Academic year 2014-15
Differential Equations
| Degree: | Code: | Type: |
| Bachelor's Degree in Computer Science | 22634 | Optional subject |
| Bachelor's Degree in Telematics Engineering | 22581 | Optional subject |
| Bachelor's Degree in Audiovisual Systems Engineering | 21602 | Compulsory subject, 1st year |
| ECTS credits: | 4 | Workload: | 100 hours | Trimester: | 3rd |
| Department: | Dept. of Information and Communication Technologies |
| Coordinator: | Josep Blat |
| Teaching staff: | Juan-Francisco Garamendi, Javier Vazquez, Josep Blat (coordinator) |
| Language: | Catalan (Javier Vazquez, Josep Blat), Castellano (Juan-Francisco Garamendi) |
| Timetable: | |
| Building: | Communication campus - Poblenou |
Ordinary differential equations (ODEs) and partial differential equations (PDEs) are used to model physical phenomena. In particular, they are especially important for simulations, and used to create visual effects in movies or video games. Examples include the simulation of waves of water or smoke and flames. In this context, visual interactive simulations are particularly interesting because the simulations as audiovisual products need "direction" or "orchestration" of creative personnel. The interactive visual simulation is the specific orientation or emphasis taken in this Differential Equations course, be them ordinary or partial.
The course combines modeling phenomena using Differential Equations, properties of their analytical and numerical solution, the error in the numerical solutions and visual simulation of solutions.
As a methodological core of the course we will explore these issues in detail for some phenomena and equations particularly interesting from the point of view of training and visual simulation: a spring, surface water waves, and where appropriate, advection.
It is highly recommended to have attained skills in mathematics and programming corresponding to the subjects Discrete Mathematics and Algebra, Calculus and Numerical Methods and Programming Fundamentals.
| General Competences | Specific Competences |
|---|---|
|
Instrumental 1. Ability to understand and analyze mathematical statements. 2. Ability to identify the appropriate methodology to analyze a problem and find its solution. 3. Ability to express mathematical ideas and concepts orally and in writing with precision. 4. Capacity for abstraction. Interpersonal 5. Ability to work in teams to solve problems so as to deepen in theoretical issues. 6. Ability to communicate ideas accurately, both orally and in writing. Systemic 7. Ability to work independently to solve a problem. 8. Ability to find the most appropriate solution, depending on the given characteristics of a particular context. 9. Ability to infer mathemetical notions. 10. Ability to check and interpret solutions, taking into account particular cases |
1. Ability to identify and justify the application of appropriate mathematical model to analyze a problem and find its solution. 2. Ability to understand and be able to reproduce demonstrations. 3.Solving skills related with those differential equations presented during the course. 4. Ability to model a problem in which a magnitude and its rate of change play a a role by meeans of a differential equation. 5. Ability to use approximation methods to solve differential equations. 6.Learn to recognize the structure of the fundamental PDE's and their meaning. |
The evaluation will be continuous on solving exercises sessions and seminars-problems-programming practices.
Both aspects must be passed in order to pass the course altogether. The weight of each is 50%.
The evaluation regarding problems is not recoverable, the assessment regarding programming is.
If the teacher deems appropriate, assessment can be completed by oral interviews.
The course combines theory, exercises solving equations and modeling, and programming / simulation. The contents following this interdependence.
1 Basics of ordinary differential equations and examples of physical systems and models using differential equations. Elementary analytical solutions through separation of variables.
Exercises elementary solutions of ODEs and models.
States, Simulation and Visualization - Basic Processing
2 Concepts and numerical solution of differential equations with initial values, finite differences, the Euler method and errors.
Exercises and solutions of ODE models.
3. Detailed development of physical model, analytical solution and numerical solution of an elastic spring: The space of solutions of a linear equation of second order, stability of the Euler method, implicit simple methods.
Programming the time step: Euler, Euler-Cromer, Waypoints. Comparison with the exact solution, stability.
4. Numerical methods using higher order interpolation approaches: Runge-Kutta of order 2 and 4. Introduction to other methods (predictor-corrector)
Exercises related to variants of an elastic spring.
Programming RK2 and RK4. Variants.
5. 1D wave equation. Analytical solutions of partial differential equations by elementary methods. Boundary value problems and their numerical approximation by finite differences, implicit and explicit methods.
Exercises analytical and numerical solutions of PDEs
Exercises elementary numerical solution of systems of linear equations by iterative methods
Explicit and implicit solution methods for the 1D wave equation, and visual simulation. Stability problems. Numerical solution of systems of linear equations by iterative methods.
6. Project: interactive simulation of the wave equation in 2D and advection.
|
|
Dilluns |
Dijous |
Divendres |
|
1 06-10 abr |
06/04
FESTIU
|
09/04
T
|
10/04
T
|
|
2 13-17 abr |
13/04
S103/S104
|
16/04
S101 /S102
|
17/04
P102 |
|
3 20-24 abr |
20/04
P101 |
23/04
FESTIU
|
24/04
T
|
|
4 27 abr-1 mai |
27/04
T |
30/04 S101/S102
|
01/05
FESTIU
|
|
5 04-08 mai |
04/05
S103/S104
|
07/05
P102
|
08/05
P101
|
|
6 11-15 mai |
11/05
T
|
14/05
T
|
15/05
S103/S104
|
|
7 18-22 mai |
18/05
S101/S102
|
21/05
P102
|
22/05
P101
|
|
8 25-29 mai |
25/05
T
|
28/05
T
|
29/05
S103/S104
|
|
9 01-05 jun |
01/06
FESTIU
|
04/06
S101/S102
|
05/06
P101
|
|
10 08-12 jun |
08/06
P102
|
11/06
T
|
12/06
S103/S104
|
|
11 15-19 jun |
15/06
S101/S102 |
18/06
NO LECTIU |
19/06
NO LECTIU |
Notes and exercises.
Basic bibliography:
Zill, Dennis G.: Ecuaciones diferenciales con aplicaciones de modelado (9ª ed), International Thomson, México, D.F., 2009. QA372 .Z5518
Golub, Gene H. i Ortega, James M. : Scientific computing and differential equations : an introduction to numerical methods, Academic Press, San Diego, 1992. QA371 .G65 1992
Shiffman, Daniel : Learning Processing : a beginner's guide to programming images, animation, and interaction, Morgan Kaufmann/Elsevier, Amsterdam & Boston, 2008
The course is inspired by the visual simulation of differential equations blog by Christopher Horvath (https://www.blogger.com/profile/04341930852316328263):
http://encinographic.blogspot.com.es/2013/05/simulation-class-intro-and-prerequisites.html
https://github.com/blackencino/SimClass
Other references:
Krantz, Steven G.: Differential equations demystified , McGraw Hill, New York, 2005. QA371 .K63 2005
Simmons, George Finlay: Ecuaciones diferenciales, Con aplicaciones y notas históricas (2ª ed.), McGraw Hill, 1993. QA372 .S5618 1993
Braun, Martin : Differential equations and their applications : short version, Springer-Verlag, New York, 1978 QA371 .B73 1978
Courant, Richard i John, Fritz : Introducción al cálculo y al análisis matemático (2 vols.), Limusa, Mexico, 1971-78. QA303 .C6818 1988 QA76.73.P75 S55 2008