Mathematical Analysis III C
Code
5004
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
José Maria Nunes de Almeida Gonçalves Gomes
Weekly hours
5
Total hours
70
Teaching language
Português
Objectives
We define as main objectives:
1) Learning of basic tools for solving first order differential equations (linear equation, separable equation, exact differential equation) as well as of fundamental theoretical results (Picard’s Existence and Uniqueness Theorem). Knowledge of classical applications and acquisition of modelling skills of certain problems using differential equations.
2) Learning of basic tools for solving higher order differential equations (Variation of constants method, Judicious Guessing Method, Laplace Transform). Knowledge of classical applications of second order differential equations.
3) Learning of basic Fourrier Analysis and its applications to Partial Differential Equations.
Prerequisites
Knowledge of he contents of a first year academic courses on Linear Algebra and Calculus.
Subject matter
1) First-order linear first differential equations. Separable equations. Substitution Methods. Exact differential equations and integrating factors. Picard''''s Existence and Uniqueness Theorem.
2) General facts about higher order linear differential equations. The second order linear equation with constant coefficients. The method of variation of parameters and the method of judicious guessing applied to non-homogeneous second order linear equations. Applications.
3) The Laplace Transform. Definition. Properties. Using Laplace Transforms to solve second order differential equations with constant coefficients.
4) Series solutions methods for solving second order differential equations.
5) Basic Fourrier Analysis and its applications on solving Partial Differential Equations.
Bibliography
Braun, Martin - Differential Equations and their Applications, Springer-Verlag.
Fernandes, Claudio; Marques, João de Deus; Nogueira, Joaquim Eurico - Introdução às Equações Diferenciais ordinárias e com derivadas parciais (gently supplied by the authors.)
Edwards, Jr; Penney, David - Elemetary Differential Equations (with boundary valur Problems), Prentice Hall.
Figueiredo, Djairo - Análise de Fourrier e Equações Diferenciais Parciais, IMPA projeto Euclides.
Teaching method
Teaching Method bases on conferences a n problems solving sessions with the support of a personal attending schedule.
Evaluation method
Important:
In order to be evaluated, the student must attend at least to 2/3 of the problem solving sessions.
Evaluation Methods.
1-Continuous evaluation
The continuos evaluation consists on three tests during the semester. One of the tests may be improved in the final examination date. The final grade is the average of the grades of the three tests. The student is aproved if the final grade is greater or equal than 9,5.
Each test has a duration of 1h 30min.
2-Final exam evaluation.
The student is aproved if the grade of the final exam is greater or equal than 9,5.
The final exam has a duration of 3h.