We consider a system of ordinary di erential equations consisting of a singularly perturbed scalar di erential equation of second order and a scalar di erential equation of rst or second order and study a Neuman-Cauchy or a Neuman-Dirichlet problem. We assume that the degenerate equation has two intersecting solutions such that the standard theory for systems of Tichonov's type cannot be applie...

Solvability of forward–backward stochastic di erential equations with nonsmooth coe cients is considered using the Four-Step Scheme and some approximation arguments. For the onedimensional case, the existence of an adapted solution is established for the equation which allows the di usion in the forward equation to be degenerate. As a byproduct, we obtain the existence of a viscosity solution t...

in this paper, the convergence of zakharov-kuznetsov (zk) equation by homo- topy analysis method (ham) is investigated. a theorem is proved to guarantee the conver- gence of ham and to nd the series solution of this equation via a reliable algorithm.

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...

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 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...