Universidade Nova de Lisboa

Universidade Nova de Lisboa

Faculdade de Ciências e Tecnologia

Mathematical Analysis III C

Code

5004

Academic unit

Faculdade de Ciências e Tecnologia

Department

Departamento de Matemática

Credits

6.0

Teacher in charge

João de Deus Mota Silva Marques

Weekly hours

5

Teaching language

Português

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.

Prerequisites

Knowledge of the  matters tought in mathematical analysis I-C, II-C. Some basic concepts in linear algebra.

Subject matter

1. Ordinary differential equations

 First-order differential equations. Exact equations. The integrating factor method. Separable equations. Homogeneous equations. The linear equation of first order. First-order equations solvable by special methods. Second-order and higher order equations. Equations where does not explicitly appear the independent variable or the unknown function. Homogeneous linear equation of order n. Nonhomogeneous linear equations of order n.­ Variation of parameters. Homogeneous linear equation of order n with constant coefficients. Special methods for determining particular solutions of the nonhomogeneous linear equation of order n, with constant coeficients. Euler´s equation. Use of power series to obtain solutions of a linear differential equation of order two. The homogeneous linear equation of second order- singular points. The Bessel equation. The Bessel functions. Systems of linear differential equations. Existence and uniqueness theorem for differential equations.
2.Laplace Transforms. The Laplace Transformation as a method for solving differential equations and corresponding initial value problems.
3.Fourier series.
4. Introduction to the study of partial differential equations.

Bibliography

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

Braun, Martin - Differential Equations and their Applications, Springer-Verlag

Freitas, A.C. - Análise Infinitesimal - Volume 1 e Volume 2 - Notas de Lições para alunos do 2º ano das Licenciaturas da FCT.

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

Kreysig - Advanced Engineering Mathematics

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

 

Zill, D. G. ; Cullen, M.R. - Differential equations with boundary value problems; 6th edition.

Teaching method

Theoretical classes consist on a theoretical exposition illustrated by examples of applications.

Practical classes consist on the solving of exercises of application of the methods and results presented in the theoretical classes. All the problems are solved by the students with the support of the teacher.

 Any questions or doubts will be adressed during the classes, during the weeekly sessions specially programmed to it  or even at special sessions previously arranged between professors and students.

Evaluation method

To get a final mark it is mandatory to obtain Frequência. This is obtained if the student attends at leats 2/3 of the problem solving classes.

The continuous evaluation consists of three tests.

More detailed rules are available in the Portuguese version.

Courses