In this note, a feedback boundary controller for an unstable heat equation is designed. The equation can be viewed as a model of a thin rod with not only the heat loss to a surrounding medium (stabilizing) but also the heat generation inside the rod (destabilizing). The heat generation adds a destabilizing linear term on the right-hand side of the equation. The boundary control law designed is ...

In this paper we will prove a maximum principle for the solutions of linear parabolic equation on complete non-compact manifolds with a time varying metric. We will prove the convergence of the Neumann Green function of the conjugate heat equation for the Ricci flow in Bk × (0, T ) to the minimal fundamental solution of the conjugate heat equation as k → ∞. We will prove the uniqueness of the f...

Recently, the author suggested a simple and composite equation of state by incorporating fundamental thermodynamic properties like heat capacities into her earlier concise equation of state for gases based on free volume and molecular association / dissociation. This work brings new results for aqueous solutions, based on the analogy of the equation of state for gases and solutions over wide ra...

We consider the stochastic heat equation with a multiplicative colored noise term on Rd for d ≥ 1. First, we prove convergence of a branching particle system in a random environment to this stochastic heat equation with linear noise coefficients. For this stochastic partial differential equation with more general non-Lipschitz noise coefficients we show convergence of associated lattice systems...

In this paper, we prove a sample-path comparison principle for the nonlinear stochastic fractional heat equation on R with measure-valued initial data. We give quantitative estimates about how close to zero the solution can be. These results extend Mueller’s comparison principle on the stochastic heat equation to allow more general initial data such as the (Dirac) delta measure and measures wit...

We introduce the standard fourth order compact finite difference formulae. We show how these formulae apply in the special case of the heat equation. It is well known that the American option pricing problem may be formulated in terms of the Black Scholes partial differential equation (PDE) together with a free boundary condition. Standard methods allow this problem to be transformed into a mov...

According to the Smolukowski-Kramers approximation, we show that the solution of the semi-linear stochastic damped wave equations μutt (t, x) = u(t, x)−ut (t, x)+ b(x, u(t, x)) + QẆ(t), u(0) = u0, ut (0) = v0, endowed with Dirichlet boundary conditions, converges as μ goes to zero to the solution of the semi-linear stochastic heat equation ut (t, x) = u(t, x)+b(x, u(t, x))+QẆ(t), u(0) = u0, end...

We study the stochastic heat equation ∂tu = Lu+σ(u)Ẇ in (1+1) dimensions, where Ẇ is space-time white noise, σ : R → R is Lipschitz continuous, and L is the generator of a Lévy process. We assume that the underlying Lévy process has finite exponential moments in a neighborhood of the origin and u0 has exponential decay at ±∞. Then we prove that under natural conditions on σ: (i) The νth absolut...