# Mathematical Analysis IV B

## Objectives

The program deals with the study of ordinary differential equations. We hope that the students be able to determine the general solution and particular solutions of various types of ordinary differential equations.

We intend the students use the Laplace Transforms for solving differential equations and corresponding initial value problems.

In the last part of the program we intend to familiarize the students with Fourier Analysis in such a way they could be able to use it as a tool in solving the most important partial differential equations occuring in engineering.

## General characterization

5006

6.0

##### Responsible teacher

Fábio Augusto da Costa Carvalho Chalub

Weekly - 5

Total - 70

Português

### Prerequisites

Linear Algebra and Mathematical Analysis I, II, III.

### Bibliography

Basic textbooks.
Online materal: Vilatte, Jaime, Equações diferenciais e equações de diferenças, FEUP http://villate.org/doc/eqdiferenciais/eqdif_20110426.pdf

Apostol, T.M., Calculus, Volume I and Volume II, Blaidsell Publishing Company.

Howard, Anton, Calculus: A New Horizon, John Wiley and Sons.

Taylor, A.E., Man, W.R., Advanced Calculus, John Wiley and Sons.

Stewart, J. Cálculo, Thomson Learning.

Ferreira, M. A. e Amaral, I, Matemática, Integrais míltiplos, equações diferenciais, Edições Síabo

Extra references:

Topics 7, 8 e 10. Butkov, E. Mathematical Physics.

Topic 9. The Mathematics of Medical Imaging: A Beginner''s Guide, Timothy G. Feeman, Springer

### Teaching method

General (3hs/week) and exercises (2hs/week) class. Homework and examples will be solved in exercises class.

### Evaluation method

The evaluations consists in three tests. If the grade othe third test is equal or superior to seven, then the final classification is the unweighted average of the three tests, rounded to the near integer (n.5 is rounded to n+1).

Students with average equal or superior to 9.5 but who fails to satisfy the minimum of 7.0 in the third test, will have "avaliação contínua" equals to 9.

For students that fail in the evaluation, there is a final exam.

## Subject matter

1. First order differential equations. Exact differentials. Integrating factors. Separation of variables. Homogeneous equations. Linear equations. Qualitative theory.

2. Second order differential equatons. Linear equations and Euler equation. Newton''s second law. The free, damped and forced harmonic oscilator. Variation of constants.

3. Solution in series. Bessel, Lagrange and Hermite functions.

4. Linear equations of higher order.

5. Systems of linear equation with constant coefficients. Differential equations in polar coordinates. Linearization of non-linear systens near equilibria.

6. Partial differential equations. Heat, wae and Laplace equations. Laplacian in spherical coordinates.

8. Laplace transform and its use in differential equations. Dirac delta.

9. Introduction to variational calculus. The law of sinus (Snell-Descartes). Catenary. The principle of minimum action. Euler-Lagrange equation and the lagrangean. Brachistochone curve.

10. Introduction to inverse problems. Radon transform.

## Programs

Programs where the course is taught: