Course teached as: B012965 - MODELLI NUMERICI PER LA SIMULAZIONE Second Cycle Degree in MATHEMATICS Curriculum APPLICATIVO
Teaching Language
Italian
Course Content
Difference equations, stability of solutions, linear multistep methods for ordinary differential equations. Functions of matrices, sequences of functions of matrices, positive matrices. Linear systems. Nonlinear systems, linearization, Lyapunov functions. Conservative problems. The method of lines, the spectrum of a family of matrices, application to partial differential equations of parabolic and hyperbolic type, and trasnport-diffusion type.
- Lecture notes.
- L. Brugnano, D. Trigiante. Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordon and Breach Science Publ., 1998.
Learning Objectives
Knowledge:
Knowledge about both continuous and discrete dynamical systems, and basic numerical methods for differential equations.
Acquired skills:
Ability to implement on a computer basic numerical methods for differential equations.
Skills acquired at the end of the course:
Ability to analyze and simulate dynamical systems, both continuous and discrete.
Prerequisites
Functions of several variables; matrices; polynomials; basic numerical methods.
Teaching Methods
Total hours of the course: 225
Number of hours for personal study and other individual learning: 153
Number of hours related to activities in the classroom: 72
Further information
Attendance of lectures, practice and lab:
Not mandatory (though recommended)
Supporting tools for learning:
http://web.math.unifi.it/users/brugnano/Corsi/index.htm
Office hours:
Office hours can be consulted at the web page
http://web.math.unifi.it/users/brugnano/Corsi/ORARIO%20RICEVIMENTO.htm
Difference equations: basic facts; the difference and shift operators; factorial powers; particular cases of difference equations; comparisons results.
Linear difference equations: general solution; the case of constant coefficients; stability of solutions; Schur's criteria; applications to economics; basic facts about linear differential equations.
Linear multistep formulae: basic facts; order and consistency conditions; 0-stability and convergence; absolute stability and A-stability; Dahlquist's barriers; boundary locus; L-stability.
Functions of matrices: minimal polynomial; functions of matrices; component matrices; sequences of functions of matrices; Jordan canonical form.
Linear systems: linear systems of ordinary differential equations and difference equations; dynamical systems in the phase-space; numerical solution of differential equations; stability, conditioning, and stiffness; positive matrices and M-matrices; Perron-Frobenius theorem; applications to population models with age structure, arms race model, and to the pagerank of Google.
Nonlinear systems: nonlinear systems of ordinary differential equations and difference equations; linearization; the case of marginal stability; Lyapunov functions; applications; again about stiffness.
From equilibria to strange attractors: prey-predator model; Lorenz' equations; logistic equation (continuous and discrete case).
Runge-Kutta methods: derivation, order and stability conditions (outline); local Fourier expansion; Hamiltonian problems.
Numerical solution of partial differential equations: Toeplitz band matrices; spectrum of a family of matrices; the heath equation; the wave equation; the transport-diffusion equation.
Auxiliary notions of linear algebra: the Kronecker's product and its applications.