On Solutions For Higher-Order Partial Differential Equations

It is generally recognized that the method of separation of variables is one of the most universal and powerfu l technique for the study of linear PDE's. Although the technique can be applied in its traditional form to any linear PDE as long as mixed derivatives are absent, it is a common belief that in their presence, variables separation is not realizab le. In what follows, a modified and simpler procedure of the corresponding generalized version of the mentioned method is presented. The effectiveness of the introduced approach, capable to explicit ly solve linear higher-order PDE's incorporating mixed derivatives, is shown providing closed form solutions for, e.g. the nn-metaharmonic equation, as well as a part icular case of the infin ity Lap lace equation.