Suppose I have a set of real valued polynomial functions:
$p_1(x)=\sum_{i=1}^la_ix^i,\;p_2(x)=\sum_{i=1}^mb_ix^i,\;p_3(x)=\sum_{i=1}^nc_ix^i$.
Are parabolic PDEs of the form $\frac{\partial u}{\partial t}+p_1(x)\frac{\partial u}{\partial x}+p_2(x)\frac{\partial^2 u}{\partial x^2}+p_3(x)=0$ exactly solvable?
If so, what is the method? If not, are there efficient numerical methods to solve the above?