Difference Equations - Discrete Models (MAE816)

Undergraduate Courses Outlines - Department of Mathematics

General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE816
Semester	8
Course Title	Difference Equations - Discrete Models
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Language of Instruction (lectures): Greek Language of Instruction (activities other than lectures): Greek and English Language of Examinations: Greek and English
Is the Course Offered to Erasmus Students	Yes
Course Website (URL)	Course description: http://math.uoi.gr Learning Management System (e.g.: Moodle): http://users.uoi.gr/kmavridi

Learning Outcomes

Learning outcomes

Remembering: The concept of the Difference Operator, the Summation Operator and the Shift Operator. The concept of Binomial Coefficient and the Gamma Function. The concept of the Generating Function. The concept of the Difference Equation. The concept of the z-Transformation. The concepts of the Stable Fixed Point and the Asymptotically Stable Fixed Point. The concepts of Liapunov Function and Strictly Liapunov Function. The concept of sensitive dependence on initial conditions. The concept of asymptotic relation between functions. The concepts of "O-big" and "O-small". The concept of the homogeneous linear Poincare-type equation. The concept of the boundary value problem for non-linear equations. The concept of Partial Difference Equations. Comprehension: Basic properties of the Difference Operator, the Summation Operator and the Shift Operator. Calculation of indefinite sums. Solving certain types of linear difference equations. Finding fundamental sets of solutions for linear difference equations. Using the Casorati determinant in order to solve linear difference equations. Using Generating Functions and z-Transformations in order to solve difference equations. Linearisation of non-linear difference equations. Studying the stability of the solutions of difference equations and the Floquet Theory. Studying the stability of non-linear systems of difference equations and chaotic behaviour. Asymptotic approximation of sums. Green Functions of boundary value problems for difference equations. Oscillation of solutions for difference equations. Studying the Sturm-Liouville problem. Studying boundary value problems for non-linear difference equations. Studying partial difference equations. Applying: Studying economy-related real world problems. Studying the growth or the decline of populations. Studying physics-related real world problems. Studying probabilities-related real world problems. Studying epidemiology-related real world problems. Evaluating: Teaching undergraduate and graduate courses.

General Competences

Creative, analytical and inductive thinking. Required for the creation of new scientific ideas. Working independently. Working in groups. Decision making.

Syllabus

The Difference Calculus, Linear difference equations, Stability theory, Asymptotic methods, The Sturm-Liouville problem, Boundary value problems for non-linear difference equations, Partial difference equations.

Teaching and Learning Methods - Evaluation

Delivery	Lectures in class. Learning Management System (e.g.: Moodle).
Use of Information and Communications Technology	- Use of Learning Management System (e.g.: Moodle), combined with File Sharing and Communication Platform (e.g.: NextCloud) for distributing teaching material, submission of assignments, course announcements, gradebook keeping for all students evaluation procedures, communicating with students. Use of Web Appointment Scheduling System (e.g.: Easy!Appointments) for organising office appointments. Use of Google services for submitting anonymous evaluations regarding the teacher.
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 78 Preparation of assignments and interactive teaching 33 Course total 150
Student Performance Evaluation	Language of evaluation: Greek and English. Methods of evaluation: Weekly written assignments. Few number of tests during the semester. Based on their grades in the aforementioned weekly assignments and tests, limited number of students can participate in exams towards the end of the semester, before the beginning of the exams period. In any case, all students can participate in written exams at the end of the semester, during the exams period. The aforementioned information along with all the required details are available through the course's website. The information is explained in detail at the beginning of the semester, as well as, throughout the semester, during the lectures. Reminders are also posted at the beginning of the semester and throughout the semester, through the course's website. Upon request, all the information is provided using email or social networks.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics.