Center of Ring Theory - Tutorial Lecture Series

This is the third Lecture in Bruno Buchberger's tutorial series for the Center of Ring Theory and its Applications.

Abstract: Boundary Value Problems are of utmost importance for science and engineering. In fact, in nature, problems describable by differential equations hardly occur without boundary conditions. However, surprisingly, even for the linear case, only ad hoc methods for the symbolic solution of such problems are known so far, see the packages provided in Mathematica and Maple. In the textbooks, the solution of linear boundary value problems is presented by Green's functions that are constructed by certain heuristics for the problems at hand. In the talk, we present a new method - based on the speaker's Groebner bases theory - for obtaining Green's functions in a uniform way for arbritrary linear boundary value problems. For the implementation of the method, we use the functor concept introduced by the speaker for the Theorema system. This allows easy adjustment of the code to various domains of coefficients and various representations of the objects in the abstract mathematical domains in which the theory lives. We conclude by giving an example of a problem that cannot be handled appropriately by the current methods in Mathematica and Maple but can be solved by our new method.