Calculus III

Code:

EQ0068

Acronym:

AM III

Keywords
Classification	Keyword
OFICIAL	Physical Sciences (Mathematics)

Instance: 2013/2014 - 1S (of 01-09-2013 to 31-07-2014)

Active?	Yes
Responsible unit:	Department of Chemical and Biological Engineering
Course/CS Responsible:	Master in Chemical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
MIEQ	93	Syllabus	2	-	6	63	162

Teaching language

Portuguese

Objectives

Background:

The theoretical modeling of transient physical-chemical phenomena is an important component of Chemical Engineering, which is set on the construction and resolution of mathematical models based on differential equations.

Specific aims:

- Acquisition of fundamental knowledge in math, namely analytical and numerical resolution of differential equations.

- Development of competences in the areas of formulation, identification and modelling of engineering problems.

- Development of creative and critical thinking and for solving engineering problems. Previous knowledge Basic knowledge on math analysis and algebra: integration and differentiation, complex numbers, matrix algebra, eigen values and eigen vectors. Basic knowledge on programming: programming structures, variable indexation.

Percentual distribution:

Scientific component: 90 %

Technological component: 10 %

 

Learning outcomes and competences

Capability to solve ordinary differential equations (first and higher orders) using analytical and numerical methods.

Capability to formulate and implement simple mathematical models describing physical-chemical phenomena.

Capability to use computational tools for numerical resolution of differential equations.

Working method

Presencial

Program

I) Analytical resolution od differential equations

1. Introduction: Definitions. Linearity. General solution and particular solution. Existence and unicity of solution.

2. First order ordinary differential equations: Equations with separable variables. Equations reduced to separable variables by change of variable. Integrating factor method. Exact equations. Bernoulli equations. 

3. Second order linear ordinary differential equations. General solution of homogeneous linear equation. D’Alembert’s method (order reduction). Homogeneous equations with constant coefficients. General solution of non-homogeneous linear equation. Method of indeterminated coefficients. Method of parameter variation.

4. Laplace transforms: Definition and properties. Inverse transform. Unit step function (Heaviside) and unit impulse function (Dirac). Application to resolution of linear ordinary differential equations. 

5. Systems of linear ordinary differential equations: Method of elimination. Laplace transform method. Matrix method. Homogeneous systems. Non-homogeneous systems

6. Introduction to partial differential equations.

 

II) Numerical resolution of differential equations

Introduction to Matlab programming. Initial value problems. Euler method. Runge-Kutta methodAnalysis of error in mumerical resolution. Resolution of systems of differential equations. Usage of Matlab's ODE45 function. Boundary value problems. Method of finite differences. Numerical stiffness.

 

Mandatory literature

Farlow, Stanley J.; An introduction to differential equations and their applications. ISBN: 0-07-113315-1
Chapra, Steven C.; Numerical Methods for Engineers. ISBN: 0-07-079984-9

Complementary Bibliography

Kreyszig, Erwin; Advanced Engineering Mathematics. ISBN: 0-471-50729-6

Teaching methods and learning activities

Exposition of theoretical concepts and resolution of practical examples during classes. Interaction with students will be emphasized. The classe dedicated to numerical methods will all take place in computer rooms. Small evaluation tests will take place along the semester, in order to stimulate and monitor the process of aquisition of knowledge and skills by the students along that time period.

Software

Matlab

keywords

Physical sciences > Mathematics > Applied mathematics > Numerical analysis
Physical sciences > Mathematics > Mathematical analysis > Differential equations
Physical sciences > Mathematics > Applied mathematics > Engineering mathematics

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Designation	Weight (%)
Exame	75,00
Teste	25,00
Total:	100,00

Eligibility for exams

Previous attendance is attributed to students who obtained a minimum grade of 6 in distributed evaluation.

Calculation formula of final grade

The final classification (CF) will be computed according to:

CF = max(0.25*AD + 0.75*EF, EF)

Where:

AD (distributed evaluation) = 0.5*(average of the 2 best grades in the 3 analytical math mini-tests + grade in the numerical methods mini-test)

EF = grade in final exam

Conditions for course approval:

• Minimum grade of 6 on distributed evaluation component (AD).
• Minimum grade of 6 on final exam and on the numerical methods component of the final exam.

The mini-tests are mandatory for students without previous attendence of the course. In case of missing a mini-test, its repetition in a later date ill only be possible after evaluation of the properly fundamented  justification.

The distributed evaluation from previous years is not maintained. Performing the mini-tests is optional for students with previous attendance. In case these students intend to perform the mini-tests, they should inform the teacher during the first week of classes, becoming in this way vinculated to teh new distributed evaluation.

Classification improvement

Improvement of classification (distributed evaluation and final exam) can be attempted in the second examination ("recurso"). The computation formula is identical to the one used in final classification described above.