Mathematics III

Code:

EBE0013

Acronym:

MAT3

Keywords
Classification	Keyword
OFICIAL	Basic Sciences

Instance: 2012/2013 - 1S

Active?	Yes
Responsible unit:	Mathematics Section
Course/CS Responsible:	Master in Bioengineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
MIB	79	Syllabus	2	-	5	56	135

Teaching language

Portuguese

Objectives

Theoretical concepts and integration techniques for ordinary differential equations and systems of diferential equations as well as its applications to simple problems.
Study of numerical, funcional and Fourier series and of Laplace transforms.


Skills and Learning Outcomes: At the end of this unit the student should be able to

1 - Classify and solve the ordinary differential equations given.
2 - Solve systems of linear ordinary differential equations of first order.
3 - Analyze the convergence of numerical series.
4 - Determine the convergence interval of power series.
5 - Obtain Fourier series to simple periodic functions.
6 - Calculate Laplace transforms and apply this technique to solve some differential equations.
7 - Find differential equation models to solve some simple problems

Program

I - Introduction to differential equations: general classification, definition of solution and of boundary value problems.

II - Ordinary differential equations of first order: the existence and uniqueness theorem; separable equations; linear equations , homogeneous and non homogeneous. The Bernoulli equations. Some problems modeled by first order equations.

III - Ordinary differential equations of nth order. Reduction of order equations: the two special cases of missing the independent variable or the dependent variable. Linear equations of order greater than one: general theory of homogeneous and non homogeneous linear nth order equations. Existence and uniqueness theorem. General solution for homogeneous linear equations with constant coefficients. Linear non homogeneous equations: the variation of parameters method.

IV -Systems of first order linear equations: introduction and its relation with an nth order linear differential equation. Some examples. Basic theory of systems of first order linear equations. Homogeneous systems of linear equations with constant coefficients. Real or complex single eigenvalues case and repeated eigenvalues case. Fundamental matrices. The method of variation of parameters for non homogeneous systems.

V - Numerical series : definition, properties and main examples. Convergence theorems and criteria. Absolute and simple convergence. Alternate series and the Leibniz theorem.
Functional series: convergence domain and its sum. Special case of power series - convergence radius and convergence interval. Taylor series - examples.
Fourier series - definition and main properties. Euler formulae. Examples and aplications to several types of periodic functions.

VI - The Laplace transform: definition and existence conditions. Laplace transform of some basic functions using the definition. Main properties of Laplace transform: first and second translation theorems and the transform of the derivative. Inverse Laplace transform. Solution of initial value problems and of differential equations with discontinuous forcing functions, using the Laplace transform. Impulse functions and Dirac δ-function. The convolution theorem..

Mandatory literature

Teresa Arede; Apontamentos de Matemática III, Publicados no SIFEUP nos conteúdos da disciplina
C. Henry Edwards, David E. Penney; Differential Equations. ISBN: 0-13-067337-4
Larson, Hostetler, Edwards; Cálculo, vol 1 e 2, 8ª ed., McGraw-Hill, 2006

Complementary Bibliography

Madureira, Luísa; Problemas de equações diferenciais ordinárias de Laplace. ISBN: 972-752-065-0
Kreyszig, Erwin; Advanced Engineering Mathematics. ISBN: 0-471-33328-X

Teaching methods and learning activities

As the classes are theoretical and practical there will be periods of theoretical exposition of the concepts and techniques with solved examples, followed by periods during which the students should work out, by themselves, proposed problems and exercises.
The therotical exposition will be done at the blackboard or with slide projection.
The students will be given texts containing the theory as well as lists of problems and exercises to solve.

Evaluation Type

Distributed evaluation with final exam

Eligibility for exams

The presence of each student, in the classes, is registered. The student cannot miss more than 25% of the previsible classes. In this unit the maximum of missing classes should be 6.
As some problems are proposed for individual and written solution, the teacher will try to assign a positive or negative information to this piece of work.

Calculation formula of final grade

The evaluation in this unit will be in the following way :

- the student should pass a 1st test, T1, at the middle of the semester, that is compulsory for all;

- if the student pass the 1st test ( with a mark equal or greater than 10 over 20), he can do a 2nd test, T2, that includes only the subjects given after the 1st test; this second test will be at the end of the semester, both tests will have the same weight ;

- if the student didn't pass the 1st test ( its mark being less than 10 over 20) then he should pass a final exam, EF, that wil be at the same time as the second test ; the final exam will have all the given subjects; nevertheless any student is free to choose todo this final exam

- there will be yet a second possibility for the final exam, ER, that can be done by all the students that have not yet pass and can serve also to improv the final mark.

The final mark, CF, will be than obtained by :

CF=(T1 mark+ T2 mark)/2 or CF=EF mark or CF=ER mark

Examinations or Special Assignments

For the students with final mark greater than 18 over 20, there will be an extra oral exam to pass in order they can mantain its classification. If such a student doen't pass the oral exam he can only have at most 18 over 20, on its final mark.

Special assessment (TE, DA, ...)

Following the evaluation rules in this School.

Classification improvement

In the second possibility of the final exam (ER) or following the evaluation rules in this School.