# 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

Ana Maria de Sousa Alves de Sá, Joaquim Eurico Anes Duarte Nogueira

##### 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

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

DENG, Y. - Lectures, Problems And Solutions For Ordinary Differential Equations, World Scientific, 2017.

NOONBURG, V. W. - Differential Equations: From Calculus to Dynamical Systems, Maa Press, 2019.

PENNEY, D., EDWARDS, C. H., - Elementary Differential Equations with Boundary Value Problems, 5ª edição, Pearson Education, Inc., 2015.

### Teaching method

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

### Evaluation method

Important:

In order to be evaluated, the student must attend at least to 3/4 of the problem solving sessions.

Evaluation Methods.

1-Continuous evaluation

The continuos evaluation consists on three tests during the semester. One of the tests may be improved in the final examination date. The final grade is the average of the grades of the three tests. The student is aproved if the final grade is greater or equal than 9,5.

Each test lasts for 1h 30min.

2-Final exam evaluation.

The student is aproved if the grade of the final exam is greater or equal than 9,5.

The final exam lasts for 3h.

## 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: