We study the Cauchy problem for linear differential equation ∂u/∂t = Fu in module of formal generalized functions over a commutative ring, where F is operator finite or infinite order. prove well-posedness problem, find its fundamental solution, and obtain representation unique solution to form convolution initial condition. These results are specified heat equation, simplest transport one-dime...

If u denotes a local, spatial average of u, then u′ = u − u is the associated fluctuation. Consider a time relaxation term added to the usual finite element method. The simplest case for the model advection equation ut + −→a · ∇u = f(x, t) is (uh,t + −→a · ∇uh, vh) + χ(uh, v′ h) = (f(x, t), vh). We analyze the error in this and (more importantly) higher order extensions and show that the added ...

Material structures and chemical equilibrium are important learning units in high school chemistry. In this study, an augmented reality (AR) system is developed to assist students Students can use AR cards conduct virtual chemistry experiments, the submicroscopic view of a reaction will be displayed according equation specified by reactants coefficients on cards. They change observe experimenta...

We apply the method of simplest equation for obtaining exact solitary traveling-wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives. We consider first the general case of presence of monomials of the both (odd and even) grades and then turn to the two particular cases of nonlinear equations that contai...

We present a method for numerically building a vortex knot state in the superfluid wave function of a Bose-Einstein condensate. We integrate in time the governing Gross-Pitaevskii equation to determine evolution and shape preservation of the two (topologically) simplest vortex knots which can be wrapped over a torus. We find that the velocity of a vortex knot depends on the ratio of poloidal an...

Abstract. The weakly interacting trapped Bose gases have been customarily described using the mean-field approximation in the form of the Gross-Pitaevskii equation. The mean-field approximation, however, has certain limitations, in particular it can not describe correlations between particles. We introduce here an alternative variational approach, based on the correlated Gaussian method, which ...