New symbolic computation methods for the exact solution of two-point boundary value problems

A linear two-point boundary value problem is given by a linear ordinary differential equation together with a set of linear boundary conditions set up at the two boundary points of a finite real interval. The given differential equation is inhomogeneous, and its right-hand side is regarded as a symbolic parameter, so solving it can be seen as finding a linear operator (the so-called “Green’s operator") that maps the symbolic parameter to the symbolic solution of the boundary value problem (assuming existence and uniqueness). Up to now, boundary value problems have not received much attention in symbolic computation since they may be subsumed under the heading of differential equations. However, standard methods from there will often not be effective due to the presence of the symbolic parameter. Moreover, it seems more natural to work directly on the level of operators since boundary value problems are, as described above, operator problems in disguise. We present a new solution approach that takes these issues into account and sheds some light on the essence of integral operators in symbolic computation.