# Partial Differential Equations (MAE713)

### General

School School of Science Department of Mathematics Undergraduate MAE713 7 Partial Differential Equations Lectures (Weekly Teaching Hours: 3, Credits: 6) Special Background - Greek, English Yes (in English) 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 Differential Equations. By this course the students become familiar with a broad area of Analysis that has many applications to other sciences. The course highlights the wealth of problems that arise and proposes methods to overcome them. These are presented exemplarily and aim to teach ways of transcending and generalizing known methods and solutions. The students learn to analyze methodically externally given problems, taking into account relevant informations and aims, and to try to apply knowledge from other areas of Pure Mathematics in order to solve these problems. Moreover, the students learn to interpret the obtained mathematical results. On the level of content, the students learn about, mainly linear, Partial Differential Equations of first and second order for functions of two variables with respect to both, their explicit solution and their qualitative behavior, and obtain an elementary overview of further problems. 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 Partial Differential Equations (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).

(In the following the focus is given on 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 characteristic curves, wave equation on the line (homogeneous and inhomogeneous), separation of variables for the Laplace and heat equations, Poisson formula.

(Alternatively: instead of the discussion of the (non-linear) 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