This paper is an extension of previous work where we laid the foundation for the kernel-based collocation solution of stochastic partial differential equations (SPDEs), but dealt only with the simpler problem of right-hand-side Gaussian noises. In the present paper we show that kernel-based collocation methods can be used to approximate the solutions of high-dimensional elliptic partial differe...

Strong dierential subordination and superordination properties are determined for some familiesanalytic functions in the open unit disk which are associated with the Komatu operator by investigatingappropriate classes of admissible functions. New strong dierential sandwich-type results arealso obtained.

The existence of random attractors for a large class of stochastic partial differential equations (SPDE) driven by general additive noise is established. The main results are applied to various types of SPDE, as e.g. stochastic reaction-diffusion equations, the stochastic p-Laplace equation and stochastic porous media equations. Besides classical Brownian motion, we also include space-time frac...

The curse of dimensionality is commonly encountered in numerical partial differential equations (PDE), especially when uncertainties have to be modeled into the equations as random coefficients. However, very often the variability of physical quantities derived from PDE can be captured by a few features on the space of random coefficients. Based on such observation, we propose using neural-netw...

The objective of this work is to provide a methodology for approximating globally optimal Fekete point con-gurations. This problem is of obvious interest in numerical mathematics and scientic modeling. Following a brief discussion of the pertinent analytical background, Lipschitz global optimization (LGO) is applied to determine {i.e., to numerically approximate{ Fekete point congurations. Next...

the present study introduces a new technique of homotopy perturbation method for the solution of systems of fractional partial differential equations. the proposed scheme is based on laplace transform and new homotopy perturbation methods. the fractional derivatives are considered in caputo sense. to illustrate the ability and reliability of the method some examples are provided. the results ob...

The discretization of linear partial differential equations with random data by means of the stochastic Galerkin finite element method results in general in a large coupled linear of system of equations. Using the stochastic diffusion equation as a model problem, we introduce and study a symmetric positive definite Kronecker product preconditioner for the Galerkin matrix. We compare the popular...

The theoretical studies conducted mainly by the author are reviewed on (1) derivation of arbitrary order moment equations and solutions of some equations, (2) scattering by many particles and the effective medium constant of random medium, (3) scattering by a conducting body in random media and (4) spatially partially-coherent wave scattering, with application to satellite communications, artif...

In this paper we analyse the almost sure exponential stability and ultimate boundedness of the solutions to a class of neutral stochastic semilinear partial delay differential equations. This kind of equations arises in problems related to coupled oscillators in a noisy environment, or in viscoeslastic materials under random or stochastic influences.

Using the semigroup product formula of P. Chernoff, a central limit theorem is derived for products of random matrices. Applications are presented for representations of solutions to linear systems of stochastic differential equations, and to the corresponding partial differential evolution equations. Included is a discussion of stochastic semigroups, and a stochastic version of the Lie-Trotter...