Partial Differential Equations

The study of linear partial differential equations is in some respects no different than that of matrices or ordinary differential equations. Symbolically, if L is a partial differential operator, to solve Lu = f we must find the operator L ⁻¹ which inverts the problem to u = L ⁻¹ f The operator L ⁻¹ is the Green's operator. The main difficulty, however, is that the construction of this Green's operator is an order of magnitude harder than before. If we have learned anything from all that precedes this section, it is that we want some way to reduce these equations to simpler ones that we already know how to solve. That is, after all, the whole point of transform theory.