GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable
ACM Transactions on Mathematical Software (TOMS)
An introduction to commutative and noncommutative Gro¨bner bases
Selected papers of the second international colloquium on Words, languages and combinatorics
The Definition of Standard ML
Groebner Bases for Non-Commutative Polynomial Rings
AAECC-3 Proceedings of the 3rd International Conference on Algebraic Algorithms and Error-Correcting Codes
Journal of Symbolic Computation
Integro-differential polynomials and operators
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
A skew polynomial approach to integro-differential operators
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
A Symbolic Framework for Operations on Linear Boundary Problems
CASC '09 Proceedings of the 11th International Workshop on Computer Algebra in Scientific Computing
A new symbolic method for solving linear two-point boundary value problems on the level of operators
Journal of Symbolic Computation
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In this paper, we summarize our recent work on establishing, for the first time, an algorithm for the symbolic solution of linear boundary problems. We put our work in the frame of Wen-Tsun Wu's approach to algorithmic problem solving in analysis, geometry, and logic by mapping the significant aspects of the underlying domains into algebra. We briefly compare this with the lines of thought of Wolfgang Groebner. For building up the necessary tower of domains in a generic and flexible way, we use the machinery of algorithmic functors introduced in our Theorema project. The essence of this concept is explained in the first section of the paper. The main part of the paper then describes our symbolic analysis approach to linear boundary problems, which hinges on three basic principles: (1) Differentiation as well as integration is treated axiomatically, setting up an algebraic data structure that can encode the problem statement (differential equation and boundary conditions) and suitable symbolic expressions for their solution (Green's operators qua integral operators). (2) Abstract boundary problems are introduced as pairs consisting of an epimorphism on a vector space (abstract differential operator) and a subspace of its dual (abstract boundary conditions). (3) Operator algebras are treated by noncommutative polynomials, modulo Groebner bases for certain relation ideals.