# 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.

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.

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.

## People |
## ORCID iD |

Markus G Rosenkranz (Principal Investigator) |

### Publications

Gao X
(2015)

*Free integro-differential algebras and Gröbner-Shirshov bases*in Journal of Algebra
GAO X
(2014)

*CONSTRUCTION OF FREE COMMUTATIVE INTEGRO-DIFFERENTIAL ALGEBRAS BY THE METHOD OF GRÖBNER-SHIRSHOV BASES*in Journal of Algebra and Its Applications
Guo L
(2014)

*On integro-differential algebras*in Journal of Pure and Applied Algebra
Rosenkranz M
(2013)

*A Noncommutative Algebraic Operational Calculus for Boundary Problems*in Mathematics in Computer Science
Zheng S
(2019)

*Classification of Rota-Baxter operators on semigroup algebras of order two and three.*in Communications in algebraDescription | 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 |

Sector | Academic/University |

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 |

Sector | Academic/University |

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 | École Centrale de Lille |

Country | France |

Sector | Academic/University |

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 |