Computer Algebra for Linear Boundary Problems

Boundary problems are arguably the most popular models for describing
nature and society. They combine a generic part (the differential
equation) with a specific part (the boundary conditions). The generic
part typically describes a law of nature or similar principle by
analysing how small changes in one quantity cause small changes in
another (measured by differential quotients). A typical example is
given by Fourier's law of heat transfer in a solid material, relating
small changes in temperature to small changes in heat energy. Since
differential equations apply uniformly to large classes of situations,
additional data is needed for mastering any particular situation, for
example heat transfer in an iron slab of a particular shape with a
particular control mechanism (like cooling) at its boundary. This is
why the specific part is called a boundary problem: It is typically
determined by the shape of a boundary and the values of certain
quantities of interest (like a uniformly cool temperature) on this
boundary.

Due to their enormous importance in applications, there is a rich
arsenal of computational methods for solving boundary
problems. However, virtually all of these methods are based on
numerical approximation. This is fully acceptable for the applied
scientist who is mostly interested in the numerical description (and
visualisation) of key quantities. For the mathematical fine analysis
it is often more advantageous to have an exact or symbolic
solution. Also parameter dependence can be studied most efficiently in
this manner.

Despite their great importance, boundary problems are rather neglected
in symbolic computation, both in theory and in practise. The deeper
reason for this is that the algebraic treatment of boundary problems
does not fit into the common frameworks (notably a branch called
differential algebra and differential Galois theory) used for the
symbolic treatment of differential equations. It is the aim of this
project to extend and generalise such algebraic frameworks to cover
boundary problems, specifically those for linear partial differential
equations.

Symbolic methods for boundary problems are not meant to compete with
numerical methods; in fact, class of the boundary problems amenable to
exact solutions is rather restricted. But symbolic methods are not
only good for solving boundary problems, they can also effect various
other operations. The ultimate goal of our research on boundary
problems is thus to achieve a tight interaction with numerical
methods: We want to analyse / manipulate / decompose boundary problems
symbolically and solve the atomic chunks numerically. In this project
we focus on the following symbolic operations: decompose higher-order
problems into lower-order ones, transform a given problem to a simpler
geometry, explore situations of solvable model problems, and combine
exact solutions.

Boundary problems form a large are of applied mathematics because
they are ubiquitous in physical, chemical, economic, and actuarial
models (an example of the latter type is exemplified in our earlier
research and should be developed further in the present project).
For example, the temperature distribution in a blast furnace for steel
production is described by a boundary problem that can be solved
by temperature measurements on its outside boundary.

In this project, we will acquire a better understanding of
certain important types of boundary problems. Our contribution is not
only of a theoretical nature but engages in actual algorithm
development and software production for treating boundary
problems. We distinguish two complementary aspects of (future)
software tools: While the numerical component is here for providing
approximate solutions to large real-world problems (and many programs
are on the market for achieving this), the symbolic component can give
exact solutions for simpler model problems and decompose them into
smaller instances. The advantage of symbolic expressions is
that they can be analysed without numerical uncertainties. Moreover,
they may often contain generic parameters---a single symbolic solution
can replace a whole series of numerical simulations.

Software support for treating boundary problems on a symbolic level is
virtually non-existent---in fact, there is also precious little on the
theoretical side. Having a combined symbolic-numerical tool, the
engineer of the future will be in a better position to make
predictions for the field he is investigating, e.g. optimising the
production of steel in a blast furnace. Since boundary problems are so
widespread in physics, chemistry, economics and actuarial mathematics,
we expect (in the long run) to enhance the efficiency,
performance and sustainability of businesses dealing in a wide
range of commodities.