Mathematical Analysis III

Code:

EC0011

Acronym:

AMAT3

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2008/2009 - 1S

Active?	Yes
Responsible unit:	Mathematics Division
Course/CS Responsible:	Master in Civil Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
MIEC	273	Syllabus since 2006/2007	2	-	5,5	60	145

Teaching language

Portuguese

Objectives

Introduction of the fundamental concepts for the study of behaviour of ordinary differential equations, using analytical, qualitative and numerical methods. Preparation of formulation engineering problems using ordinary differential equations.

Program

1. First Order Differential Equations.
Differential Equations and Mathematical Models. Integrals as General and Particular Solutions. Slope Fields and Solution Curves. Separable Equations and Applications. Linear First Order Equations. Substitution Methods and Exact Equations.

2. Mathematical Models and Numerical Methods.
Numerical Approximation: Euler's Method. The Runge-Kutta Method.

3. Linear Equations of Higher Order.

Introduction. General Solutions of Linear Equations. Homogeneous Equations with Constant Coefficients. Mechanical Vibrations. Nonhomogeneous Equations and Undetermined Coefficients. Forced Oscillations and Resonance.

4. Introduction to Systems of Differential Equations.
First-Order Systems and Applications. Linear Systems and Matrices. The Eigenvalue Method for Homogeneous Systems. Second Order Systems and Mechanical Applications. Multiple Eigenvalue Solutions. Matrix Exponentials and Linear Systems. Nonhomogeneous Linear Systems.

5. Qualitative Techeniques
Stability and the Phase Plane. Linear and Almost Linear Systems.

Mandatory literature

Maria do Carmo Coimbra; Equações Diferenciais, Uma Primeira Abordagem, 2008

Complementary Bibliography

Edwards, Charles Henry; Differential Equations. ISBN: 0-13-067337-4
Zill, Dennis G.; Equações diferenciais com aplicações em modelagem. ISBN: 85-221-0314-3
Madureira, Luísa; Problemas de equações diferenciais ordinárias de Laplace. ISBN: 972-752-065-0

Teaching methods and learning activities

Formative with special care to mathematical formulation of engineering problems. Fundamental theoretical knowledge coordinated with subjects placed ahead in the course. Enhancement of intuitive knowledge, as well as computational capacities. Frequent use of physical and geometrical examples. Encouragement of the use of software.

keywords

Physical sciences > Mathematics > Mathematical analysis > Differential equations

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Subject Classes	Participação presencial	56,00
	Total:	-	0,00

Eligibility for exams

Limitations are as fixed by the school regulations (Artº 4-nº1).

Calculation formula of final grade

E: classification in the final exam
P: classification in the coursework

Final classification = max(E, 0.80*E + 0.20*P)

Special assessment (TE, DA, ...)

Final Exam

SPECIAL RULES FOR MOBILITY STUDENTS:
Proficiency in Portuguese; Previous attendance of introductory graduate courses in the scientific field addressed in this module; Evaluation by exam and/or coursework(s) defined in accordance with student profile.

Classification improvement

Final Exam

Observations

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Working time estimated out of classes: 3 hours