In this paper we study the Local Discontinuous Galerkin scheme for solving the stochastic heat equation driven by the space white noise. We begin by giving a brief introduction to stochastic processes, stochastic differential equations, and their importance in the modern mathematical context. From there, using an example stochastic elliptic partial differential equation, we approximate the whit...

in this paper, the exp-function method, with the aid of a symbolic computation system such as maple, is applied to the (2+1) -dimensional calogero bogoyavlanskii schiff equation. exact and explicit generalized solitary solutions are obtained in more general forms. the free parameters can be determined by initial or boundary conditions. the method is straightforward and concise, and its applicat...

we focus on the use of two stable and accurate explicit finite difference schemes in order to approximate the solution of stochastic partial differential equations of it¨o type, in particular, parabolic equations. the main properties of these deterministic difference methods, i.e., convergence, consistency, and stability, are separately developed for the stochastic cases.

‎In the present paper‎, ‎optimal heating of temperature field which is modelled as a boundary optimal control problem‎, ‎is investigated in the uncertain environments and then it is solved numerically‎. ‎In physical modelling‎, ‎a partial differential equation with stochastic input and stochastic parameter are applied as the constraint of the optimal control problem‎. ‎Controls are implemented ...

The focus of this work is on a two-dimensional stochastic vorticity equation for an incompressible homogeneous viscous fluid. We consider a signed measure-valued stochastic partial differential equation for a vorticity process based on the Skorohod–Ito evolution of a system of N randomly moving point vortices. A nonlinear filtering problem associated with the evolution of the vorticity is consi...

White noise-driven nonlinear stochastic partial differential equations (SPDEs) of parabolic type are frequently used to model physical systems in space dimensions d = 1, 2, 3. Whereas existence and uniqueness of weak solutions to these equations are well established in one dimension, the situation is different for d ≥ 2. Despite their popularity in the applied sciences, higher dimensional versi...

Complex systems display variability over a broad range of spatial and temporal scales. Some scales are unresolved due to computational limitations. The impact of these unresolved scales on the resolved scales needs to be parameterized or taken into account. One stochastic parameterization scheme is devised to take the effects of unresolved scales into account, in the context of solving a nonlin...

This paper proposes a stochastic finite difference approach, based onhomogenous chaos expansion (SFDHC).The said approach can handle time dependent nonlinear as well as linear systems with deterministic or stochastic initial and boundary conditions. In this approach, included stochastic parameters are modeled as second-order stochastic processes and are expanded using KarhunenLoève expansion, w...

Stochastic partial differential equations (SPDE) on graphs were recently introduced by Cerrai and Freidlin (Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 865–899). This class of stochastic in infinite dimensions provides a minimal framework for the study effective dynamics much more complex systems. However, how they emerge from microscopic individual-based models is still poorly understood...

We consider the perturbation of parabolic operators of the form ∂t + P (x,D) by large-amplitude highly oscillatory spatially dependent potentials modeled as Gaussian random fields. The amplitude of the potential is chosen so that the solution to the random equation is affected by the randomness at the leading order. We show that, when the dimension is smaller than the order of the elliptic pseu...