ACIT4310 Applied and Computational Mathematics Emneplan

The course will provide the students with an understanding of what a mathematical model is and how we use models to gain insights into systems and processes in science and engineering. The course will train the students in using analytical and computational methods for analyzing and solving differential equations and prepare them for developing, analyzing and simulating mathematical models in their own projects. The models and methods taught in this course are generic and applicable not only in science, but also in various industrial contexts.

No formal requirements over and above the admission requirements.

A student who has completed this course should have the following learning outcomes defined in terms of knowledge, skills and general competence:

Knowledge

On successful completion of the course the student:

• knows the relevance of a selection of mathematical models to real-world phenomena
• has a thorough understanding of how mathematical modelling and scientific computing are utilized in various industrialized settings
• has a repertoire of methods to solve and/or analyze ordinary and partial differential equations (ODEs and PDEs)
• knows how to analyze the dynamics of an ODE system
• has a thorough understanding of the definitions of a smooth manifold and the tangent space
• knows the definitions and algebra of tensors and differential forms on a smooth manifold

Skills

On successful completion of this course the student:

• is able to derive mathematical models from facts and first principles for a selection of dynamical systems
• can apply mathematical modelling techniques on scenarios relevant to industry
• can implement mathematical models within the context of applied computer and information technology
• is able to analyse ODE systems and use bifurcation theory to elucidate the qualitative behavior of the systems
• is able to implement and use a selection of numerical methods for solving ODEs and PDEs
• is able to give examples of smooth manifolds and prove their smooth manifold property from the definition
• is able to use the geometric concepts and tools associated with smooth manifolds in the analysis of mathematical problems within mathematics, physics and engineering

General competence

On successful completion of this 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
• can explain and use numerical methods and interpret results of numerical simulations
• is aware of the role of smooth manifolds as one of the most fundamental concepts in mathematics and physics

• Approaches to scientific computing and implementation of mathematical models
• Principles of modelling and derivation of mathematical models
• Analysis, numerical solution and bifurcations of ODEs
• Numerical methods for computation of solutions of ODEs and PDEs

The course is organized as a series of lectures and seminars where the subject material is presented and discussed. Between these sessions the students should work with problem solving, implementation of numerical methods and model simulations. The last part of the semester students will work with a compulsory individual project supervised by the course lecturer. The project will involve studies and analyses of a mathematical model and a rather extensive implementation of the numerical solution of the model. This will result in a report that should be 2000 - 4000 words of length plus figures.

None

The exam consists of two parts:

1. An individual project report of about 2000 - 4000 words. The report counts 50% towards the final grade.

2. A 30 minute individual oral exam, which includes a 10 minute presentation of the candidates project. The oral exam counts 50% towards the final grade. Both exams must be passed in order to pass the course. The oral exam cannot be appealed.

New/postponed exam

In case of failed exam or legal absence, the student may apply for a new or postponed exam. New or postponed exams are offered within a reasonable time span following the regular exam. The student is responsible for applying for a new/postponed exam within the time limits set by OsloMet. The Regulations for new or postponed examinations are available in Regulations relating to studies and examinations at OsloMet.

In the event of a postponed examination in this course the exam may be held as an oral exam. Oral exams cannot be appealed.

The student can bring his/her own project report. The student is also allowed to make use of his/her own computer for the presentation.

For the final assessment a grading scale from A to E is used, where A denotes the highest and E the lowest pass grade, and F denotes a fail.

Two internal examiners will assess the individual report and the oral presentation. External examiner is used periodically.

Associate Professor Leiv Øyehaug