Solving linear boundary value problems via non-commutative Groebner bases

A new approach for symbolically solving linear boundary

value problems is presented. Rather than using general-purpose tools

for obtaining parametrized solutions of the underlying ODE and fitting

them against the specified boundary conditions (which may be quite

expensive), the problem is interpreted as an operator inversion

problem in a suitable Banach space setting. Using the concept of the

oblique Moore-Penrose inverse, it is possible to transform the

inversion problem into a system of operator equations that can be

attacked by virtue of non-commutative Groebner bases. The resulting

operator solution can be represented as an integral operator having

the classical Green's function as its kernel. Although, at this stage

of research, we cannot yet give an algorithmic formulation of the

method and its domain of admissible inputs, we do believe that it has

promising perspectives of automation and generalization; some of these

perspectives are discussed.