اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدInitial-Boundary Value Problem for the heat equation - A stochastic algorithm
101
0
0.0
(
0
)
تأليف
Madalina Deaconu
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
The Initial-Boundary Value Problem for the heat equation is solved by using a new algorithm based on a random walk on heat balls. Even if it represents a sophisticated generalization of the Walk on Spheres (WOS) algorithm introduced to solve the Dirich-let problem for Laplaces equation, its implementation is rather easy. The definition of the random walk is based on a new mean value formula for the heat equation. The convergence results and numerical examples permit to emphasize the efficiency and accuracy of the algorithm.
قيم البحث
اقرأ أيضاً
We consider a family of nonlinear stochastic heat equations of the form $partial_t u=mathcal{L}u + sigma(u)dot{W}$, where $dot{W}$ denotes space-time white noise, $mathcal{L}$ the generator of a symmetric Levy process on $R$, and $sigma$ is Lipschitz
continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure $u_0$. Tight a priori bounds on the moments of the solution are also obtained. In the particular case that $mathcal{L}f=cf$ for some $c>0$, we prove that if $u_0$ is a finite measure of compact support, then the solution is with probability one a bounded function for all times $t>0$.
We consider the stochastic heat equation with a multiplicative white noise forcing term under standard intermitency conditions. The main finding of this paper is that, under mild regularity hypotheses, the a.s.-boundedness of the solution $xmapsto u(
t,,x)$ can be characterized generically by the decay rate, at $pminfty$, of the initial function $u_0$. More specifically, we prove that there are 3 generic boundedness regimes, depending on the numerical value of $Lambda:= lim_{|x|toinfty} |log u_0(x)|/(log|x|)^{2/3}$.
The solution of Kardar-Parisi-Zhang equation (KPZ equation) is solved formally via Cole-Hopf transformation $h=log u$, where $u$ is the solution of multiplicative stochastic heat equation(SHE). In earlier works by Chatterjee and Dunlap, Caravenna, Su
n, and Zygouras, and Gu, they consider the solution of two dimensional KPZ equation via the solution $u_varepsilon$ of SHE with flat initial condition and with noise which is mollified in space on scale in $varepsilon$ and its strength is weakened as $beta_varepsilon=hat{beta} sqrt{frac{2pi varepsilon}{-log varepsilon}}$, and they prove that when $hat{beta}in (0,1)$, $frac{1}{beta_varepsilon}(log u_varepsilon-mathbb{E}[log u_varepsilon])$ converges in distribution to a solution of Edward-Wilkinson model as a random field. In this paper, we consider a stochastic heat equation $u_varepsilon$ with general initial condition $u_0$ and its transformation $F(u_varepsilon)$ for $F$ in a class of functions $mathfrak{F}$, which contains $F(x)=x^p$ ($0<pleq 1$) and $F(x)=log x$. Then, we prove that $frac{1}{beta_varepsilon}(F(u_varepsilon(t,x))-mathbb{E}[F(u_varepsilon(t,x))])$ converges in distribution to Gaussian random variables jointly in finitely many $Fin mathfrak{F}$, $t$, and $u_0$. In particular, we obtain the fluctuations of solutions of stochastic heat equations and KPZ equations jointly converge to solutions of SPDEs which depends on $u_0$. Our main tools are It^os formula, the martingale central limit theorem, and the homogenization argument as in the works by Cosco and the authors. To this end, we also prove the local limit theorem for the partition function of intermediate $2d$-directed polymers
We present a non-iterative algorithm to reconstruct the isotropic acoustic wave speed from the measurement of the Neumann-to-Dirichlet map. The algorithm is designed based on the boundary control method and involves only computations that are stable.
We prove the convergence of the algorithm and present its numerical implementation. The effectiveness of the algorithm is validated on both constant speed and variable speed, with full and partial boundary measurement as well as different levels of noise.
A reaction-diffusion equation with power nonlinearity formulated either on the half-line or on the finite interval with nonzero boundary conditions is shown to be locally well-posed in the sense of Hadamard for data in Sobolev spaces. The result is e
stablished via a contraction mapping argument, taking advantage of a novel approach that utilizes the formula produced by the unified transform method of Fokas for the forced linear heat equation to obtain linear estimates analogous to those previously derived for the nonlinear Schrodinger, Korteweg-de Vries and good Boussinesq equations. Thus, the present work extends the recently introduced unified transform method approach to well-posedness from dispersive equations to diffusive ones.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
