# Mathematical Analysis III D

## Objectives

- The student should acquire basic skills on the theory of complex variable functions.

- The studentsshould be able to determine the general solution and particular solutions of various types of ordinary differential equations.

- The student is also supposed to learn the essential about Fourier series and their application to the resolution of partial differential equations.

## General characterization

##### Code

7544

##### Credits

6.0

##### Responsible teacher

José Maria Nunes de Almeida Gonçalves Gomes

##### Hours

Weekly - 4

Total - 56

##### Teaching language

Português

### Prerequisites

Background in Linear Algebra and Mathematical Analysis I and II.

### Bibliography

Complex Analysis:

Basic Complex Analysis, J. Marsden and M. Hoffman, Freeman ed., 1987

Complex Analysis, L. Alfhors, McGraw-Hill int. ed., 1979.

Ordinary Differential Equations and Partial Differential Equations:

M. Braun. Differential Equations and their applications (4th edition). Springer-Verlag, 1993.

Análise de Fourier e Equações Diferenciais Parciais, ed. IMPA, projecto Euclides, 1977.

### Teaching method

Theoretical and Practical cllasses.

Students can ask for any questions either in class or in the professor''s office ours.

### Evaluation method

**Assessment Method - Mathematical Analysis III-D**

In accordance with the Regulation of Knowledge Evaluation of the Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa, the discipline of Mathematical Analysis III-D has the following method of evaluation:

Continuous Evaluation:

The continuous assesment of the disciplin consists on two written tests, of 1h30min, quoted up to 20, with grades respectively T1 and T2.

The student is aproved if (T1+T2)/2 is greater or equal than 9.5.

Exam evaluation:

The exam evaluation consists on a 3h00 proof, quoted up to 20, whose content is equivalento to the contiunuou evaluation tests. The student is aproved if the grade N of the exam is greater than 9.5

IMPORTANT:

1) Inscription in tests via clip is mandatory.

2) Accordingly to FCT Regulation, the responsible teacher of the curricular unit may require individually complementary proofs to the evaluation process.

## Subject matter

**1. Functions of a Complex Variable**

1. Complex Functions. Algebra of complex numbers.Definition of the elementary complex functions. Limits and continuity. Differentiability - holomorfic functions. Harmonic functions. Differentiability of the elementary functions. Conformal mappings; fractional linear transformations.

**2. Ordinary Differential Equations (EDO)**

2.1 First Order Differential Equations: Field of directions associated with a first order ODE; integral curves and solutions. Some results of existence and uniqueness of solutions: the theorems of Picard and Peano. Notion of implicit solution of a differential equation. Autonomous equations and equilibrium solutions. Linear equations, separable variable equations and Bernoulli equations. Exact differential equations and notion of integrating factor.

2.2 Second order differential equations. Homogeneous equations: characteristic polynomial and base of the solution vector space. Generalization to homogeneous linear differential equations of order greater than or equal to three. The Wronskian and linear independence of solutions. The structure of the set of solutions of a second order linear ODE. The method of d'' Alembert. Method of the variation of the parameters. Method of indeterminate coefficients. Notion of resonance.

2.3 Systems of linear differential equations of constant coefficients: General and structure of the solutions. Base space vector solution; relationship between the spectrum of the associated linear system and the stability of the solutions.

**3. Partial differential equations (EDP)**

3. 1 Representation of periodic functions in Fourier series . Generalities about periodic functions. Modes sin (2πt / n) and cos (2πt / n); Fourier series associated to a sufficiently regular periodic function (real and complex formalism); Study of the convergence of a Fourier series; discontinuity points and Gibbs phenomenon. Representation of a regular function in series of sines / cosines in a given interval.

3.2 Applications of Fourier series to EDP: General about EDP. The method of separation of variables. Applications to the parabolic case (heat equation), hyperbolic (wave equation) and elliptic (Laplace equation).

3.3 Some examples of application to the study of Navier-Stokes equation.

## Programs

Programs where the course is taught: