Section: New Results

Boundary value problems for linear ordinary integro-differential equations

Participants : Alban Quadrat, Georg Regensburger.

In [61] , we study algorithmic aspects of linear ordinary integro-differential operators with polynomial coefficients. Even though this algebra is not noetherian and has zero divisors, Bavula recently proved in [81] that it is coherent, which allows one to develop an algebraic systems theory. For an algorithmic approach to linear systems theory of integro-differential equations with boundary conditions, computing the kernel of matrices is a fundamental task. As a first step, we have to find annihilators, which is, in turn, related to polynomial solutions. We present an algorithmic approach for computing polynomial solutions and the index for a class of linear operators including integro-differential operators. A generating set for right annihilators can be constructed in terms of such polynomial solutions. For initial value problems, an involution of the algebra of integro-differential operators also allows us to compute left annihilators, which can be interpreted as compatibility conditions of integro-differential equations with boundary conditions. These results are implemented in Maple based on the IntDiffOp and IntDiffOperations packages. Finally, system-theoretic interpretations of these results are given and illustrated on integro-differential equations.

In [78] , we develop linear algebra results needed for generalizing the composition of boundary problems to singular ones. We consider generalized inverses of linear operators and study the question when their product in reverse order is again a generalized inverse. This problem has been studied for various kinds of generalized inverses, especially for matrices. Motivated by our application to boundary problems, we use implicit representation of subspaces via “boundary conditions” from the dual space and this approach gives a new representation of the product of generalized inverses. Our results apply to arbitrary vector spaces and for Fredholm operators, the corresponding computations reduce to finite-dimensional problems, which is crucial for our implementation for boundary problem for linear ordinary differential equations.

In collaboration with Li Guo and Markus Rosenkranz [77] , we study algebraic aspects of integro-differential algebras and their relation to so-called differential Rota-Baxter algebras. We generalize this concept to that of integro-differential algebras with weight. Based on free commutative Rota-Baxter algebras, we investigate the construction of free integro-differential algebras with weight generated by a regular differential algebra. The explicit construction is not only interesting from an algebraic point of view but is also an important step for algorithmic extensions of differential algebras to integro-differential algebras (compare with the related construction and the implementation of integro-differential polynomials in [72] ). In this paper, we review also the construction of integro-differential operators, the algorithms for regular boundary problems and a prototype implementation in the Theorema system.

In [11] , we adapt our factorization technique for boundary problems to study ruin probabilities and related quantities in renewal risk theory. The analysis is based on boundary problems for linear ordinary differential equations (on the half bounded interval from zero to infinity) with variable coefficients and the corresponding factorization of Green's operators. With this approach, we obtain closed-form and asymptotic expressions for discounted penalty functions under the more realistic assumption that the premium income depends on the present surplus of the insurance portfolio.