An Averaged-Separation of Variable Solution to the Burger Equation

The Burger Partial Differential Equation (PDE) provides a nonlinear model that incorporates several of the important properties of fluid behavior. However, no general solution to it is known for given arbitrary initial and/or boundary conditions. We propose a ``new'' method for determining approximations for the solutions. Our method combines the separation of variables technique, combined with an averaging over the space variable. A test of this procedure is made for the following problem, where u = u(x,t): \begin{center} 0 $\le $ x $\le $ 1, t $>$ 0, \end{center} \begin{center} u(0,t) = 0, u(1,t) = 0, \end{center} \begin{center} u(x,0) = x(1-x), \end{center} \begin{center} u$_{t}$ + uu$_{x}$ = Du$_{xx}$, \end{center} where D is a non-negative parameter. The validity of the calculated solution is made by comparing it to an exact analytic solution, as well as an accurate numerical solution for the special case where D = 0.