Dirichlet problem on the half-line for a forced Burgers equation with time-variable coefficients and exactly solvable models

Abstract

We consider a forced Burgers equation with time-variable coefficients and solve the initial-boundary value problem on the half-line 0 < x < infinity with inhomogeneous Dirichlet boundary condition imposed at x = 0. Solution of this problem is obtained in terms of a corresponding second order ordinary differential equation and a second kind singular Volterra type integral equation. As an application of the general results, we introduce three different Burgers type models with specific damping, diffusion and forcing coefficients and construct classes of exactly solvable models. The Burgers problems with smooth time-dependent boundary data and an initial profile with pole type singularity have exact solutions with moving singularity. For each model we provide the solutions explicitly and describe the dynamical properties of the singularities depending on the time-variable coefficients and the given initial and boundary data. (C) 2019 Elsevier B.V. All rights reserved.