# Differential equations

## Objectives

The student is supposed to learn the basic ideas about Ordinary Differential Equations and Partial Differential Equations: methods and applications.

## General characterization

##### Code

7814

##### Credits

6.0

##### Responsible teacher

Luís Manuel Trabucho de Campos

##### Hours

Weekly - 5

Total - 28

##### Teaching language

Português

### Prerequisites

Background in Analysis (I A, II A and III A) and Algebra (Linear Algebra I and II).

### Bibliography

**M. Braun. Differential Equations and their applications (4th edition). Springer-Verlag, 1993.**

**1. **A. Bivar Weinholtz, Equações Diferenciais – uma introdução, Textos de Matemática, Vol. 7, Departamento de Matemática da FCUL, 2000.

**2. **E. Kreyszig. Advanced engineering mathematics (8th edition). John Wiley & Sons, 1999.

**3. **C. Póvoas,** **Métodos Matemáticos da Física-Uma Introdução, Textos de Matemática, Vol. 17, Departamento de Matemática da FCUL, 2002.

**4. **S. Ross. Differential Equations (3rd edition). John Wiley & Sons, 1984.

**5.** C. Fox — An Introduction to the Calculus of Variations, Dover, 1987.

**6.** H. Sagan — Introduction to the Calculus of variations, McGraw-Hill, 1969.

### Teaching method

Classes consist on two different aspects: an oral explanation which is illustrated by examples and the resolution, by the students, of proposed exercises. Most results are proven.

Students can ask for any questions either in class or in the professor''''s office ours.

### Evaluation method

1. The knowledge assessment will be done through two tests (T1, T2), in person. Students must enroll in the test. If due to circumstances outside the FCT it will not possible to carry out the tests, these will be replaced by a final exam, in person.

2. The tests will all have the same weight. Each test will be graded 0-20 values.

3. To pass, students must obtain a final classification equal to, or higher than 10. The final classification (CF) is obtained by rounding the following value (V), to the units

V = (C1 + C2) / 2,

where C1 and C2 represent the grades of tests T1 and T2, respectively.

Example: if V = 12.4 then CF = 12; if V = 12.5 is then CF = 13.

4. An exam will take place after the end of the course for those who did not succeed. The classification will be an integer from 0 to 20. The student will pass if the classification is 10 or higher, in a maximum of 20.

## Subject matter

1. Ordinary Differential Equations (ODE)

1.1. Generalities. Exemples and applications.

1.2. Existence and unicity of solution.

1.3. Systems of ordinary differential equations.

2. Partial Differential Equations (PDE)

Method of Characteristics.

2.1. First examples of PDE.

Initial and boundary conditions. Well posed problems.

2.2. First order linear equations.

2.3. First order quai-linear equations.

2.4. Classification of second order equations.

2.5. Reduction to the canonical form.

3. Partial Differential Equations (PDE)

Fourier Method.

3.1. Separation of variables and sobreposition principle.

3.2. Fourier Series (revision).

3.3 Sturm-Liouville Theory.

3.4. Wave equation

3.5. Heat equation.

3.6. Poisson equation.

3.7 Existence and unicity of solution. Maximum Principle.

4. Introduction to the Calculus of Variations

4.1 Classical problems in the Calculus of Variations.

4.2 Euler-Lagrange Equations.

4.3 Hamilton''''''''''''''''''''''''''''''''s Principle.