Mathematical Analysis III

Code:

EC0011

Acronym:

AMAT3

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2010/2011 - 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	284	Syllabus since 2006/2007	2	-	5,5	60	145

Teaching language

Portuguese

Objectives

This course aims to acquaint students with concepts and analytical, numerical and qualitative techniques, which are essential to study the behaviour of engineering problems modulated by differential equations.

Skills to be developed according to CDIO (Conceive- Design- Implement- Operate- an engineering education initiative that was formally founded by Massachusetts Institute of Technology) :
- technical knowledge of underlying sciences (differential equations);
- to know how to deal with new problems and unfamiliar situations in diverse and multidisciplinary contexts;
- To be capable of dealing with complex situations, finding solutions or giving an opinion in situations where information is limited or incomplete;
- To develop competences that lead to a life long learning process in a self oriented and autonomous way;
- To be capable of communicating and presenting knowledge clearly and unambiguously

Program

1- ODEs (Ordinary differential equations): fundamental concepts; formulation and solutions; initial conditions; initial value problem; first order ordinary differential equations; separable equations; exact equations and linear equations; direction fields; orthogonal curves; qualitative analysis; autonomous equations, base lines, bifurcations; uniqueness of solutions; convergence of numerical methods; Euler’s Method and Runge-Kutta method; Applications on Civil Engineering
2- First order ordinary differential equations; Linear systems; Method of eigenvalues; IVPs (Initial Value Problems); Qualitative analysis; Points of equilibrium, phase portraits; Non-linear systems and linearization; global and local behaviour;
3- Second order (and superior) ordinary differential equations; Linear equations with constant coefficients; methods of undetermined coefficients and parameter variation; Application to oscillatory phenomena; singular points;
4- Laplace transforms; Special case of ordinary differential equations with Dirac and Heaviside functions; Application to discontinuous processes;
5- Partial derivative equations; Wave and heat equations; Solution using Fourier series

Mandatory literature

Maria do Carmo Coimbra; Equações diferenciais: uma primeira abordagem, 2009

Complementary Bibliography

Victor G. Sousa; Apontamentos de Análise Matemática 3, Mestrado Integrado em Engenharia Civil, 2009
Colecção de exercícios, AM3, MIEC, 2009
George F. Simmons, Steven G. Krantz ; trad. Helena Maria de Ávila Castro; Equações diferenciais. ISBN: 978-85-86804-64-9
Figueiredo, Djairo; Neves, Aloisio; Equações Diferenciais Aplicadas, IMPA, 2002. ISBN: 85-7028-014-9
Stewart, James 1908-1997; Cálculo. ISBN: 85-211-0484-0

Teaching methods and learning activities

This course is mostly instructive and it has a special focus on mathematical formulation and engineering problems. There is going to be a relation between the essential theoretical knowledge of this course and the other courses of this degree. An intuitive understanding of the concepts, as well as computer skills will be valued. Subjects will be presented in a clear and objective way and examples of physical and geometrical nature will be given. Students will be encouraged to use software (Matlab and Maxima) and calculating machines.

Software

Maxima

keywords

Physical sciences > Mathematics > Mathematical analysis > Differential equations

Evaluation Type

Distributed evaluation with final exam

Assessment Components

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

Eligibility for exams

According to General Evaluation Rules of FEUP, students must attend to 75% of the classes

Calculation formula of final grade

E: grade of the final exam
A: grade of the practical assignment

Final Mark = max { E, 0.8 E + 0.2 A} (*)

(*) Students who do not do the practical assignment will not achieve a higher grade than 16 out of 20.

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