Boundary Problems via Symbolic Integral Operators

This project is concerned with Symbolic Computation for boundary problems via computable algebras of integral operators. More specifically, we will address the following goals (dealing with linear boundary problems / integral operators except the fourth): (1) Identifying computable classes of boundary problems / integral operators (2) Operations on boundary problems and representation of solutions (3) Factorization of boundary problems into lower-order problems (4) Algebraic aspects of linear and nonlinear integral operators (5) Prototype implementation of key algorithms (6) Combination with numerical methods (7) Applications in actuarial mathematics Despite their obvious significance in applications, boundary problems-in contrast to differential equations per se- have as yet received little attention in Symbolic Computation. Nevertheless the treatment of boundary conditions is important since they come naturally in many applications, and they are crucial for constraining the solution space for partial differential equations. We apply methods from computer algebra for representing and manipulating boundary problems and integral operators: Following a novel approach developed by the authors, integral operators are represented either explicitly by a certain type of noncommutative polynomials or implicitly by the boundary problems they solve. This principle of dual representation enhances the usual constructive approaches to operator algebras; it is particularly helpful for various algebraic problems like composition and factorization of boundary problems. For the key algorithms to be developed in this project, we plan to create prototype implementations in a computer algebra system. The overall aim is to advance the exact and versatile treatment of boundary problems in Symbolic Computation.