Many physical and engineering phenomena are modeled by partial differential equations which often contain some levels of uncertainty. The advantage of modeling using these so-called stochastic partial differential equations (SPDEs) is that SPDEs are able to more fully capture the behavior of interesting phenomena; it also means that the corresponding numerical analysis of the model will require...

This paper is concerned with linear parabolic partial diierential equations in divergence form and their discrete analogues. It is assumed that the coeecients of the equation are stationary random variables, random in both space and time. The Green's functions for the equations are then random variables. Regularity properties for expectation values of Green's functions are obtained. In particul...

In this paper we implement the unified rational expansion methods, which leads to find exact rational formal polynomial solutions of nonlinear partial differential equations (NLPDEs), to the (1+1)dimensional dispersive long wave and Clannish Random Walker’s parabolic (CRWP) equations. By using this scheme, we get some solutions of the (1+1)-dimensional dispersive long wave and CRWP equations in...

This paper is concerned with sample path properties of anisotropic Gaussian random fields. We establish Fernique-type inequalities and utilize them to study the global and local moduli of continuity for anisotropic Gaussian random fields. Applications to fractional Brownian sheets and to the solutions of stochastic partial differential equations are investigated.

We provide necessary and sufficient conditions for stochastic invariance of finite dimensional submanifolds with boundary in Hilbert spaces for stochastic partial differential equations driven by Wiener processes and Poisson random measures.

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 this study, we aim to construct a traveling wave solution for nonlinear partial differential equations. In this regards, a cosine-function method is used to find and generate the exact solutions for three different types of nonlinear partial differential equations such as general regularized long wave equation (GRLW), general Korteweg-de Vries equation (GKDV) and general equal width wave equ...

In this paper we present the theoretical framework needed to justify the use of a kernelbased collocation method (meshfree approximation method) to estimate the solution of highdimensional stochastic partial differential equations (SPDEs). Using an implicit time stepping scheme, we transform stochastic parabolic equations into stochastic elliptic equations. Our main attention is concentrated on...

This work is concerned about the asymptotic behavior of solutions two and three dimensional stochastic convective Brinkman-Forchheimer (SCBF) equations driven by white noise with nonlinear diffusion terms. We prove existence uniqueness weak pullback mean random attractors for 2D SCBF (for $r\geq1$) as well 3D $r>3$, any $\mu,\beta>0$ $r=3$, $2\mu\beta\geq1$) in Bochner spaces, when terms are Li...