Introduction to Applied Mathematics

Code:

M1027

Acronym:

M1027

Level:

100

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2019/2020 - 2S

Active?	Yes
Web Page:	https://moodle.up.pt/course/view.php?id=481
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:EG	2	The study plan from 2019	3	-	6	56	162
L:M	83	Official Study Plan	1	-	6	56	162

Teaching language

Suitable for English-speaking students

Objectives

Application of mathematical concepts, namely the ones studied in other first-year courses, to the analytical and numerical treatment of mathematical models in Physics, Biology, Ecology, Economics, Medicine and other fields of knowledge.

Models given by recursion relations

Students are expected to treat models with one or more dependent variables, in which the independent variable is time, varying in uniformly spaced increments. These models are given by a recursion relation.
Students are supposed to use the program wxMaxima to compute values and represent them graphically. They should also interpret the calculation results according to the model.

More specifically, students should be able to treat mathematically the following concepts, in the context of a recursion relation:
orbit of a point; fixed point of the relation; attracting fixed point; interpretation of limit behaviour of an orbit.

In the particular case of a single dependent variable, students should represent graphically an orbit using a cobweb. Students should also know the general form of the solution of a dynamical system that is given by an affine map, and should use it to draw conclusions about the orbits.

For recursion relations with more than one dependent variable and given by an affine map, students are expected to use eigenvalues and eigenvectors to obtain dynamical information.

In the treatment of specific models, students should know and treat models for the following situations: compound interest and applications to problems in finance; stratified population models; propagation of infectious diseases.
For other given applications students should be able to formulate an affine model for a specific situation and use it to draw conclusions.
For non- affine models, students should be able to analyse a model using the program wxMaxima.

Curve fitting

Students are expected to estimate the parameters that adjust a model in a given class to a set of data points and to analyse the error incurred. More specifically, it is expected that:

A) given an expression of the form $Y=f_{a,b}(X)$, students should be able to find a change of the variables $X$ and $Y$ that transform $Y=f_{a,b}(X)$ into an affine relation.

For a given table of values for a set of data points, students are expected to:

B) find graphically an affine relation that could serve as model for the data;

C) use the minimum squares method to find the affine relation that best represents the data;

D) use together either the methods A) and B) above, or the methods A) and C) above, to obtain a relation $Y=f_{a,b}(X)$ that could serve as model for the data;

E) compute the error incurred by a model, with respect to data.

Models by continuous-time dynamical systems

Students are expected to treat models where the independent variable is time, varying continuously, and where there is a single dependent variable, i.e. models given by an autonomous first order differential equation $x'=F(x)$.
Students should be able to describe the geometric properties of the solution for a given initial condition, using information from the graph of $F$. They should find equilibrium points and discuss whether they are attractors. Students are also expected to draw the phase portrait indicating the concavity of the graph of the solution. In the particular case where $F$ is an affine function $F(x)=ax+b$ students are expected to find the explicit form of the solution for a given initial condition.


Models with level curves

Students are expected to treat models given by a conservative system with one degree of freedom, to wit, models given by an autonomous second order differential equation of the form $x"=F(x)$.
Students should obtain the expressions of the kinetic and potential energies and use them to obtain the phase portrait in the plane $(x,x')$, indicating the orientation of the solution curves.
In other words, students should obtain the geometric properties of solutions for each initial condition $(x(t_0),x'(t_0))$ using the graph of $F(x)$. They should also find equilibrium points of the equation and describe the phase curves near them.
Students should also be able to use level curves to discuss the geometrical properties of  solutions to the ordinary differential equation model SIR for propagation of contagious diseases that confer immunity. They should also be able to compute maximum numbers of infected population, limit behaviour and provide epidemiological interpretations of the results. 

Learning outcomes and competences

The student should be able to translate the proposed problems in mathematical language, classify them, propose an adequate model and test such model. 

Whenever possible, the student should solve the problem analytically as well as obtaining a graphic representation of it. He should also be capable of using the Maxima software for graphic representation and simulation of solutions to the problem. 

Working method

À distância

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Prerequisites: Real Analysis I, Linear Algebra and Analytic Geometry I, Mathematical Laboratory (or equivalent courses).
Co-requirements: Real Analysis II.

Prerequisites: basic knowledge of:
Differential Calculus of real functions of a real variable;
 Linear Algebra and Analytic Geometry: linear and affine maps, matrices, determinants;
algebraic manipulaton language Maxima.

Co-requirements: basic knowledge of:
Differential Calculus of real functions of several real variables, level curves.

