Partial Differential Equations (MAE713): Διαφορά μεταξύ των αναθεωρήσεων

Τελευταία αναθεώρηση της 15:44, 15 Ιανουαρίου 2025

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 course introduces the students to Partial Differential Equations (PDE). The importance of the fact that their solutions are scalar or vector-valued functions of more than one independent variables is stressed and that, in contrast to Ordinary Differential Equations, this has significant consequences, in the sense that for PDEs, next to the analytical properties of the solutions, also the algebraic structure of the equations plays a distinguished role, which implies also geometric properties of the solutions. Also, the connection with and origin from the Natural Sciences and Geometry for many of them is stressed and that this implies not only that the focus mainly on certain types of equations results from the questions posed by other scientific fields but also that the latter dictate to a big extend in a natural way the methods for solving and studying the properties of the various classes of PDEs. In this way, the course strengthens the ability of the students to examine a problem from several perspectives and to take into account knowledge and results from other scientific areas. In particular, the course introduces the students to the main classes of PDEs, highlights the fact that each class relies on its own analysis techniques, that their solutions have properties which are characteristic for the class to which they belong, and that results which are obtained for one class can be used partly also for the analysis of equations of a different class, although under essential restrictions. In this introductory course initially only classical solutions are studied and an emphasis is given on the explicit solving of prototypical equations for each class and on a first study of their characteristic properties.
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

Introduction: definition of a PDE, definition of classical solutions. Classification with respect to (non) linearity. Examples of equations and systems. First order equations. Method of characteristics. Transport equation. Linear PDEs of second order. Laplace and Poisson equation, heat equation, wave equation: representation formulas of solutions and energy method.

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 course introduces the students to Partial Differential Equations (PDE). The importance of the fact that their solutions are scalar or vector-valued functions of more than one independent variables is stressed and that, in contrast to Ordinary Differential Equations, this has significant consequences, in the sense that for PDEs, next to the analytical properties of the solutions, also the algebraic structure of the equations plays a distinguished role, which implies also geometric properties of the solutions. Also, the connection with and origin from the Natural Sciences and Geometry for many of them is stressed and that this implies not only that the focus mainly on certain types of equations results from the questions posed by other scientific fields but also that the latter dictate to a big extend in a natural way the methods for solving and studying the properties of the various classes of PDEs.
-<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.
+In this way, the course strengthens the ability of the students to examine a problem from several perspectives and to take into account knowledge and results from other scientific areas.
-<br/>
+In particular, the course introduces the students to the main classes of PDEs, highlights the fact that each class relies on its own analysis techniques, that their solutions have properties which are characteristic for the class to which they belong, and that results which are obtained for one class can be used partly also for the analysis of equations of a different class, although under essential restrictions. In this introductory course initially only classical solutions are studied and an emphasis is given on the explicit solving of prototypical equations for each class and on a first study of their characteristic properties.
-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 70: @@
 * 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/>
+Introduction: definition of a PDE, definition of classical solutions. Classification with respect to (non) linearity. Examples of equations and systems. First order equations. Method of characteristics. Transport equation. Linear PDEs of second order. Laplace and Poisson equation, heat equation, wave equation: representation formulas of solutions and energy method.
-(for the following we focus on the case of two independent variables)
-<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.
-<br/>
-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.
-<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)
 === Teaching and Learning Methods - Evaluation ===
 {| class="wikitable"
@@ Γραμμή 111: / Γραμμή 107: @@
 ! Student Performance Evaluation
 |
-* Final written exam (obligatory)
+* Written exam (mandatory)
-* Home work (optional)
+* Homework (optional)
 |}
 === Attached Bibliography ===
-See the official [https://service.eudoxus.gr/public/departments#20 Eudoxus site] or the [https://cloud.math.uoi.gr/index.php/s/62t8WPCwEXJK7oL local repository] of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Additionally:
+<!-- In order to edit the bibliography, visit the webpage -->
-* L. C. Evans: Partial Differential Equations (2 edition), AMS, 2010
+<!-- https://wiki.math.uoi.gr/index.php/%CE%A0%CF%81%CF%8C%CF%84%CF%85%CF%80%CE%BF:MAE713-Biblio -->
+See the official [https://service.eudoxus.gr/public/departments#20 Eudoxus site] or the [https://cloud.math.uoi.gr/index.php/s/62t8WPCwEXJK7oL local repository] of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:
+{{MAE713-Biblio}}