Studieinfo emne PENG9570 2022 HØST
PENG9570 Applied Mathematical Modelling and Analysis Emneplan
- Engelsk emnenavn
- Applied Mathematical Modelling and Analysis
- Studieprogram
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PhD Programme in Engineering SciencePhD Programme in Engineering Science
- Omfang
- 10.0 stp.
- Studieår
- 2022/2023
- Pensum
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VÅR
2023
- Emnehistorikk
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Innledning
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.
Anbefalte forkunnskaper
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.
Forkunnskapskrav
None.
Læringsutbytte
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:
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knows how mathematical models can be derived from facts and first principles.
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has a repertoire of methods to solve and/or analyse both ordinary differential equation (ODE) systems and certain partial differential equations (PDEs).
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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.
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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:
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can determine steady states of ODE systems and use linear approximation to elucidate the stability properties of these states.
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can solve and/or analyse selected PDE models.
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is able to implement and use some numerical methods for solving relevant PDEs.
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can devise the solution of certain composite quantitative problems.
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can disseminate results and findings in an accessible manner – both orally and in writing.
General competence
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is aware of the usefulness and limitations of mathematical modelling as well as of pitfalls frequently encountered in modelling and simulation.
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is able to discuss properties of a system using the equations of the mathematical model that describes the system.
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can explain and use numerical methods, know their strengths and weaknesses and interpret results of numerical simulations.
Innhold
Introductory module:
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Principles of modelling and derivation of mathematical models
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Analysis of ordinary differential equations (ODEs)
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Linear partial differential equations (PDEs)
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Prominent results from functional analysis and their application to ODEs and PDEs
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Numerical methods for computing of solutions of PDEs
Functional analysis:
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Completeness for normed spaces
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Hilbert spaces, compact and diagonalisable operators
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Theory of topological vector spaces
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Test functions, distributions and the Fourier transform
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Sobolev spaces and fundamental solutions of partial differential equations
Biosystems:
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Mathematical models for biological systems
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Analytical and numerical methods for simulation of system response
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Actuators and sensors for stimulation and measurements of biological systems
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Interaction of biological and measurement system
Modern physics:
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Monte Carlo techniques
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Splines and other expansion techniques
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Applications of expansions in spherical harmonics
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Numerical problems in general relativity and quantum physics
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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.
Arbeids- og undervisningsformer
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.
Arbeidskrav og obligatoriske aktiviteter
The following required coursework must be approved before the student can take the exam:
- Completion of an extensive individual project in the specialised module.
Vurdering og eksamen
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.
Hjelpemidler ved eksamen
The student's own project.
Vurderingsuttrykk
Pass or fail.
Sensorordning
Two examiners. External examiner is used periodically.