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 + (u2 / 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 = 2θx / 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 = 2θx / 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) = F1(t) and u(L, t) = F2(t), 0 ≤ x ≤ L. 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.