Program

1) Discrete time mathematical modeling with classical examples of application: 

a) modeling in one dimension: discrete dynamical system and its variation, resolution of linear and affine model; fixed points, phase portrait and graph; models in Economics, Biology and Social Sciences; 

b) modeling in dimensions two and three: discrete dynamical system and its variation, resolution of the dynamical system in the linear case; fixed points, phase portrait (in dimension two) and graph; models in Ecology and Epidemiology.

2) Adapting a model to a set of points: transformation into an affine model, graphical method and least square method to determine an affine model. 

3) Continuous time mathematical modeling with classical examples of application: 

a) first order autonomous differential equation (o.d.e.): resolution in the linear and affine case as well cases where an explicit solution can be obtained; continuous time  models in Pharmacy, Physics and Biology; phase portrait of an autonomous o.d.e.: equilibria, monotonicity intervals, concavity; graph of the solutions obtained from the analysis of the phase portrait.

b) conservative systems with one degree of freedom, stability of equilibria, phase portrait on the plane, applications to Physics. 
Other systems with first integrals, SIR model for epidemics.

 

 

Mandatory literature

Giordano Frank R.; A first course in mathematical modeling. ISBN: 978-0-495-55877-4

Complementary Bibliography

Britton Nicholas F.; Essential mathematical biology. ISBN: 1-85233-536-X
Brauer Fred; Mathematical models in population biology and epidemiology. ISBN: 0-387-98902-1
Burghes D. N.; Modelling with differential equations. ISBN: 0-85312-286-5
Arnold V. I.; Equações diferenciais ordinárias
Fred Brauer; Mathematical models in population biology and epidemiology. ISBN: 0-387-98902-1

Teaching methods and learning activities

Up to March 12, 2020
Theoretical classes: exposition of theory and discussion of examples.
Proposed problems to be solved out of class and to be treated in practical classes.
Practical classes, with evaluation: resolution of concrete problems with use of computer and appropriate software for the resolution of problems in class time. Individual discussion of classwork.

Adapted to distance learning due to covid19 epidemics, from March 12, 2020

Online texts with exposition of theory and discussion of examples.
Proposed problems to be solved, with solutions provided some days later.
Online resolution of exercises with evaluation: resolution of problems in applications with use of computer and adequate software for the resolution of problems in prescribed time. The sessions occur in predefined times and are evaluated in dialogue.
Online forum for discussion of difficulties.

Software

Maxima

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	55,00
Trabalho prático ou de projeto	45,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	104,00
Trabalho escrito	2,00
Apresentação/discussão de um trabalho científico	8,00
Trabalho laboratorial	48,00
Total:	162,00

Eligibility for exams

(adapted to distance learning due to covid19 epidemics)

Participation in at least 4 online assessments during term.
Correct solving and adequate discussion, during term, of the practical exercises of at least 4 of the practical modules.

Approval of a practical exercise means that the exercise is completely correct.

New practical exercises of a module may be done as many times as necessary to obtain approval.

Each student only needs to take the number of practical sessions online necessary to obtain approval in each one of the practical modules.

Students that have taken the course last year may apply to have last year's practical exercises considered for this year.

Calculation formula of final grade

(adapted to distance learning due to covid19 epidemics)

1. The final grade will be the sum T+P of the points obtained in two partial components:
T - theoretical assessment in exam: 11 points, minimum 3.5 points.
P - practical assessment using computer, in the practical classes: 9 points (1.5 point per complete and discussed exercise).

2. The grading of component P will only take place during term time (see exception for working students below).

2.The component T is the result of the exam taken through the platform moodle in either the first or in the second call.

4. Exception: if the sum of the points obtained in the components T and P is greater than or equal to 15, then an oral complementary assessment will be required, to take place in a date to fix with the student. The oral assessment will take place by vidoeconference.
The final grade can take any value between 15 and 20 and will depend only on the student's performance in this complementary assessment. If the student fails to take part in the complementary assessment, the final mark will be 15.

Special assessment (TE, DA, ...)

Students that, due to special conditions, are exempted from practical classes may take an online practical exam using the computer to obtain the partial grade P.

Classification improvement

The improvement of final grade for students who took the exam in the first call will be done in the exam in seond call, keeping the results of the practical assessment P.

For students who took the course in the previous year, the improvement will be done in the exam only, keeping the results of the practical assessment of the previous year.
Student intending to improve the practical assessment should enroll in a practical group and do the missing practical exercises.

Students that have failed the course last year may apply to have last year's practical exercises considered for this year.