In a recent interesting publication Subramanian and Mathew I'1] propose a "collocation by derivatives" approach as a technique suitable for obtaining approximate solutions to differential equations. They assume that all the boundary conditions are satisfied and provide examples dealing with the determination of the fundamental frequency of vibration of certain structural elements (beams and plates).
It is important to emphasize that in his now classical treatise on approximate methods, Collatz 1'2] specifically mentioned a "collocation with derivatives" technique when describing some error distribution principles.
In reference [3] (published almost 15 years ago in this Journal) Collatz' approach was modified by requiring "that the integral of the error function e and the integral of the first (p-1) derivatives of the error function over the domain .... be equal to zero". This leads to a system of p-algebraic equations in the unknowns ai which can be readily solved for the arbitrary parameters. In the case Of an eigenvalue problem the requirement generates a determinantal equation in the desired eigenvalues.
High accuracy was achieved in the results presented in reference . Admittedly the approach followed by Subramanian and Mathew [1] is simpler since no integration is required. Also, their work [1] constitutes probably the first numerical investigation dealing with a two dimensional problem in which a "collocation by derivatives" approach has been used.