Supported by the Austrian Science Fund (FWF), Project P 30052 (2017-2020)

Many of the problems usually considered in science and engineering - but also in economics and actuarial mathematics - find their adequate expression in the language of so-called boundary problems. One translates the process of interest into a differential equation along with additional boundary conditions; the former typically describes a law of nature or some model relation while the latter corresponds to measurements or adjustments that identify the process uniquely. There is a vast array of powerful numerical methods for solving boundary problems.

The numerical treatment necessarily involves approximations and fixing a single numerical instantiation out of an infinite manifold of possible ones. For investigating solutions algebraically - like studying dependence on parameters or possible decomposition schemes (so-called factorizations) or a suitable transformation to a simpler domain - one has to represent the problem as well as the solution in an algebraic manner, even if this needs simplification of the model (for example linearization, which in practical terms usually means limitation to "small" regions or oscillations).

In this project we would like to propose such algebraic methods that will allow us to directly represent, decompose and analyze the solution operators. Here these methods should be understood from a practical perspective, meaning as accessible computer programs that we shall provide in the frame of a suitable computer algebra package. In the long run we have the vision of an integrated workbench that will bring together numerical and algebraic methods with modern visualization techniques.

There will be two concrete applications that we want to study in this project: (1) A key problem in actuarial mathematics concerns the risk of accrued premium capital in the view of incoming insurance claims from the clients. For measuring the probability of ruin and other crucial stochastic parameters of the risk process one utilizes the so-called Gerber-Shiu function, which in turn can be characterized by a boundary problem. While we have investigated the latter with algebraic operator methods in a simplified model, leading to the great satisfaction of actuarial mathematicians, we hope to incorporate tax payments in a more accurate model that we will subject to the new methods of his project. (2) A typical problem in engineering mechanics consists in determining the distribution of stress in Kirchhoff plates, a certain model of elastic strain often applicable in technical materials. We have treated such a case under simplifying symmetry assumptions but would like to cover the full 2D model case with the tools to be developed in this project.

Research Institute for Symbolic Computation

Johannes Kepler University Linz

Altenbergerstrasse 69

A-4040 Linz, Austria

marcus@rosenkranz.or.at