Partial Differential Equations (MAE713): Διαφορά μεταξύ των αναθεωρήσεων

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[[Undergraduate Courses Outlines]] - [https://math.uoi.gr  Department of Mathematics]
* [[Μερικές Διαφορικές Εξισώσεις (ΜΑΕ713)|Ελληνική Έκδοση]]
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=== General ===
=== General ===
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! Independent Teaching Activities
! Independent Teaching Activities
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Lectures, laboratory exercises (Weekly Teaching Hours: 3, Credits: 6)
Lectures (Weekly Teaching Hours: 3, Credits: 6)
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! Course Type
! Course Type
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! Course Website (URL)
! Course Website (URL)
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| See [https://ecourse.uoi.gr/ eCourse], the Learning Management System maintained by the University of Ioannina.
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! Learning outcomes
! Learning outcomes
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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.
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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.
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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.  
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! General Competences
! General Competences
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* Production of free, creative and inductive thinking
* Production of free, creative and inductive thinking
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=== Syllabus ===
=== 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)
 
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* 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)
 
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(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.
 
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* 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.  
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(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 ===
=== Teaching and Learning Methods - Evaluation ===
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! Student Performance Evaluation
! Student Performance Evaluation
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* Final written exam (obligatory)
* Written exam (mandatory)
* Home work (optional)
* Homework (optional)
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=== Attached Bibliography ===
=== Attached Bibliography ===


See [https://service.eudoxus.gr/public/departments#20 Eudoxus]. 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}}

Τελευταία αναθεώρηση της 12:31, 15 Ιουνίου 2023

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.