Course syllabus - Applied mathematics

Objectives

The aim of the course is to provide a broad introduction to the concepts and methods of Applied Mathematics and through a mixture of individual and group assignments practice skills in calculation, reasoning, modeling and problem solving both independently and in collaboration with others.

Learning outcomes

After completing the course the student should be able to:

1. describe some mathematical models based on both linear and non-linear differential equations used in application areas such as heat transfer, biology, economics and finance
2. explain the ideas behind scaling and other changes of variable in differential equations
3. solve ordinary differential equations that separable, first order linear or linear with constant coefficient
4. describe and calculate with Fourier series, the Fourier transform and the Laplace transform for solution and analysis of differential equations and linear systems (input–output signal models)
5. describe basic ideas in distribution theory and perform simple calculations with distributions
6. describe and use separation of variables to solve linear partial differential equations
7. use the Jacobian to classify equilibria for autonomous systems of ODEs, both in the general case and for linear systems
8. describe some example of how periodic or chaotic properties can manifest in a non-linear dynamical system

Course content

- Partial derivatives, with the chain rule
- Series expansions, definition of convergence and divergence
- Ordinary differential equations, integrating factor, separable equations, characteristic equation
- Definition of partial differential equation. The linearity property.
- Scaling and change of variables
- Fourier series and other series expansions. Introductory Sturm-Liouville theory
- The Fourier- and Laplace transforms, with the most important rules for calculation. Convolutions
- Separation of variables for linear PDEs. Especially for the heat equation, wave equation, Laplace equation and the Schrödinger equation
- Discussion of solutions of some simple non-linear PDE with connection to engineering or science, that demonstrates similarities and differences in methodlogy. E.g. the Korteweg-de Vries equation or the minimal surface equation
- Continuous dynamical systems. Chaos, stability and bifurcations. The Jacobian and its use in classifying equilibria. Phase portraits
- Discussion of periodic or chaotic properties of some non-linear dynamical system with connection to engineering or science, e.g. the Lorenz system

Tuition

Lectures and or classes.

Specific requirements

120 credits in one of several of the following subjects: Engineering, Natural Science, Business Administration or Economics, at least 60 credits must be in the field of Engineering including courses in Mathematics/Applied Mathematics with a total sum of at least 30 credits which must include Basic Calculus Continuation Course, 7.5 credits or equivalent. In addition, Swedish B/Swedish 3 and English A/English 5 are required. In cases when the course is offered in English, the requirement for Swedish B/Swedish 3 is excluded.

Examination

INL1, Written assignment, 3 credits, oral and written presentation of problem-solving concerning learning outcomes 1-8, grades Fail (U) or Pass (G).
TEN1, Examination, 4.5 credits, written and/or oral examination concerning learning outcomes 1-8, grades Fail (U), 3, 4 or 5.

A student who has a certificate from MDU regarding a disability has the opportunity to submit a request for supportive measures during written examinations or other forms of examination, in accordance with the Rules and Regulations for Examinations at First-cycle and Second-cycle Level at Mälardalen University (2020/1655). It is the examiner who takes decisions on any supportive measures, based on what kind of certificate is issued, and in that case which measures are to be applied.

Suspicions of attempting to deceive in examinations (cheating) are reported to the Vice-Chancellor, in accordance with the Higher Education Ordinance, and are examined by the University’s Disciplinary Board. If the Disciplinary Board considers the student to be guilty of a disciplinary offence, the Board will take a decision on disciplinary action, which will be a warning or suspension.

Grade

Pass with distinction, Pass with credit, Pass, Fail

Interim Regulations and Other Regulations

The course overlaps with MAA508 Applied Mathematics, 7.5 credits.