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Numerical Solution of Ordinary Differential Equations (MAE744): Διαφορά μεταξύ των αναθεωρήσεων

Numerical Solution of Ordinary Differential Equations (MAE744): Διαφορά μεταξύ των αναθεωρήσεων

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[[Undergraduate Courses Outlines]] - [https://math.uoi.gr  Department of Mathematics]
* [[Αριθμητική Επίλυση Συνήθων Διαφορικών Εξισώσεων (ΜΑΕ744)|Ελληνική Έκδοση]]
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=== General ===
=== General ===
Γραμμή 34: Γραμμή 36:
! Course Type
! Course Type
|
|
Special Background
Special background, skills development.
|-
|-
! Prerequisite Courses
! Prerequisite Courses
Γραμμή 56: Γραμμή 58:
! Learning outcomes
! Learning outcomes
|
|
The course is an introduction to the basic methods for the numerical solution of ordinary differential equations. The objectives of the course are:
Upon successful completion of the course, students will be able to:
* Development of theoretical background in matters concerning the numerical solution of ordinary differential equations (ODEs) and ODE systems.
# describe the basic characteristics of single-step and multi-step methods and recognize their differences.
* Ability of using numerical methods for solving ODEs with computational programs that will help with the implementation, e.g. Mathematica, Matlab etc.
# apply a variety of techniques for the construction of single-step and multi-step methods.
* Upon completion of this course the student will be able to use numerical methods for solving mathematical problems that may not have analytical solution and further deepen the understanding of such methods.
# apply numerical analysis techniques to show consistency, stability, and convergence of numerical methods.  
# be aware of the optimal order of accuracy of key numerical methods as well as the limitations that may be required in the discretization parameters to ensure stability.
# write code (in Python or Octave) for the implementation of explicit and implicit numerical methods and calculate their experimental order of convergence.
# write code in Python or Octave for the numerical approximation of the solution of ODEs models that describe a variety of real-world situations.
|-
|-
! General Competences
! General Competences
|
|
The course aims to enable undergraduate students to develop the ability to analyze and synthesize basic knowledge of Numerical Analysis with the help of computers to numerically solve difficult problems in mathematics and/or physics. This will give to the student the opportunity to work in an international environment.
* Search for, analysis and synthesis of data and information, with the use of the necessary technology.
* Adapting to new situations.
* Decision-making.
* Teamwork.
* Production of free, creative, and inductive thinking.
* Promotion of analytical and synthetic thinking.
|}
|}
=== Syllabus ===
=== Syllabus ===
Difference Equations, Initial Value Problems, One step methods (Euler - explicit and implicit, Runge Kutta methods), Multiple steps methods (Adams-Bashforth, Adams-Moulton, Predictor-Corrector methods). Convergence, Stability, Compatibility, Stiff ODE systems, Boundary Value Problems, Shooting method, Finite differences, Eigenvalue problems.
 
# Initial Value Problems
# Explicit Euler and Implicit Euler.
# Consistency, stability, and convergence of Runge-Kutta methods.
# Consistency, stability, and convergence of multistep methods.
# Applications to ODEs systems arising from Physics and Biology.
 
