The main goal of this note is to introduce another second-order differenceequation where every nontrivial solution is of minimal period 5, namelythe difference equation:xn+1 =1 + xn−1xnxn−1 − 1, n = 1, 2, 3, . . .with initial conditions x0 and x1 any real numbers such that x0x1 6= 1.

We put a direct new method to construct the rational Jacobi elliptic solutions for nonlinear differential difference equations which may be called the rational Jacobi elliptic functions method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential difference equations in mathematical physics via the lattice equation and the dis...

Accurate, reliable, efficient, and robust simulation of groundwater flow in the unsaturated zone for the problems that characterized by sharp fronts in both space and time is computationally expensive. The accurate numerical solution of these problems by standard approaches with uniform spatial and temporal discretization usually inefficient and simulation is too costly. Moreover, it is very di...

In this paper, a new implicit nonstandard finite difference scheme for conservation laws, which preserving the property of TVD (total variation diminishing) of the solution, is proposed. This scheme is derived by using nonlocal approximation for nonlinear terms of partial differential equation. Schemes preserving the essential physical property of TVD are of great importance in practice. Such s...

A Miura chain is a (closed) sequence of differential (or difference) equations that are related by Miura or Bäcklund transformations. We describe such a chain for the sixth Painlevé equation (PVI), containing, apart from PVI itself, a Schwarzian version as well as a second-order second-degree ordinary differential equation (ODE). As a byproduct we derive an auto-Bäcklund transformation, relatin...

The fully nonlinear and weakly dispersive Green-Naghdi equations for shallow water waves of large amplitude is studied. An hybrid finite volume and finite difference splitting approach is proposed. Numerical validations are then performed in one horizontal dimension.

A method is proposed for analysis of arbitrarily loaded lossy and dispersive nonuniform single or coupled transmission lines. In this method, the transmission lines are subdivided to several uniform sections, at first. Then the voltage and current distributions are obtained using second order step-by-step numerical integration (second order finite difference method). The accuracy of the method ...

In this paper the buckling coefficient of FGM rectangular plates calculated by the new version of differential quadrature method (DQM). At the first the governing differential equation for plate has been calculated and then according to the new version of differential quadrature method (DQM) the existence derivatives in equation , convert to the amounts of function in the grid points inside of ...

We investigate L → L∞ dispersive estimates for the three dimensional Dirac equation with a potential. We also classify the structure of obstructions at the thresholds of the essential spectrum as being composed of a two dimensional space of resonances and finitely many eigenfunctions. We show that, as in the case of the Schrödinger evolution, the presence of a threshold obstruction generically ...