Academic year 2013-14

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: Juan Calvo
Teaching staff:

Juan Calvo (coordinator), Mariella Dimiccoli, Ernest Montbrió
Language:

Juan Calvo (spanish), Mariella Dimiccoli (spanish), Ernest Montbrió
Timetable:
Building: Communication campus - Poblenou

 

Introduction

This course will provide students with the basic concepts related to ordinary differential (ODE's) and partial differential equations (PDE's). Particular emphasis is given to modeling applications,  to highlight the importance of these types of equations beyond their theoretical interest.

The summary of the topics that will be taught during the course is presented below:

 
1. The concept of differential equation: definitions, terminology, initial value problems.
 
2. ODE's in separate variables: underlying theory and exercises. Applications and examples of modeling are:

- Population dynamics and logistic models

- Fall of a body in a vacuum and in air

- Economy: simple interest and compound interest

 

3. Linear ODE's of first order: underlying theory and exercises. Modeling examples:

 - Radioactive decays

- Radiocarbon dating of materials 

- RL and RC electrical circuits in series

 
 
4. Linear ODE's of second order with constant coefficients: Structural theorems. Solution of homogeneous equations through the associated characteristic polynomial. Solution of non-homogeneous equations: method of similarity.
 
Modelling examples:

- Spring and mass systems: free, damped motion (over, under and critical), forced

- Series RLC circuits
 
 
5. The Laplace transform: definition and properties. Applications to  ODE solving.
 
6. Numerical methods for solving ODE's:

- Method (s) of Euler;

- Heun method (predictor-corrector);

- Runge-Kutta method.
 
7. Introduction to EDP's: generalities. The heat equation and its properties. Boundary value problems,  separation of variables.
 
The main objective of the course is to introduce students to modeling techniques through algebraic-differential and integral calculus that they have already learnt in  previous calculus courses.
 
 
These techniques have a wide domain of  application, which the student will appreciate during his studies.

 

Prerequisites

It is highly advisable to have successfully completed the following subjects: Algebra and Discrete Mathematics, Calculus and Numerical Methods.

 

Associated competences

Competencias generalesCompetencias específicas

Instrumentales

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.

Interpersonals

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.

Sistèmiques

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 express mathematical ideas and concepts orally and in writing with precision.

3.Ability to understand and be able to reproduce theoretical demonstrations.

4. Solving skills related with those differential equations presented during the course.

5. Ability to model a problem in which a magnitude and its rate of change play a a role by meeans of a differential equation.

6. Ability to use approximation methods to solve differential equations that can not be solved analytically. 

7.Learn to recognize the structure of the fundamental PDE's and their meaning.

 

Assessment

The course is assessed through a written examination and two class tests (a written test and a computer test). The written exam can be recovered in July; class tests cannot. The exam will take place during the dates of June examinations. The two tests will take place during weeks 6 and 9 (approximately). Both the exam and class tests consist of theoretical questions and exercises related to the different types of differential equations presented during the course. The score for each exercise shall be notified on the sheet of examination / test.

 

Both tests and exam score over 10. You need to get at least 5 out of 10 in the exam to pass the course. Should you meet the above condition, the final grade will be calculated using the following formula:

Final_grade = maximum between

1) the exam qualification 

2) + 0.75 · Exam_qualification +0.15 · Written_test +0.10 · Computer_test.
 
The student has the chance of recovering the exam in July  (test scores are kept).

 

Contents

Bloc 1.

Tema 1.

The concept of differential equation: definitions, terminology, initial value problems.

Tema 2.

ODE's in separate variables: underlying theory and exercises. Modeling examples:Prueba_Ordenador

- Population dynamics and logistic models

- Fall of a body in a vacuum and in air

- Economy: simple interest and compound interest

- Newton's Law of cooling / heating 

 

Bloc 2.

Tema 3. 

Linear ODE's of first order: underlying theory and exercises. Modeling examples:

- RL and RC electrical circuits in series

- Radioactive decays

- Radiocarbon dating of materials 

 

Tema 4. 

ODE's linear second order with constant coefficients: structural theorems. Solution of homogeneous equations through the associated characteristic polynomial. Solution of non homogeneous equations: method of similarity.

- Spring and mass systems: free, damped (over, under and critical), forced movement.

- Series RLC circuits

Bloc 3.

Tema 5.

The Laplace transform: definition and properties. Application to solve ODE's.

Bloc 4.

Tema 6. 

Numerical methods for solving ODE's:

- Euler's method(s);

- Heun's method (predictor-corrector);

- Runge-Kutta method.

Bloc 5.

Tema 7.

Introduction to PDE's: generalities. The heat equation and its properties. Boundary value problems, separation of variables.

 

Methodology

 
During each theory block review exercises will be proposed.
 Solving these exercises will serve the students to test their understanding of the theory. Dedication may vary from person to person.
 
During seminar hour students will be invited to present solutions to exercises and discuss with the teacher any concerns or difficulties they have encountered during resolution of the exercises. We consider this interaction to be quite important, therefore we encourage students to attend seminars with the exercises already solved or, if they have met difficulties, with their attempted solution.
 
 
Practice hours will be mainly devoted to modeling problems using ODE's previously introduced in theory lectures. The student will appreciate the versatility of differential equations examining practical problems from physics, engineering, biology, electronics and economics.

 

Resources

• Notes by the teachers.

• D. G. ZILL : Ecuaciones diferenciales con aplicaciones de modelado , International Thomson Editores, 1997.

• S. G. KRANTZ : Differential equations demystified , Ed. McGraw Hill, 2005.

• G. F. SIMMONS : Ecuaciones diferenciales, Con aplicaciones y notas históricas , Ed. McGraw Hill, 1993.

• F. DIACU : An introduction to differential equations: order and chaos, Freedman and company, 2000.

• T. M. APOSTOL : Calculus (vols. 1 y 2), Reverté, 1990.

• M. R. SPIEGEL : Transformadas de Laplace, Ed. McGraw Hill, 1998.