Recently, I try to solve some PDE through numerical way. Then, I found that it was hard to judge whether the method performed good or bad because in many cases, it seems to be hard to obtain a special solution. For example, the Burgers equation:
\begin{equation}
u\_t + uu\_x = \delta u\_{xx},\ \ \delta\ \mbox{is constant.}
\end{equation}

How can I obtain a special solution for this equation?

And, I wonder that if there is a good way to find a special solution for any PDE easily.