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.
- Dispense del docente.
- L. Brugnano, D. Trigiante. Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordon and Breach Science Publ., 1998.
Learning Objectives
1 Knowledge acquired:
Knowledge about both continuous and discrete dynamical systems, and basic numerical methods for differential equations.
Competence acquired:
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 sysstems, both continuous and discrete.
Teaching Methods
Total hours of the course (including the time spent in attending lectures, seminars, private study, examinations, etc...): 225
Hours reserved to private study and other indivual formative activities: 153
Hours for lectures: 72
Hours for laboratory: 0
Hours for laboratory-field/practice: 0
Seminars (hours): 0
Stages (hours): 0
Intermediate examinations (hours): 0
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 and pacifist's model, 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. Conservative problems.
Polynomials and Toeplitz matrices: Toeplitz banded matrices, spectrum of a family of matrices.