A New Symbolic Method for Linear Boundary Value Problems Using Groebner Bases
چکیده
Boundary value problems are of utmost importance for science and engineering. In fact, most differential equations come along with boundary conditions of some sort. It is therefore surprising that such problems—even in the linear case—have gained little attention in Symbolic Computation. Consequently, their coverage in computer algebra systems is rather unsystematic and unpredictable. The proper consideration of boundary conditions leads to a substantial revision of the algebraic structures currently used in established symbolic methods like differential algebra or differential Galois theory. One important ingredient in an algebraic approach to boundary value problems is the interaction of differential, integral and boundary operators. We present one such approach, based on Buchberger’s powerful concept of Groebner bases. For the implementation of the method we use the functor concept introduced by Buchberger for the Theorema system. This allows for easy adjustment of the code to various coefficient domains and different representations of the underlying objects. Holonomic functions revisited Toshinori Oaku Tokyo Woman’s Christian University 2-6-1, Zempukuji, Suginami-ku, Tokyo, 167-8585 Japan oaku@lab.twcu.ac.jp