term in a mo…
In this paper, we are concerned with the stochastic time-fractional diffusion-wave equations in a Hilbert space. The main objective of this paper is to establish properties of the stochastic weak solutions of the initial-boundary value…
In this article, we consider the reconstruction of $\rho(t)$ in the (time-fractional) diffusion equation $(\partial_t^\alpha-\triangle)u(x,t)=\rho(t)g(x)$ ($0<\alpha \le 1$) by the observation at a single point $x_0$. We are mainly…
Duhamel's principle reduces the Cauchy problem for an inhomogeneous partial differential equation to the corresponding homogeneous problem. In the fractional-order setting, the classical principle does not apply directly because fractional…
We study the numerical solution of nonlinear partially observed optimal stopping problems. The system state is taken to be a multi-dimensional diffusion and drives the drift of the observation process, which is another multi-dimensional…
In this work we investigate an inverse problem of recovering point sources and their time-dependent strengths from {a posteriori} partial internal measurements in a subdiffusion model which involves a Caputo fractional derivative in time…
Of primary interest in this paper is the numerical approximation of a time dependent fractional, in space, diffusion equation where the domain is assumed to be nonhomogeneous, having different axial diffusion coefficients. This work is…
Diffusion models are powerful tools for sampling from high-dimensional distributions by progressively transforming pure noise into structured data through a denoising process. When equipped with a guidance mechanism, these models can also…
A quality-Bayesian approach, combining the direct sampling method and the Bayesian inversion, is proposed to reconstruct the locations and intensities of the unknown acoustic sources using partial data. First, we extend the direct sampling…
We focus here on a class of fourth-order parabolic equations that can be written as a system of second-order equations by introducing an auxiliary variable. We design a novel second-order fully discrete mixed finite element method to…
We consider a two-component semilinear reaction-diffusion system in a bounded spatial domain $\Omega$ over a time interval $(0,T)$, which governs the water density $u(x,t)$ and the vegetation biomass density $v(x,t)$ for $x\in\Omega$ and…
We first formulate an inverse problem for a linear fractional Lam\'e system. We determine the Lam\'e parameters from exterior partial measurements of the Dirichlet-to-Neumann map. We further study an inverse obstacle problem as well as an…
Inverse problems have many applications in science and engineering. In Computer vision, several image restoration tasks such as inpainting, deblurring, and super-resolution can be formally modeled as inverse problems. Recently, methods have…
This article deals with the uniqueness in identifying multiple parameters simultaneously in the one-dimensional time-fractional diffusion-wave equation of fractional time-derivative order $\in (0,2)$ with the zero Robin boundary condition.…
Two main aims of this paper are to develop a numerical method to solve an inverse source problem for parabolic equations and apply it to solve a nonlinear coefficient inverse problem. The inverse source problem in this paper is the problem…
The diffusion system with time-fractional order derivative is of great importance mathematically due to the nonlocal property of the fractional order derivative, which can be applied to model the physical phenomena with memory effects. We…
Anomalous diffusion in the presence or absence of an external force field is often modelled in terms of the fractional evolution equations, which can involve the hyper-singular source term. For this case, conventional time stepping methods…
Inverse design refers to the problem of optimizing the input of an objective function in order to enact a target outcome. For many real-world engineering problems, the objective function takes the form of a simulator that predicts how the…
In this paper, direct and inverse problems for a space fractional advection dispersion equation on a finite domain are studied. The inverse problem consists in determining the source term from a final observation. We first drive the…
This article is devoted to the simultaneous resolution of three inverse problems, among the most important formulation of inverse problems for partial differential equations, stated for some class of diffusion equations from a single…
Inverse problems involve making inference about unknown parameters of a physical process using observational data. This paper investigates an important class of inverse problems -- the estimation of the initial condition of a…