A new transformation of Burger’s equation for an exact solution in a bounded region necessary for certain boundary conditions

Abstract

In this work, the transient analytic solution is found for the initial-boundary-value Burgers equation u t = u xx + u 2 2 x in 0 ⩽ x ⩽ L . The boundary conditions are a homogeneous Dirichlet condition at x = 0 and a constant total flux at x = L . The technique used consists of applying the transformation u = 2 θ x θ - 1 that reduces Burgers equation to a linear diffusion–advection equation. Previous work on this equation in a bounded region has only applied the Cole–Hopf transformation u = 2 θ x θ , which transforms Burgers equation to the linear diffusion equation. The Cole–Hopf transformation can only solve Burgers equation with constant Dirichlet boundary conditions, or time-dependent Dirichlet boundary conditions of the form u ( 0 , t ) = F 1 ( t ) and u ( L , t ) = F 2 ( t ) , 0 ⩽ x ⩽ L . In this work, it is shown that the Cole–Hopf transformation will not solve Burgers equation in a bounded region with the boundary conditions dealt with in this work.