Self-consistent boundary conditions with blurred derivatives

It is shown in this paper that self-consistent boundary conditions for numerical methods based on blurred derivatives can be derived from a suitable change of variables of the fundamental blurred approximation of the differential equation, followed by application of Leibnitz theorem for differentiation of an integral. The simplest scheme obtained in this way resembles the weak Local Petrov-Galerkin approximation, although interpretation of the operators appearing in the final equations is quite different-as is the derivation itself. Subsequent transformation leads to integral equations similar to the starting point for boundary integral methods of solution. In this way, a number of well-known computational methods are shown to be derivable from adequate manipulation of the blurred derivative technique. However, other approximations, which are not derivable with standard methods can also be obtained, hinting at a greater generality of blurred derivatives.