Partial Differential Equations (MAE713): Διαφορά μεταξύ των αναθεωρήσεων

Τελευταία αναθεώρηση της 12:31, 15 Ιουνίου 2023

Ελληνική Έκδοση
Undergraduate Courses Outlines
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Department of Mathematics
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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 (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 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.
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 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

Activity	Semester Workload
Lectures (13X3)	39
Working independently	78
Exercises-Homeworks	33
Course total	150

Student Performance Evaluation

Written exam (mandatory)
Homework (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.

@@ Γραμμή 1: / Γραμμή 1: @@
-[[Undergraduate Courses Outlines]] - [https://math.uoi.gr  Department of Mathematics]
+* [[Μερικές Διαφορικές Εξισώσεις (ΜΑΕ713)|Ελληνική Έκδοση]]
+{{Course-UnderGraduate-Top-EN}}
+{{Menu-OnAllPages-EN}}
 === General ===
@@ Γραμμή 30: / Γραμμή 32: @@
 ! Independent Teaching Activities
 |
-Lectures, laboratory exercises (Weekly Teaching Hours: 3, Credits: 6)
+Lectures (Weekly Teaching Hours: 3, Credits: 6)
 |-
 ! Course Type
@@ Γραμμή 56: / Γραμμή 58: @@
 ! 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 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.
-<br/>
-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.
-<br/>
-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
@@ Γραμμή 69: / Γραμμή 67: @@
 * 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)
-<br/>
+* 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).
-(for the following we focus on the case of two independent variables)
-<br/>
+(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.
-<br/>
+* 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.
+* 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.
-<br/>
-(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)
+(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 ===
 {| class="wikitable"
@@ Γραμμή 111: / Γραμμή 111: @@
 ! Student Performance Evaluation
 |
-* Final written exam (obligatory)
+* Written exam (mandatory)
-* Home work (optional)
+* Homework (optional)
 |}