In this survey we recall the results obtained in [16] where we gave a representation theorem for the solutions of stochastic di¤erential equations in Hilbert spaces. Using this representation theorem and the deterministic characterizations of exponential stability and uniform observability obtained in [16], [17], we will prove a result of Datko type concerning the exponential dichotomy of stoch...

This report discusses a mathematical software package DKLAG5 which consists of a suite of FORTRAN subroutines for the numerical solution of systems of functional di erential equations with state dependent delays. The package implements continuously imbedded Runge{ Kutta methods which are based on C polynomial interpolants. These interpolants are exploited by the software to handle the necessary...

We address the enumeration of q-coloured planar maps counted by the number of edges and the number of monochromatic edges. We prove that the associated generating function is di erentially algebraic, that is, satis es a non-trivial polynomial di erential equation with respect to the edge variable. We give explicitly a di erential system that characterizes this series. We then prove a similar re...

We consider the construction of geometric integrators in the class of RKMK methods. Any di erential equation in the form of an in nitesimal generator on a homogeneous space is shown to be locally equivalent to a di erential equation on the Lie algebra corresponding to the Lie group acting on the homogenous space. This way we obtain a distinction between the coordinate-free phrasing of the di er...

We de ne a discrete (integer-valued) Lyapunov function V for cyclic nearest neighbor systems of di erential delay equations possessing a feedback condition. This extends analogous de nitions for cyclic systems of ODE's, and for scalar di erential delay equations. We relate the values of V to the real parts of the Floquet multipliers for such linear periodic systems, and thereby prove all Floque...

We give a short introduction to the stochastic calculus for Itô-Lévy processes, and review brie‡y the two main methods of optimal control of stochastic systems described by such processes, namely: (i) Dynamic programming and the Hamilton-Jacobi-Bellman (HJB) equation (ii) The stochastic maximum principle and its associated adjoint backward stochastic di¤erential equation (BSDE). The two methods...

This paper presents a comparison between variational iteration method (VIM) and modfied variational iteration method (MVIM) for approximate solution a system of Volterra integral equation of the first kind. We convert a system of Volterra integral equations to a system of Volterra integro-di®erential equations that use VIM and MVIM to approximate solution of this system and hence obtain an appr...

In this study we analyze a non-overlapping domain decomposition method for the solution of elliptic Partial Di erential Equation (PDE) problems. This domain decomposition method involves the solution of Dirichlet and Neumann PDE problems on each subdomain, coupled with smoothing operations on the interfaces of the subdomains. The convergence analysis of the method at the di erential equation le...

Prediction and ltering of continuous-time stochastic processes require a solver of a continuous-time di erential Lyapunov equation (cdle). Even though this can be recast into an ordinary di erential equation (ode), where standard solvers can be applied, the dominating approach in Kalman lter applications is to discretize the system and then apply the discrete-time di erence Lyapunov equation (d...

Let us consider the di erential equation