On the Solutions of Viscous Burgers Equations

Abstract

The viscous Burgers equation ut + uux = νuxx, is a nonlinear partial differential equation, named after the great physicist Johannes Martinus Burgers (1895-1981). This equation can be linearized to the heat equation through ColeHopf transformation. First, we study asymptotic behavior of solutions to an initial value problem posed for heat equation. For which, we construct an approximate solution to the initial value problem in terms of derivatives of Gaussian by incorporating the moments of initial function. Spatial shifts are introduced into the leading order term as well as penultimate term of the approximation. We extend these results to observe asymptotic behavior of solutions to the viscous Burgers equation. Secondly, we deal with a forced Burgers equation (FBE) subject to the initial function, which is continuous and summable on R. Large time asymptotic behavior of solutions to the FBE is determined with precise error estimates. To achieve this, we construct solutions for the FBE with a different initial class of functions using the method of separation of variables and Cole-Hopf like transformation. These solutions are constructed in terms of Hermite polynomials with the help of similarity variables. The constructed solutions would help us to pick up an asymptotic approximation and to show that the magnitude of the difference function of the true and approximate solutions decays algebraically to zero for large time.