We consider an elliptic PDE in two variables. As one parameter approaches zero, this PDE collapses to a parabolic one, that is forward parabolic in a part of the domain and backward parabolic in the remainder. Such problems arise naturally in various stochastic models, such as fluid models for data-handling systems and Markovmodulated queues. We employ singular perturbation methods to study the...

In this study, the optimal control problem of nonlinear parabolic partial differential equations (PDEs) is investigated. Optimal control of nonlinear PDEs is an open problem with applications that include fluid, thermal, biological, and chemically-reacting systems. Model predictive control with a fast numerical solution method has been well established to solve the optimal control problem of no...

ABSTRACT. The aim of this article is to study the asymptotic behaviour for large times of solutions to a certain class of stochastic partial differential equations of parabolic type. In particular, we will prove the backward uniqueness result and the existence of the spectral limit for abstract SPDEs and then show how these results can be applied to some concrete linear and nonlinear SPDEs. For...

We consider the backward parabolic equation { ut +Au = f(t, u(t)), 0 < t < T, u(T ) = g, where A is a positive unbounded operator and f is a nonlinear function satisfying a Lipschitz condition, with an approximate datum g. The problem is severely ill-posed. Using the truncation method we propose a regularized solution which is the solution of a system of differential equations in finite dimensi...

We derive the precise limit of SHS in the high activation energy scaling suggested by B.J. Matkowksy-G.I. Sivashinsky in 1978 and by A. Bayliss-B.J. Matkowksy-A.P. Aldushin in 2002. In the time-increasing case the limit coincides with the Stefan problem for supercooled water with spatially inhomogeneous coefficients. In general it is a nonlinear forward-backward parabolic equation with disconti...

We propose in this paper a technique to accelerate the convergence of the Newton method to solve non-linear unsteady CFD problems. The principle relies on an effective way to obtain a better solution for the current time step than the solution of the previous time step. This is performed on a reduced model computed with a Proper Orthogonal Decomposition of some solution computed at some previou...

The impact of forward and backward amplification schemes on the generation of optical similaritons has been numerically and experimentally investigated around 1550 nm using a fiber based Raman amplifier. The pumping configuration does not affect the similariton nature of the output pulse, but it significantly influences the characteristics of the pulse. Shorter and higher peak-power similariton...

We address the self-similar solvability of a singular parabolic problem and show that solutions to parabolic fully nonlinear equations are not expected to be C.

We consider an initialand Dirichlet boundaryvalue problem for a fourth-order linear stochastic parabolic equation, in two or three space dimensions, forced by an additive space-time white noise. Discretizing the space-time white noise a modeling error is introduced and a regularized fourthorder linear stochastic parabolic problem is obtained. Fully-discrete approximations to the solution of the...