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

In this work, the transient analytic solution is found for the initial-boundary-value Burgers equation ut=uxx+u22x 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=2θxθ-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=2θxθ, 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)=F1(t) and u(L,t)=F2(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.