Lecturer: Piero Colli Franzone   Giancarlo Sangalli  

Course name: Differential Models: Numerical Methods And Applications
Course code: 504011
Degree course: Bioingegneria
Disciplinary field of science: MAT/08
The course relates to:
University credits: ECTS 9
Course website: n.d.

Specific course objectives

The aim of 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.

The second part of the course will be devoted to the introduction of the variation formulation of the stationary problema and to their numerical approximation by the finite element method.

Course programme

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 dimension.

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 development.

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.

FINITE ELEMENT METHOD AND APPLICATIONS
Basic notions of functional analysis. Sobolev spaces. Variational formulation of elliptic problems (Poisson).

Ritz-Galerkin method
Mesh in one and more dimemsions -- Some finite elements -- Approximation properties -- Error estimates for elliptic problems of second order.

MATLAB solver implementation
Solution of the Poisson problem in one dimension. Solution of the Poisson problem in two dimension: assembling the linear system, numerical quadrature, system solving. Mesh refinement.

Course entry requirements

Basic mathematical courses of the "laurea triennale" or " undergraduate degree" and or "bachelor degree"

Course structure and teaching

Lectures (hours/year in lecture theatre): 44
Practical class (hours/year in lecture theatre): 57
Practicals / Workshops (hours/year in lecture theatre): 0

Suggested reading materials

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.

M. Crouzeix, A.L. Mignot. Analyse Numèriques des Èquations Diffèrentielles. Masson, Paris 1984.

A.M. Stuart , A.R. Humphries. Dynamical Systems and Numerical Analysis. Cambridge University Press 1998.

A. Quarteroni, R. Sacco, F. Saleri. Matematica Numerica. Springer 3ra ed., 2008.

Quarteroni A.. Modellistica numerica per problemi differenziali. Springer Verlag, 2009.

Braess D.. Finite Elements. Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press.

Testing and exams

Oral examination with discussion and interpretation of the models simulations developed in the laboratory.