=== Teaching and Learning Methods - Evaluation ===
=== Teaching and Learning Methods - Evaluation ===
{| class="wikitable"
{| class="wikitable"
Γραμμή 72: Γραμμή 89:
! Delivery
! Delivery
|
|
In class
Face-to-face.
|-
|-
! Use of Information and Communications Technology
! Use of Information and Communications Technology
| Use of computer (Mechanics) lab
|
* Use of a tablet device to deliver teaching.  Lecture materials in pdf-format are made available to students, for later review, on Moodle e-learning platform.
* Provision of study materials in Moodle e-learning platform.
* Provision of model solutions for some exercises in podcast format.
* Communication with students through e-mails, Moodle platform and Microsoft Teams.
* Use of sophisticated software (Python or Octave) for the implementation of the numerical algorithms.
|-
|-
! Teaching Methods
! Teaching Methods
Γραμμή 86: Γραμμή 108:
| 39
| 39
|-
|-
| Study of theory
| Study and analysis of bibliography
| 78
| 75
|-
| Directed study of Computer-based exercises
| 6
|-
|-
| Home exercises
| Non-directed study of Computer-based exercises
| 33
| 30
|-
|-
| Course total  
| Course total  
Γραμμή 98: Γραμμή 123:
! Student Performance Evaluation
! Student Performance Evaluation
|
|
* Weekly assignments
* Computer-based exercises (organised in teams of 2) with oral examination (Weighting 30%, addressing learning outcomes 4-6)
* Final project
* Written examination (Weighting 70%, addressing learning outcomes 1-4)
* Written examination at the end of the semester
|}
|}
=== Attached Bibliography ===
=== 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.
<!-- In order to edit the bibliography, visit the webpage -->
<!-- https://wiki.math.uoi.gr/index.php/%CE%A0%CF%81%CF%8C%CF%84%CF%85%CF%80%CE%BF:MAE744-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:
 
{{MAE744-Biblio}}

Τελευταία αναθεώρηση της 12:36, 15 Ιουνίου 2023

General

School

School of Science

Academic Unit

Department of Mathematics

Level of Studies

Undergraduate

Course Code

MAE744

Semester

7

Course Title

Numerical Solution of Ordinary Differential Equations

Independent Teaching Activities

Lectures (Weekly Teaching Hours: 3, Credits: 6)

Course Type

Special background, skills development.

Prerequisite Courses -
Language of Instruction and Examinations

Greek

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

Upon successful completion of the course, students will be able to:

  1. describe the basic characteristics of single-step and multi-step methods and recognize their differences.
  2. apply a variety of techniques for the construction of single-step and multi-step methods.
  3. apply numerical analysis techniques to show consistency, stability, and convergence of numerical methods.
  4. be aware of the optimal order of accuracy of key numerical methods as well as the limitations that may be required in the discretization parameters to ensure stability.
  5. write code (in Python or Octave) for the implementation of explicit and implicit numerical methods and calculate their experimental order of convergence.
  6. write code in Python or Octave for the numerical approximation of the solution of ODEs models that describe a variety of real-world situations.
General Competences
  • Search for, analysis and synthesis of data and information, with the use of the necessary technology.
  • Adapting to new situations.
  • Decision-making.
  • Teamwork.
  • Production of free, creative, and inductive thinking.
  • Promotion of analytical and synthetic thinking.

Syllabus

  1. Initial Value Problems
  2. Explicit Euler and Implicit Euler.
  3. Consistency, stability, and convergence of Runge-Kutta methods.
  4. Consistency, stability, and convergence of multistep methods.
  5. Applications to ODEs systems arising from Physics and Biology.

Teaching and Learning Methods - Evaluation

Delivery

Face-to-face.

Use of Information and Communications Technology
  • Use of a tablet device to deliver teaching. Lecture materials in pdf-format are made available to students, for later review, on Moodle e-learning platform.
  • Provision of study materials in Moodle e-learning platform.
  • Provision of model solutions for some exercises in podcast format.
  • Communication with students through e-mails, Moodle platform and Microsoft Teams.
  • Use of sophisticated software (Python or Octave) for the implementation of the numerical algorithms.
Teaching Methods
Activity Semester Workload
Lectures 39
Study and analysis of bibliography 75
Directed study of Computer-based exercises 6
Non-directed study of Computer-based exercises 30
Course total 150
Student Performance Evaluation
  • Computer-based exercises (organised in teams of 2) with oral examination (Weighting 30%, addressing learning outcomes 4-6)
  • Written examination (Weighting 70%, addressing learning outcomes 1-4)

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:

  • “Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations”, E. Hairer, & C. Lubich, Springer, 2010.
  • “Numerical Methods for Ordinary Differential Equations: Initial Value Problems”, D.F. Griffiths, & D. J. Higham, Springer, 2010.
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