Sharif Digital Repository

Stochastic Partial Differential Equations have many applications in other area of science. In this thesis we investigate two pproaches in SPDE.The first approach is semigroup and the second is variational method.Our main purpose is stability of these equations  

n this thesis we have investigated stochastic evolution equations by variational method. For these equations, explicit and implicit numerical schemes are presented. We have performed these numerical schemes for stochastic heat equation. We have investigated 2-D Navier-Stokes equation too  

In this thesis, we consider an implicit approximation scheme for the stochastic heat equation with additive and multiplicative space-time white noise. we use the spectral Galerkin method in space combined with the linear implicit Euler method in time to simulate weak approximation error  

We study linear and semilinear stochastic evolution partial differential equations driven by additive noise. We present a general and flexible framework for representing the infinite dimensional Wiener process, which drives the equation. The equation is discretized in space by a standard piecewise linear finite element method. We show how to obtain error estimates when the truncated expansion is used in the equation. We show that the orthogonal expansion of the finite-dimensional Wiener process, that appears in the discretized problem, can be truncated severely without losing theasymptotic order of the method, provided that the kernel of the covariance operator of the Wiener process is... 

We obtain a large deviation principle describing the small time asymptotics of the solution of a stochastic evolution equation with multiplicative noise. Our assumptions are a condition on the linear drift operator that is satisfied by generators of analytic semigroups and Lipschitz continuity of the nonlinear coefficient functions. Methods originally used by Peszat.
For the small noise asymptotics problem are adapted to solve the small time asymptotics problem. The results obtained in this way improve on some results of Zhang  

We establish the existence and uniqueness of the mild solution for stochastic Volterra equation with a non-self-adjoint operator. The specific Volterra equation that we consider is a generalization of the fractional differential equation. To obtain the mild solution for the case of multiplicative problem, the resolvent property of the linear perturbation of a sectorial operator will be considered. Moreover, we establish the existence and uniqueness of the mild solution for semilinear stochastic Volterra equation involving a demicontinuous and semimonotone nonlinearity. The Volterra equation in this case, has a positive-type memory kernel. To obtain the mild solution of the multiplicative... 

Semilinear stochastic evolution equations with multiplicative Lévy noise and monotone nonlinear drift are considered. A novel method of proof for establishing existence and uniqueness of the mild solution is proposed. We also prove the continuous dependence of the mild solution with respect to initial conditions and also on coefficients. As corollaries of the continuity result, we derive sufficient conditions for asymptotic stability of the solutions, we show that Yosida approximations converge to the solution and we prove that solutions have Markov property. Examples on stochastic partial differential equations and stochastic delay differential equations are provided to demonstrate the... 

In this thesis we study sharp spatial and temporal mean-square regularity results for a class of semi-linear parabolic stochastic partial differential equations (SPDEs) driven by infinite dimensional fractional Brownian motion with the Hurst parameter greater than one-half. In addition, the mean-square numerical approximations of such problems are investigated, performed by the spectral Galerkin method in space and the linear implicit Euler method in time. We see that by using the obtained sharp regularity properties of the problems one can identify optimal mean-square convergence rates of the full discrete scheme. At the end, these theoretical findings are accompanied by several numerical... 

In this thesis we study Galerkin methods for semilinear stochastic partial differential equations (SPDEs) with multiplicative noise and Lipschitz continuous nonlinearities. The strong error of convergence for spatially semidiscrete approximations as well as a spatio-temporal discretization which is based on a linear implicit Euler–Maruyama method, are also investigated. We see that the obtained error estimates in both cases as well as the regularity results for the mild solution of the SPDE are optimal. The results hold for different Galerkin methods such as the standard finite element method or spectral Galerkin. At the end, these theoretical findings are accompanied by several numerical...