# Mathematical Analysis III C

## Objectives

At the end of this unit, students must

- solve by different methods ordinary differential equations (ODE);

- extract qualitative information of ODEs and system of ODEs;

- understand the difference between a linear and a nonlinear equation;

- have basic skills in solving partial differential equations.

## General characterization

##### Code

5004

##### Credits

6.0

##### Responsible teacher

João de Deus Mota Silva Marques

##### Hours

Weekly - 4

Total - 56

##### Teaching language

Português

### Prerequisites

Knowledge of he contents of a first year academic courses on Linear Algebra and Calculus.

### Bibliography

Alves de Sá, A.; Louro, B. - Sucessões e Séries, Escolar Editora, 2ª Edição, 2014.

Apostol, T.M. - Calculus - Volume I e Volume II - Blaidsell Publishing Company.

Boyce, W. E., Diprima, R., - Elementary Differential Equations and Boundary Value Problems, 11ª edição, John Wiley and Sons, Inc., 2017.

Braun, Martin - Differential Equations and their Applications, Springer-Verlag

Freitas, A.C. - Análise Infinitesimal - Volume 1 e Volume 2 - Notas de Lições para alunos do 2º ano das Licenciaturas da FCT.

Howard, Anton - Calculus: A New Horizon -John Wiley and Sons.

Kreysig - Advanced Engineering Mathematics

Taylor, A.E.;-- Man, W.R. - Advanced Calculus - John Wiley and Sons.

Zill, D. G. ; Cullen, M.R. - Differential equations with boundary value problems; 6th edition.

., 2015.

### Teaching method

Teaching Method bases on conferences a n problems solving sessions with the support of a personal attending schedule.

### Evaluation method

**Assessment Method - Mathematical Analysis II-C**

In accordance with the Knowledge Assessment Regulation (RAC) of the Faculty of Science and Technology, Universidade Nova de Lisboa, approved on February 7, 2017, the discipline of Mathematical Analysis II-C has the following method of assessment:

**Attendance**

According to points three and four six of article six of the RAC, attendance will be attributed to students who have attended at least two thirds of the number of actual practical classes. Students who have obtained attendance in the second semester of the academic year 2022/2023 or who have a status that exempts them from attendance are exempt from attendance.

Continuous evaluation

The continuous evaluation of the discipline is carried out through two written tests (theoretical and practical evaluation) with a duration of two hours each. The tests will take place on 08/11 and 14/02 respectively (subjected to confirmation). Each written test will be assigned a classification (t1, t2). Regarding t2, there is a minimum grade requirement of 7.5 points

The final classification of the Continuous Assessment "AC" is calculated by rounding

(t1 + t2) / 2

units, by the usual conventions.

The student is approved by continuous assessment if he obtained frequency according to the rules explained above, if t2 is greater than or equal to 7.5 and if AC = 10.

Exam

Students who fail continuous assessment to whom attendance has been assigned, or have been dismissed from it, may take the exam.

The exam consists of a written test lasting no less than 3 hours covering the entire contents of the course.

The exam will be assigned an entire classification between 0 and 20 values, with the student being approved to the discipline, with that classification, if it is greater than or equal to 10 values.

The use of calculating machines or any other calculation support instruments is prohibited at all times of assessment.

Note Defense

Students with a final classification greater than or equal to 18 values must take a defense test. Failure to take this test leads to a final classification of 17 values in the discipline. The final grade of a student who takes the defense test will never be lower than 17 values.

Rating Improvement

Students approved by continuous assessment may request, upon compliance with all the provisions imposed by FCT-UNL, to improve their classification. In that case, they can take the exam. The final grade will be the maximum between the grades obtained in continuous assessment and in exam.

Conditions for the written tests (tests and exam)

Students who are regularly enrolled in the Clip can attend each of the Continuous Assessment tests and the Exam. Registration must be done at least one week before the test (test or exam). On the day of the test, each student must have a blank exam book which will be handed over at the beginning of the test to the teacher who is conducting the surveillance. During the test, the student must carry an official identification document with a recent photograph.

The General Regulations of FCT-UNL apply in everything that this Regulation does not contain.

## Subject matter

1. Numerical Series

1.1 Convergence of Numerical Series. Required Convergence Condition. Telescopic Series. Geometric series.

1.2 Series of nonnegative terms. Dirichlet series. Comparison Criteria. Criterion of Reason. D''''''''Alembert''''''''s criterion. Root Criterion. Cauchy''''''''s criterion.

1.3 Simple and Absolute Convergence. Alternating Series and Leibniz Criterion.

2. Power Series

2.1 Power Series. Taylor series of analytic functions.

3. Ordinary Differential Equations

3.1 First-order Differential Equations: Field of Directions associated with a first-order EDO; integral field curves and solutions. Some results of Existence and Uniqueness of solutions: the Picard and Peano Theorems. Notion of implicit solution of a differential equation. Autonomous equations and equilibrium solutions. Bernoulli, separable and linear equations. Exact equations and notion of integral factor.

3.2 Second Order Differential Equations. Case of homogeneous equations: characteristic polynomial and base of the vector space solution. Generalization in the case of homogeneous linear differential equations of order n≥3. Wronskian determinant and notion of linear independence of a family of functions; related structure of the solution set of a 2nd order linear ODE. D''''''''Alembert''''''''s method. Method of variation of constants. Method of undetermined coefficients. Notion of resonance.

3.3 Resolution of Ordinary Differential Equations through the use of power series.

3.4 Systems of constant differential linear equations: Generalities and structure of the solutions. Space base vector solution; Relationship between the associated linear system spectrum and the stability of equilibrium solutions.

4. Laplace Transform

4.1 Definition. Laplace transform of the usual functions: Polynomials, exponential and trigonometric functions.

4.2 Effect on Laplace Transform of multiplication by an exponential and a linear function. Laplace transform of the derivative of a function and the translated function.

4.3 Laplace transform of Heaviside function and Dirac distribution.

4.4 Laplace Transform and Convolution. Inverse Laplace Transform.

4.5 Applications for solving linear differential equations.

5. Partial Derivative Equations

5.1 Fourier series decomposition of a periodic function: Generalities about periodic functions; sen (2πt / n) and cos (2πt / n) modes; the Fourier series associated with a sufficiently regular periodic function; sufficient conditions of equality between a function and its Fourier series; points of discontinuity and Gibbs phenomenon. Decomposition of a regular series / sine function in a given interval.

5.2 Fourier Series Applications to EDP: Generalities about EDP; variable separation method. Applications to the parabolic (heat equation), hyperbolic (wave equation) and elliptical (Laplace equation) case.

## Programs

Programs where the course is taught: