# Mathematical Methods in Biomedical Engineering

## Objectives

Students will acquire skills related to various aspects of numerical analysis and complex analysis. They should be

able, among others, to apply numerical techniques in the calculation of integrals in the complex domain, and in

the real domain using complex domain tools.

At the end of this curricular unit, the student will have acquired knowledge that allows skills and competences that

allows him to understand and apply numerical methods in solving mathematical problems, namely: 1.

approximation of functions; 2. approximation of integrals; 3. resolution of non-linear equations; 4. resolution of

systems of linear equations; 5. approximation of the solution of ordinary differential equations with initial condition.

## General characterization

##### Code

12577

##### Credits

6.0

##### Responsible teacher

Jorge Felizardo Dias Cunha Machado

##### Hours

Weekly - 4

Total - 56

##### Teaching language

Português

### Prerequisites

Basic knowledge of Mathematical Analysis aquired in Mathematical Analysis I and II.

### Bibliography

- J. P. Santos e M. F. Laranjeira, Métodos Matemáticos para Físicos e Engenheiros, Fundação da Faculdade de Ciências e Tecnologia, Lisboa, 2004.
- Atkinson K., An Introduction to Numerical Analysis, Wiley, Second Edition, 1989.
- Burden R. e Faires J., Numerical Analysis, Brooks-Cole Publishing Company, 2011.
- Pina H., Métodos Numéricos, Mc Graw Hill, 1995.

### Teaching method

The course is divided in two kinds of lectures: general classes, where the theory and examples are presented and problem-

solving, where specific exercises are worked in full detail.

### Evaluation method

Evaluations consists in two tests. The average is the so

called "avaliação continua". For students who fail in the "avaliação contínua", there is a general exam in the end

of the term ("exame de recurso")

## Subject matter

- Resolution of non-linear equations - Methods of bisection, fixed point and Newton
- Resolution of systems of linear equations - iterative methods: iterative methods: Jacoby, Gauss-Seidel and general case. Vector and matrix norms, conditioning of a system.
- Interpolation and Polynomial Approximation - Interpolation, Lagrange Polynomial, Cubic Splines, Least Square MethodNumeric Integration - Simple and compound Newton-Cotes formulas: midpoint rule, trapezoid rule, Simpson''''s rule, Gauss method
- Numerical resolution of ordinary differential equations with initial condition: Euler, Taylor and Runge-Kutta methods

- Integrals of Complex Functions - Curves or Lines in the Complex Plane, Linear Integrals, Cauchy Theorem, Cauchy-Goursat Theorem, Cauchy Integral Formula, Cauchy Integral Formula for Derivative of Order n, Morera''''s Theorem, Cauchy''''s Inequality, Liouville''''s Theorem
- Residue Theorem - Residues, Calculation of Residues, Residue Theorem, Calculus of Integrals
- Integral Transform - Introduction, Integral Transform Definition, Convolution Definition, Laplace Transform, and Inverse Laplace Transform