PENG9570 Anvendt matematisk modellering og analyse Emneplan

The course Applied Mathematical Modelling and Analysis focuses on the modelling, derivation, and analysis of mathematical models. The course is organised into an introductory module covering relevant models and methods and a specialization module, where the student works on a project relevant to the syllabus of the course.

The course covers the following topics:

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

Specialization module:

• Project work where the project topic may be adjusted to accommodate for the research area of each PhD candidate.

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.

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.

None.

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 diffusion-convection and discuss the relevance of these models to real-world 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 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

On successful completion of the course, the student:

• 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.

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.

The following required coursework must be approved before the student can take the exam:

• Completion of an extensive individual project in the specialised module.

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.

The student's own project.

Pass or fail.

Two examiners. External examiner is used periodically.

Leiv Øyehaug.