2018/2019

2018/2019

DM270

MAT/08 (NUMERICAL ANALYSIS)

DEPARTMENT OF ELECTRICAL,COMPUTER AND BIOMEDICAL ENGINEERING

BIOENGINEERING

Tecnologie per la salute

1°

1st semester (01/10/2018 - 18/01/2019)

6

56 lesson hours

Italian

WRITTEN AND ORAL TEST

PAVARINO LUCA FRANCO (titolare) - 6 ECTS

Differential and integral calculus for function of many variables, vector calculus, matrices. MATLAB programming

The course is divided in two parts. DYNAMICAL SYSTEMS: theory and numerical methods (6CFU) and FINITE ELEMENT METHOD AND APPLICATIONS (3CFU).

The first part of the course introduces the main concepts related to qualitative and quantitative study of solutions of ordinary differential systems providing the main analytical and numerical methods for the investigation of the dynamics of mathematical models and the critical interpretation of the numerical results.

DYNAMICAL SYSTEMS: theory and numerical methods.
The course is an introduction to the solvability of initial value problem for ordinary differential systems and to the investigation of the qualitative properties of solutions and of equilibrium points with their asymptotic behaviour. The course develops the numerical methods for the numerical simulation of dynamical systems with applications to population dynamics and bistable models.

FINITE ELEMENTS METHODS AND APPLICATIONS
The course introduces the basic notions of the Finite Element Method and its theoretical grounds. Moreover, the practical part of the course will be devoted to the implementation of a MATLAB solver for elliptic problems in two dimensions.

DYNAMICAL SYSTEMS: theory and numerical methods
Basic notion of linear algebra and analysis
Vectorial spaces, matrices, eigenvalues, eigenvectors, linear differential equations, differential and integral calculus, vectorial Taylor series.
Introduction to initial value problems for ordinary differential equations
Local and global solvability, continuous dependence on the initial data, parameters and right hand side perturbations
Asymptotic Stability
Stability of solutions and of equilibrium points. Linear systems. Stability of the linear autonomous systems based on the spectral abscissa. Nonlinear system: linearization. Nonlinear system: Liapunov function. Two dimension linear system and global analysis of the phase plane.
Basic notions of numerical analysis
Polynomial interpolation and remainder terms. Numerical integration: Newton-Cotes formulae and Gausian quadrature. Functional iteration for a system of nonlinear equations: explicit iteration scheme and Newton method.
Numerical methods for ordinary differential systems
One step methods: consistency, zero-stability and convergence. Runge-Kutta methods based on numerical quadratures, Runge-Kutta methods based on collocation methods. Linear multistep methods: consistency, zero-stability and convergence. Adams Bashforth and Moulton methods, Predictor-Corrector methods, backwords differentiation formulae. Estimators of the local discretization error and adative strategy of the time step. Test problems and region of absolute stability. Stiff problems.
Introduction to bifurcation involving fixed points and limit cycles in biological systems.
Analysis and Simulation of dynamical systems: Lotka-Volterra model, FitzHugh-Nagumo model.

Frontal lectures +
programming labs using MATLAB and xppaut

F. Verhulst. Nonlinear differential equations and dynamical systems. Springer-Verlag,Heidelberg, 2006.
R. Mattheij, J. Molenaar. Ordinary differential equations in theory and practice. SIAM, Philadelphia, 2002.
A. Quarteroni, R. Sacco, F. Saleri. Matematica Numerica. Springer 3ra ed., 2008.
M. Crouzeix, A.L. Mignot. Analyse Numeriques des Equations Differentielles. Masson, Paris 1984.
A.M. Stuart , A.R. Humphries. Dynamical Systems and Numerical Analysis. Cambridge University Press 1998.

Written examination with possible oral exam with discussion and interpretation of the models and simulations developed in the course.