B012965 - NUMERICAL MODELS FOR SIMULATION

Italian

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, Liapunov's functions. Conservative problems. The method of lines, spectrum of a family of matrices, application to partial differential equations of parabolic and hyperbolic type. Applications.

- Professor’s lecture notes.

- L. Brugnano, D. Trigiante. Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordon and Breach Science Publ., 1998.

Knowledge acquired:

Knowledge about both continuous and discrete dynamical systems, and basic numerical methods for differential equations.

Skills acquired (at the end of the course):

Ability to analyze and simulate dynamical sysstems, both continuous and discrete.

Functions of several variables; linear algebra; Matlab programming.

Lectures

Teaching tools:

http://web.math.unifi.it/users/brugnano/Corsi/index.htm

By selecting the academic year, it is possible to download the lecture notes.

Oral exam plus a written homework, on the subjects of the lecture notes. The written homework has to contain the Matlab simulation of an adequate number of models, among those studied in the lectures.

• Difference equations: preliminary notions, the difference and shift operators, factorial powers, particular cases, comparison principle.
• Linear difference equations: general solution, the constant coefficients case, stability of solutions, cobweb model in economy amd model of economy of a nation, linear multistep methods, consistency, zero-stability, and convergence, absolute stability, Dahlquist's barriers.
• Functions of matrices: minimal polynomial, functions of matrices, component matrices, sequences of functions of matrices, analysis through the Jordan canonical form, positive matrices, theorem of Perron-Frobenius.
• Linear systems: linear systems of ordinary differential equations and linear systems of difference equations, model of arms race, stiffness of a linear problem and role of A-stable methods.
• Nonlinear systems: nonlinear systems of difference equations and nonlinear systems of ordinary differential equations, linearization process, Liapunov functions, applications. Generalization of the concept of stiffness for nonlinear problems.

(End of the program, for the course of Approximation Methods).

Conservative problems.
• Polynomials and Toeplitz matrices: Toeplitz banded matrices, spectrum of a family of matrices. Outlines on the solution of partial differential equations.