ACIT4310 Applied and Computational Mathematics Course description

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

Skills

On successful completion of this course the student can:

• derive mathematical models from facts and first principles
• apply mathematical modelling techniques on scenarios relevant to industry
• can implement mathematical models on a computer
• analyse ODE systems and use bifurcation theory to elucidate the qualitative behavior of the systems
• implement and use a selection of numerical methods for solving ODEs and PDEs

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

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 course, together with Mathematics 1000, will give the students an understanding of mathematical concepts, problems and solution methods with the focus on application, particularly in engineering subjects.

No requirements over and above the admission requirements.

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

Knowledge

The student is capable of:

• explaining how functions can be approximated by taylor polynomials, power series and/or fourier series, explain what it means that a series converge, and differentiate and integrate powerseries.
• explaining what a frequency spectrum is, and explaining the principle of filtering signals in the frequency domain.
• describing and explaining how a sequence of numbers can originate by sampling, by using a formulae or as the solution of a difference equation.
• explaining how to interpolate sampled data.
• explaining partial differentiation and using different graphical ways to describe functions of two variables
• calculating eigenvalues and eigenvectors of matrixes and giving a geometrical interpretaions of these values

Skills

The student is capable of:

• discussing the connection between fourier series and fourier transforms
• discussing pro and cons using interpolating polynomials, splines and least squares method to interpolate sampled data
• discussing error barriers when using polynomials to approximate functions
• using simple tests of convergence of series, for example the ratio test
• giving a geometrical interpretation of gradient and directional derivative and using linear approximation and total differential of functions of two variables to calculate uncertainty
• using partial differentiation to calculate and classify critical points of functions of two variables
• using eigenvalues and eigenvectors to solve systems of differential equations with constant coeffisients;

General competence

The student is capable of:

• identifying the connection between mathematics and their own field of engineering
• translating a practical problem from their own field into mathematical form, so that it can be solved analytically or numerically
• using mathematical methods and tools that are relevant to their field of engineering
• assessing the results of mathematical calculations and using basic numerical algorithms

The course is taught through joint lectures and exercises. In the exercise sessions, the students work on assignments, both individually and in groups, under the supervision of a lecturer.

Students will be able to evaluate their own and others' professional work, and formulate assessments of these in such a way that the assessment provides advice on further study work. Exercise in this takes place in the hourly planned part of the work sessions. Students will therefore conduct weekly assessments of assignments based on weekly assignments. Information on how the weekly assessments will be conducted will be given in the lectures.

None.

Individual written exam, 3 hours.

The exam result can be appealed.