Mathematical Analysis III

Code:

EQ205

Acronym:

AM3

Instance: 2003/2004 - 1S

Active?	Yes
Web Page:	http://elearning.fe.up.pt/lrcportal
Responsible unit:	Department of Chemical and Biological Engineering
Course/CS Responsible:	Chemical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
LEQ	105	Plano Oficial EQ a partir 2003	2	3,5	6	-
		Plano para Bachareis EQ a partir de 1996	1	3,5	6	-

Teaching language

Portuguese

Objectives

Familiarize students with the basic techniques for resolution of differential equations. Develop analytical capabilities towards practical problems and to obtain mathematical models for its resolution. Consolidate the knowledge of mathematical analysis and linear algebra acquired in other courses.

Program

1. Introduction to differential equations
Ordinary differential equation
Partial differential equation
Explicit and implicit solutions
Linearity
General and particular solutions
Existence and uniqueness of solution; Picard?s theorem

2. First order differential equations
Separable equations
Reduction to separable equations by change of variable
Integrating factor
Exact equations
Bernoulli equation
Riccati equation

3. Applications of first order differential equations
Radiocative decay
Newton?s second law
Chemical kinetics
Material balances
Other examples

4. Linear homogeneous equations of second order or higher
Second order linear equation
General solution of the linear homogeneous equation
D?Alembert?s method (reduction of order)
Constant coefficients homogeneous equations
Euler equation

5. Linear nonhomogeneous equations of second order or higher
General solution of the linear nonhomogeneous equation
Method of undetermined coefficients
Method of variation of parameters

6. Laplace transform
Definition
Properties
Inverse transform
Application to the resolution of differential equations
Unit step function (Heaviside function)
Unit impulse function (Dirac delta function)
Convolution product

7. Systems of differential equations
Reduction of a linear EDO of order n to a system of n first order linear EDOs
Elimination method
Matricial method
Homogeneous systems
Nonhomogeneous systems
Applications of systems of differential equations

8. Non linear differential equations
Classification of equilibrium points on the phase plane
Stability of equilibrium points
Non linear differential equations
Obtaining the trajectories of a first order autonomous system
Obtaining the trajectories of a second order autonomous EDO
Linearization of a nonlinear system in the neighborhood of equilibrium points

9. Partial differential equations
Classification
Analytical solution
Direct integration
Separation of variables
Laplace transform
Apendix: Fourier series

Main Bibliography

- Stanley J. Farlow, ?An Introduction to Differential Equations and Their Applications?, McGraw-Hill, 1994.
- Notes supplied by the teacher on the course's web page.

Complementary Bibliography

- I.E. Kreyszig, ?Advanced Engineering Mathematics?, J. Wiley.
- M. Spiegel e L. Abellanas, "Manual de Fórmulas e Tabelas de Matemática Aplicada", McGraw-Hill.

Teaching methods and learning activities

Theoretical concepts and practical examples are exposed by the teacher in the theoretical classes. The theoretical-practical classes are for clarifying doubts and discuss practical problems.

Software

MS Excel

Evaluation Type

Distributed evaluation with final exam

Eligibility for exams

Article 4 of FEUP's General Evaluation Rules applies.

Calculation formula of final grade

CF = 0.10 x TC + 0.9 x EF
where:
CF = final classification
TC = homework
EF = final exam

Examinations or Special Assignments

Not applicable

Special assessment (TE, DA, ...)

Exam in the special exam seasons.

Classification improvement

Article 10 of FEUP's General Evaluation Rules applies.