# Computer Algebra for Linear Boundary Problems

Lead Research Organisation: University of Kent
Department Name: Sch of Maths Statistics & Actuarial Scie

### Abstract

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.

### Planned Impact

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.

### ORCID iD

Markus G Rosenkranz (Principal Investigator)

### Publications

10 25 50

Description Boundary problems are an important means for modelling reality (physics, engineering, economics), allowing the transformation of problems from an application domain into the realm of mathematics - after solving it there, the solution can be mapped back to the applications.

In this project, we have treated a certain class of boundary problems by algebraic methods. This amounts to a kind of division of labour: While Analysis provides tools for probing e.g. existence and uniqueness of solutions, algebra can build up and investigate their structure - how they are built up from basic components.

In the class of boundary problems for so-called linear partial differential equations, we have built up a new language for describing such problems and their solution operators in an algebraic framework. Moreover, we have put together algorithms for decomposing larger boundary problems into smaller ones, especially in the so-called constant coefficients case. As far as we are aware, this is the first time this has been achieved on firm algebraic foundations.

Dually, as the "other side of the coin", we have made careful studies of certain integral operators, which are the prototypical solution operators of linear boundary problems. The algebraic study of integral operators is still in its infancy, particularly in their relation to differential equations. We have achieved a foundational result that describes the prototypical algebraic model for this situation, which in algebra is known as the free object in the integro-differential category.
Exploitation Route The algebraic foundation for linear boundary problems and integral operators provides a flexible framework for studying various important special classes of boundary problems from an algebraic perspective. In the long run, this should lead to robust computer algebra packages in pertinent software packages.
Sectors Chemicals,Energy,Environment

Description By its nature, our foundational results will need some time to consolidate. As detailed in the Key Findings, we expect and want our results to find their way into pertinent computer algebra packages. This will be the content of another, more implementation-oriented project. Moreover, we have already started a collaboration with people from a mechanical engineering department of Tennessee Tech to investigate applications of our methods for boundary problems in mechanics. There will be a meeting in Linz/Austria in January 2015, hosted by Prof Buchberger, for kicking off a collaboration along these lines.
First Year Of Impact 2014
Impact Types Cultural,Societal

Description Applications of algebraic boundary problem techniques to mechanical engineering
Organisation Tennessee Technological University
Country United States
PI Contribution We have described, in some details, our algebraic methods for solving boundary problems. We plan to have a first detailed meeting in January 2015 in Linz/Austria, hosted by Prof. Buchberger.
Collaborator Contribution Our partner from Tennesse, Prof Liu Jane, will provide custom-tailored applications from Mechanical Engineering, to which we wan to apply our algebraic boundary problem methods.
Impact As detailed above, we will have our first meeting early 2015.
Start Year 2014

Description Parameter estimation in linear and nonlinear control theory via integro-differential algebra
Organisation The National Institute for Research in Computer Science and Control (INRIA)
Country France
Sector Public
PI Contribution Submission of an outline proposal for an ANR project with M. Rosenkranz being an external partner.
Collaborator Contribution Decision of acceptance or rejection to be expected in February 2015. If accepted a full proposal has to be submitted within the ensuing months.
Impact See above.
Start Year 2014

Description Parameter estimation in linear and nonlinear control theory via integro-differential algebra
Organisation University of Lille
Country France