In this paper, we consider the following linear Korteweg–de Vries–Benjamin Bona Mahony (KdV-BBM) equation on a finite interval.$ \left\{ \begin{array}{ll} u_t - a^2u_{xxt} + u_x u_{xxx} = 0, & (x, t)\in(0, L)\times (0, \infty), \\ u(0, t) u(L, u_x(L, &t\in u(x, 0) u_0(x), x \in L). \end{array} \right. $We show well-posedness by semigroup theory. A set of critical length $ \mathcal{L} is...

We consider an evolution equation similar to that introduced by Vese in [12] and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time.

Let b and c be fixed coprime odd positive integers with min{b, c} > 1. In this paper, a classification of all positive integer solutions (x, y, z) of the equation 2 (x) + b (y) = c (z) is given. Further, by an elementary approach, we prove that if c = b + 2, then the equation has only the positive integer solution (x, y, z) = (1,1, 1), except for (b, x, y, z) = (89,13,1, 2) and (2 (r) - 1, r + ...

This article presents and analyzes an approximated exponential integrator for the (inhomogeneous) stochastic Manakov system. system of SPDE occurs in study pulse propagation randomly birefringent optical fibers. For a globally Lipschitz-continuous nonlinearity, we prove that strong order time is $ 1/2 $. then used to has convergence probability almost sure 1/2^{-} $, case cubic nonlinear coupli...

‎we study the existence of soliton solutions for a class of‎ ‎quasilinear elliptic equation in $mathbb{textbf{r}}^2$ with critical exponential growth‎. ‎this model has been proposed in the self-channeling of a‎ ‎high-power ultra short laser in matter‎.

‎‎In this paper, Solvability nonlinear Volterra integral equations with general vanishing delays is stated. So far sinc methods for approximating the solutions of Volterra integral equations have received considerable attention mainly due to their high accuracy. These approximations converge rapidly to the exact solutions as number sinc points increases. Here the numerical solution of nonlinear...

We study retarded parabolic non{autonomous evolution equations whose coeecients converge as t ! 1 such that the autonomous problem in the limit has an exponential dichotomy. Then the non{autonomous problem inherits the exponential dichotomy and the solution of the inhomogeneous equation tends to the stationary solution at innnity. We use a generalized characteristic equation to deduce the expon...

The aim of this paper is to present a new numerical method for solving the Bagley-Torvik equation. This equation has an important role in fractional calculus. The fractional derivatives are described based on the Caputo sense. Some properties of the sinc functions required for our subsequent development are given and are utilized to reduce the computation of solution of the Bagley-Torvik equati...