Application of the method of additional conditions to the Neumann problem for the Laplace equation in the wedge

• Authors: Igor Neygebauer
• Venue: F-and-B'11 Proceedings of the 4th WSEAS international conference on Finite differences - finite elements - finite volumes - boundary elements
• Year: 2011

The method of additional conditions (MAC) gives a well characteristic of the finite stresses near the tip of a crack of the Griffth's problem in fracture mechanics, where is supposed of a finite elastic potential which entails the zero value of the integral. MAC uses special additional conditions to correct the given traditional problems. It could be considered as a method introducing a special variation to smooth the contradiction between the given problem and introduced additional condition. In this paper we apply this method to the probably most frequently occurring partial differential equation governing the behavior of certain physical quantities. The Laplace equation is considered in the wedge. For the given Neumann problem the derivative of the solution has a singularity at the origin for an obtuse angle. The solution is unique. The singularity can be considered as impossible from the physical point of view. There is suggested an additional condition which follows from the classical Green's formula and from the supposed to be bounded solution and its derivatives, and that can characterize the nonsingular solution at the origin. The first approximation is considered.