Partial Differential Equations (MAE713)

Undergraduate Courses Outlines - Department of Mathematics

General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE713
Semester	7
Course Title	Partial Differential Equations
Independent Teaching Activities	Lectures, laboratory exercises (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek, English
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The aim of the course is an introduction to Partial Differnential Equations (PDE). By this course the students become familiar with a broad area of Analysis that, moreover, has the most applications in other Sciences. The course highlights the wealth of problems that arise in PDE and proposes methods to overcome them. These are presented exemplarily and aim to show the students ways of generalizing known methods and solutions. The student learns to analyze step-by-step externally posed problems, taking into account relevant informations and aims, and to apply knowledge from “pure” mathematics in order to solve these problems. Moreover, the student learns to interpret the obtained mathematical results. Concerning specific knowledge, the student learns about (mostly linear) PDE of first and second order for functions of two variables with respect to both, their explicit solution and their qualitative behavior, and obtains an elementary overview of further problems.

General Competences

Search for, analysis and synthesis of data and information, with the use of the necessary technology Working independently Working in an interdisciplinary environment Production of free, creative and inductive thinking

Syllabus

Overview of PDE and Systems: Classification with respect to their (non-)linearity, description of the arising problems and of the various kinds of solutions (classical and weak; general and with boundary values)
(for the following we focus on the case of two independent variables)
First order PDE (linear, semi-linear, quasi-linear): Geometric and algebraic observations concerning their qualitative behavior; Initial Value Problems and Method of Characteristics; discussion of the Burgers equation; shock waves and weak solutions; Rankine-Hugoniot condition.
Second order PDE: classification, characteristic directions and curves; wave equation on the line (homogeneous and non-homogeneous); separation of variables for the Laplace and heat equations; Poisson formula.
(alternatively: instead of the discussion of the Burgers equation and of weak solutions, an introduction to the Fourier transform may be given and the heat equation on the line may be discussed)

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	The students may contact the lecturer by e-mail
Teaching Methods	Activity Semester Workload Lectures (13X3) 39 Working independently 78 Exercises-Homeworks 33 Course total 150
Student Performance Evaluation	Final written exam (obligatory) Home work (optional)

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. Books and other resources, not provided by Eudoxus:

• Δάσιος, Γ., Κυριάκη, Κ., & Βαφέας, Π. (2023). Μερικές Διαφορικές Εξισώσεις [Προπτυχιακό εγχειρίδιο]. Κάλλιπος, Ανοικτές Ακαδημαϊκές Εκδόσεις. http://dx.doi.org/10.57713/kallipos-317
• L. C. Evans. Partial Differential Equations. Second edition. AMS, 2010.
• G. B. Folland. Introduction to Partial Differential Equations. Princeton University Press, 1995.