Related papers: A Two Dimensional Backward Heat Problem With Stati…
We consider the problem of determining a pair of functions $(u,f)$ satisfying the heat equation $u_t -\Delta u =\varphi(t)f (x,y)$, where $(x,y)\in \Omega=(0,1)\times (0,1)$ and the function $\varphi$ is given. The problem is ill-posed.…
We consider the problem of reconstructing, from the interior data $u(x,1)$, a function $u$ satisfying a nonlinear elliptic equation $$ \Delta u = f(x,y,u(x,y)), x \in \RR, y > 0. $$
We consider the recovery of a source term $f(x,t)=p(x)q(t)$ for the nonhomogeneous heat equation in $\Omega\times (0,\infty)$ where $\Omega$ is a bounded domain in $\mathbb{R}^2$ with smooth boundary $\partial\Omega$ from overposed lateral…
Let $\Omega$ be a two-dimensional heat conduction body. We consider the problem of determining the heat source $F(x,t)=\varphi(t)f(x,y)$ with $\varphi$ be given inexactly and $f$ be unknown. The problem is nonlinear and ill-posed. By a…
The goal of the present note is to study intermittency properties for the solution to the fractional heat equation $$\frac{\partial u}{\partial t}(t,x) = -(-\Delta)^{\beta/2} u(t,x) + u(t,x)\dot{W}(t,x), \quad t>0,x \in \bR^d$$ with initial…
We investigate the inverse problem of numerically identifying unknown initial temperatures in a heat equation with dynamic boundary conditions whenever some overdetermination data is provided after a final time. This is a backward parabolic…
We consider the stochastic heat equation $\partial_{s}u =\frac{1}{2}\Delta u +(\beta V(s,y)-\lambda)u$, with a smooth space-time stationary Gaussian random field $V(s,y)$, in dimensions $d\geq 3$, with an initial condition…
In this paper we will study a stiff problem in two-dimensional space and especially its probabilistic counterpart. Roughly speaking, the heat equation with a parameter $\varepsilon>0$ is under consideration: \[ \partial_t…
In this paper, we investigate a nonlinear inverse problem aimed at recovering a coefficient $a(t, x)$, dependent on both time and a subset of spatial variables, in a diffusion equation \( u_t - \Delta_x u - u_{yy} +a(t, x) u = f(t,x,y) \),…
We consider the stochastic heat equation of the following form \frac{\partial}{\partial t}u_t(x) = (\sL u_t)(x) +b(u_t(x)) + \sigma(u_t(x))\dot{F}_t(x)\quad \text{for}t>0, x\in \R^d, where $\sL$ is the generator of a L\'evy process and…
It is well-known that the backward heat conduction problem of recovering the temperature $u(\cdot, t)$ at a time $t\geq 0$ from the knowledge of the temperature at a later time, namely $g:= u(\cdot, \tau)$ for $\tau>t$, is ill-posed, in the…
The problem of recovering coefficients in a diffusion equation is one of the basic inverse problems. Perhaps the most important term is the one that couples the length and time scales and is often referred to as {\it the\/} diffusion…
In this paper, we study the local backward problem of a linear heat equation with time-dependent coefficients under the Dirichlet boundary condition. Precisely, we recover the initial data from the observation on a subdomain at some later…
For every $R>0$, consider the stochastic heat equation $\partial_{t} u_{R}(t\,,x)=\tfrac12 \Delta_{S_{R}^{2}}u_{R}(t\,,x)+\sigma(u_{R}(t\,,x)) \xi_{R}(t\,,x)$ on $S_{R}^{2}$, where $\xi_{R}=\dot{W_{R}}$ are centered Gaussian noises with the…
We consider an inverse problem governed by the initial-boundary value problem for the thermoviscoelastic Kelvin-Voigt system \begin{align*}\left\{ \begin{array}{l} \rho(z,t) u_{tt}- \left(\Gamma(\Theta) u_{zt} +p(z,t) u_z…
We consider non-linear time-fractional stochastic heat type equation $$\frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u=I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h) \stackrel{\cdot}{\tilde N }(t,x,h)\bigg]$$…
In the present paper we study inverse problems related to determining the time-dependent coefficient and unknown source function of fractional heat equations. Our approach shows that having just one set of data at an observation point…
The main goal of this work is to prove that every non-negative {\it strong solution} $u(x,t)$ to the problem $$ u_t+(-\Delta)^{\alpha/2}u=0 \ \quad\mbox{for } (x,t)\in\mathbb{R}^{n}\times(0,T), \quad 0<\alpha<2, $$ can be written as…
We consider a semi-infinite one-dimensional phase-change material with two unknown constant thermal coefficients among the latent heat per unit mass, the specific heat, the mass density and the thermal conductivity. Aiming at the…
The Fourier method is used to find conditions on the right-hand side and on the initial data in the Rayleigh-Stokes problem, which ensure the existence and uniqueness of the solution. Then, in the Rayleigh-Stokes problem, instead of the…