Solutions of the equation \documentclass{article}\pagestyle{empty}\begin{document}$$ \partial _t^n f(x,t) = \hat L(x,t)f(x,t) + S(x,t) $$\end{document}. Exact solutions of some partial differential equations in mechanics and physics

The solutions of the equation $ \partial _t^n f(x,t) = \hat L(x,t)f(x,t) + S(x,t) $, for L? a linear operator are derived. Different forms for L? whether it is time independent or time dependent and self-commutative (or not) at different times are considered separately. By using the results obtained, exact solutions of some partial differential equations are found for the first time.