The generalized Burgers equation with and without a time delay

Abstract

We consider the generalized Burgers equation with and without a time delay when the boundary conditions are periodic with period 2π. For the generalized Burgers equation without a time delay, that is, we show that the equation is exponentially stable under small delays. Using a pseudospectral method, we present some numerical results illustrating and reinforcing the analytical results.

Nonstandard finite difference method for solving complex-order fractional Burgers’ equations

“…Consider the Burgers’ equation in 1-D as follows ( [12] , [23] ): with the initial and the boundary conditions given as follows: where, and are constants, is the variation term, is the velocity component, is diffusion coefficient, is the nonlinear convective term and is the diffusion term, and are known functions. is the initial time.…”

“…Burgers’ equations can describe the communication between acoustic waves, reaction apparatuses, convection effects, heat conduction, diffusion transports, and modeling of dynamics, for more details see [11] , [12] , [13] , [14] , [16] , [17] . Several authors have investigated studied Burgers’ model for various physical flow problem in fluid dynamics.…”

Fitted Galerkin spectral method to solve delay partial differential equations

“…On the other hand, the application of GSM for solving DPDEs receives less attention. Up to our knowledge, there are a few works in this direction, see for example , in which the authors investigated the dynamical behavior of the generalized Burgers equation with (also without) a time delay.…”

“…On the other hand, the application of GSM for solving DPDEs receives less attention. Up to our knowledge, there are a few works in this direction, see for example [14], in which the authors investigated the dynamical behavior of the generalized Burgers equation with (also without) a time delay.The Galerkin method refers to a general strategy for solving nonlinear partial differential equations. The idea is to expand the solution u.t, x/ in a linear combination of a product of unknown functions in time (time-dependent coefficients) with known basis functions over infinite dimensional space (spatial components).…”

Preliminary group classification for generalized inviscid Burger's equation

“…which is a prototype for equations for which the solution can develop discontinuities (shock waves). The generalized non-homogeneous inviscid Burgers equation had introduced in [12] has the form…”