We present an α-regularization of the Birkhoff-Rott equation, induced by the two-dimensional Euler-α equations, for the vortex sheet dynamics. We show the convergence of the solutions of Euler-α equations to a weak solution of the Euler equations for initial vorticity being a finite Radon measure of fixed sign, which includes the vortex sheets case. We also show that, provided the initial densi...

We develop in this paper a new framework for discrete calculus of variations when the actions have densities involving an arbitrary discretization operator. We deduce the discrete Euler-Lagrange equations for piecewise continuous critical points of sampled actions. Then we characterize the discretization operators such that, for all quadratic lagrangian, the discrete Euler-Lagrange equations co...

The main aim of this paper is to investigate the exponential stability of the Euler method for a stochastic age-dependent population equations with Poisson random measures. It is proved that the Euler scheme is exponentially stable in mean square sense. An example is given for illustration. Keywords—Stochastic age-dependent population equations, Poisson random measures, Numerical solutions, Exp...

Homotopy Analysis Method (HAM) is applied to find the critical buckling load of the Euler columns with continuous elastic restraints. HAM has been successfully applied to many linear and nonlinear, ordinary and partial, differential equations, integral equations, and difference equations. In this study, we presented the application of HAM to the critical buckling loads for Euler columns with fi...

We study a distance between shapes defined by minimizing the integral of kinetic energy along transport paths constrained to measures with characteristic-function densities. The formal geodesic equations for this shape distance are Euler equations for incompressible, inviscid potential flow of fluid with zero pressure and surface tension on the free boundary. The minimization problem exhibits a...

We present an α-regularization of the Birkhoff-Rott equation, induced by the two-dimensional Euler-α equations, for the vortex sheet dynamics. We show the convergence of the solutions of Euler-α equations to a weak solution of the Euler equations for initial vorticity being a finite Radon measure of fixed sign, which includes the vortex sheets case. We also show that, provided the initial densi...

We study the dynamics of the generalized Euler equations on Virasoro groups D̂(S1) with different Sobolev H metric (k ≥ 2) on the Virasoro algebra. We first prove that the solutions to generalized Euler equations will not blow up in finite time and then study the stability of the trivial solutions.

The main aim of this paper is to investigate the exponential stability of the Euler method for a stochastic age-dependent population equations with Poisson random measures. It is proved that the Euler scheme is exponentially stable in mean square sense. An example is given for illustration. Keywords—Stochastic age-dependent population equations, Poisson random measures, Numerical solutions, Exp...

In this paper we describe the classical methods used to solve the Euler equations. Special attention is devoted to the iterative method based on a contraction mapping derived from these equations in Maldonado and Moreira (2003). We test the numerical robustness of this method when it is used in models with sensitiveness to initial conditions. Finally we extend the method to the case of stochast...

Implicit Euler approximations of the equations governing the porous flow of two imiscible incompressible fluids are shown to be the Euler–Lagrange equations of a convex function. Tools from convex analysis are then used to develop robust fully discrete algorithms for their numerical approximation. Existence and uniqueness of solutions control volume approximations are established.