Convergence to Homogenized or Stochastic Partial Differential Equations

Convergence to Homogenized or Stochastic Partial Differential Equations

We consider the behavior of solutions to parabolic equations with large, highly oscillatory, possibly time dependent, random potential with Gaussian statistics. The Gaussian potential fluctuates in the spatial variables and possibly in the temporal variable. We seek the limit of the solution to the parabolic equation as the scale ε at which the random medium oscillates converges to zero. Depend...

Stochastic Partial Differential Equations

On the convergence of the homotopy analysis method to solve the system of partial differential equations

One of the efficient and powerful schemes to solve linear and nonlinear equations is homotopy analysis method (HAM). In this work, we obtain the approximate solution of a system of partial differential equations (PDEs) by means of HAM. For this purpose, we develop the concept of HAM for a system of PDEs as a matrix form. Then, we prove the convergence theorem and apply the proposed method to fi...

Introduction to stochastic partial differential equations

We introduce the Hilbert space-valued Wiener process and the corresponding stochastic integral of Itô type. This is then used together with semigroup theory to obtain existence and uniqueness of weak solutions of linear and semilinear stochastic evolution problems in Hilbert space. Finally, this abstract theory is applied to the linear heat and wave equations driven by additive noise.

Homogenized Dynamics of Stochastic Partial Differential Equations with Dynamical Boundary Conditions

A microscopic heterogeneous system under random influence is considered. The randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium. This system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes (obstacles or heterogeneities), together with random dynamical boundary condi...