PENG9570 Applied Mathematical Modelling and Analysis Course description

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.

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.

The point of departure for the course is the scientific and humanistic basis for well-functioning health care services and paramedical professional practice. The course will particularly focus on knowledge and skills that promote respect, empathy, reflection and relational competence. Practical training in interaction, basic infection control and lifesaving first aid are part of the course. The course also emphasises learning in an academic context, critical use of sources and written and oral presentation.

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:

• 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.

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.

After completing the course, the student should have the following learning outcomes defined in terms of knowledge, skills and general competence:

Knowledge

The student

• can describe the history and academic tradition of emergency medical treatment, and their own profession
• can explain the structure of the health service and the ambulance service’s place in the health service
• can refer to relevant laws, regulations and rules that regulate the professional practice of paramedics
• can explain the principles of documentation and use of patient care report forms
• can explain theories and ethical and communicative perspective that are relevant for professional practice
• can explain simulation and skills training as a method
• can describe basic models for decision-making
• can explain the basic principles behind lifesaving first aid
• can describe the basic theories for good hygiene

Skills

The student

• can apply basic hygiene and infection control techniques
• can communicate and cooperate with fellow students and others in teams/groups
• can administer lifesaving first aid
• can perform cardiopulmonary resuscitation with a semi-automatic defibrillator
• can document a clinical examination
• can write a reflection note based on observational practice
• can make use of protective gloves, wash hands correctly and disinfect hands

General competence

The student

• can reflect on relevant issues in prehospital work and on the student’s role in the health service
• can present their own work to an audience
• can reflect on ethical and communicative challenges in relation to professional practice
• can explain their own need for knowledge and learning and acquire new knowledge and skills

The work and teaching methods on the programme comprise lectures, seminars, group work in student groups of five to eight students, observational practice and simulation and skills training.

The students will take part in a two to three-day long observational practice in the ambulance service. The students also carry out simulation and skills training in lifesaving first aid.

The following must have been approved in order for the student to take the exam:

• Minimum attendance of 90% in observational practice placement
• Minimum attendance of 90% in simulation and skills training
• Minimum attendance of 80% at seminars and study groups
• One individual written reflection note from observational practice placement, 800 words.
• Individual practical test in lifesaving first aid

Oral exam in groups of 4-6 students.

The exam consists of a presentation (approx. 20-30 min.) and subsequent examination (approx. 10-20 min.) The students will be assigned a topic for their presentation one week before the exam.

All exams are assessed by an internal and an external examiner.

Five credits overlap with the course PARA1000Communication, Ethics and Culture.