Studyinfo subject PENG9570 2023 HØST
PENG9570 Applied Mathematical Modelling and Analysis Course description
 Course name in Norwegian
 Applied Mathematical Modelling and Analysis
 Study programme

PhD Programme in Engineering Science
 Weight
 10 ECTS
 Year of study
 2023/2024
 Curriculum

SPRING
2024
 Schedule
 Programme description
 Course history

Introduction
Students taking the course must have a thorough knowledge of advanced calculus, including ordinary and partial differential equations. The student should also be familiar with linear algebra and Fourier and Laplace transform theory. In terms of programming, the candidate should have some experience in implementing numerical methods, including schemes for solving partial differential equations.
The candidate should also have a certain knowledge of mathematical analysis, modern physics or physiology – depending on specialization.
The course will be offered once a year, provided 3 or more students sign up for the course. If less than 3 students sign up for a course, the course will be cancelled for that year.
Recommended preliminary courses
A thorough knowledge of advanced calculus, including ordinary and partial differential equations. It is a great advantage if students are familiar with linear algebra and Fourier and Laplace transform theory. In terms of programming, some experience in implementing various numerical methods, including schemes for solving partial differential equations is recommended. Some knowledge of mathematical analysis, modern physics or physiology is recommended, depending on their specialisation.
Required preliminary courses
None.
Learning outcomes
Students who complete the course are expected to have the following learning outcomes, defined in terms of knowledge, skills and general competence:
Knowledge
On successful completion of the course, the student:

knows how mathematical models can be derived from facts and first principles.

has a repertoire of methods to solve and/or analyse both ordinary differential equation (ODE) systems and certain partial differential equations (PDEs).

is able to apply analytical and/or numerical solution methods for PDEs to models of heat transfer, wave propagation and diffusionconvection and discuss the relevance of these models to realworld phenomena.

is able to construct and develop relevant models and discuss the validity of the models.
Skills
On successful completion of the course, the student can:

can determine steady states of ODE systems and use linear approximation to elucidate the stability properties of these states.

can solve and/or analyse selected PDE models.

is able to implement and use some numerical methods for solving relevant PDEs.

can devise the solution of certain composite quantitative problems.

can disseminate results and findings in an accessible manner – both orally and in writing.
General competence

is aware of the usefulness and limitations of mathematical modelling as well as of pitfalls frequently encountered in modelling and simulation.

is able to discuss properties of a system using the equations of the mathematical model that describes the system.

can explain and use numerical methods, know their strengths and weaknesses and interpret results of numerical simulations.
Content
Introductory module:

Principles of modelling and derivation of mathematical models

Analysis of ordinary differential equations (ODEs)

Linear partial differential equations (PDEs)

Prominent results from functional analysis and their application to ODEs and PDEs

Numerical methods for computing of solutions of PDEs
Functional analysis:

Completeness for normed spaces

Hilbert spaces, compact and diagonalisable operators

Theory of topological vector spaces

Test functions, distributions and the Fourier transform

Sobolev spaces and fundamental solutions of partial differential equations
Biosystems:

Mathematical models for biological systems

Analytical and numerical methods for simulation of system response

Actuators and sensors for stimulation and measurements of biological systems

Interaction of biological and measurement system
Modern physics:

Monte Carlo techniques

Splines and other expansion techniques

Applications of expansions in spherical harmonics

Numerical problems in general relativity and quantum physics

Manifolds with geometric structures central to physics and engineering.
Within all specializations, the content may be adjusted to accommodate for the research area of each PhD candidate.
Teaching and learning methods
The teaching is organised as sessions where the subject material is presented, and as sessions where the students solve problems using analytical and/or numerical methods. Between these sessions, the students should work individually with literature studies and problem solving.
In the last, specialised part, the students are required to complete and present a rather extensive individual project involving theoretical and practical/implementational aspects.
Course requirements
The following required coursework must be approved before the student can take the exam:
 Completion of an extensive individual project in the specialised module.
Assessment
An individual, oral examination. The examination will address both general topics from within the course and the specific project developed by the student.
The oral examination cannot be appealed.
Permitted exam materials and equipment
The student's own project.
Grading scale
Pass or fail.
Examiners
Two examiners. External examiner is used periodically.