DIFFERENTIAL MODELS: NUMERICAL METHODS AND APPLICATIONS
Stampa
Enrollment year
2019/2020
Academic year
2019/2020
Regulations
DM270
Department
DEPARTMENT OF ELECTRICAL,COMPUTER AND BIOMEDICAL ENGINEERING
Course
BIOENGINEERING
Curriculum
Cellule, tessuti e dispositivi
Year of study
Period
1st semester (30/09/2019 - 20/01/2020)
ECTS
9
Language
Italian
Prerequisites
Differential and integral calculus for function of many variables, vector calculus, matrices. MATLAB programming
Learning outcomes
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.
Course contents
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.
Teaching methods
Frontal lectures +
programming labs using MATLAB and xppaut
Reccomended or required readings
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.
Assessment methods
Written examination with possible oral exam with discussion and interpretation of the models and simulations developed in the course.


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