Approximation theory: numerical and geometrical models
Research focus
The Peer review has evaluated this group as Average
Non linear equations systems Several scientific problems require to solve a non linear equation or systems of non linear equations. In recent years, much attention has been given to develop iterative methods, with high order of convergence, which require few function evaluations. By using an integral interpretation of the Newton method, a class of new iterative methods has been constructed. The new methods have, for simple roots, order of convergence three and don't require the second derivative like the classical methods of Halley and Chebyshev do. Two methods, in the new class, have the best efficiency index which is bigger than the ones of Newton, Halley and Chebyshev methods. Volterra, Fredholm and Fredholm-Volterra equations Since the beginning of years ’90 the research group has been involved in activities relevant to the application of approximation techniques based on the spline functional class in order to deal with integral and differential problems. In particular, results have been obtained for Fredholm integral equations, even with singular kernel, and for integral differential Prandtl-like equations simulating special aerodynamic effects. For such models we build and analyze Nystrom like methods based on the quasi-interpolating spline approximation. More recently some applications of the above method are involved in the research for solutions of special mixed Fredholm-Volterra equations also with weakly singular kernel and delayed input. Further work is being carried out on the study of convergence and conditioning of such a mathematical model. In the last four years the spline functional class is used for numerical solution of delay differential equations by means of a local collocation method. Convergence and stability is analyzed and the approximation quality is verified for stiff and rough equations. The previously consolidated techniques have been used for non linear and delay Volterra equations with weakly singular kernel. Applications and a theoretical analysis of this method have been carried out. Dynamical systems The first problem taken under consideration is the D-stability. This notion arises in systems of ordinary equations exhibiting different time scales. and refers to the characherization of a linearly 111 stable equilibrium point. The second problem taken under consideration is the global structure of the solutions of a three-dimensional, autonomous ordinary differential system depending on two parameters. Using graphical, heuristic and rigorous arguments it is shown that, as the parameters vary, a wide range of dynamical behaviors is displayed. The third problem taken under consideration is the numerical study of Hamiltonian systems. Representation of the analytical solution and new approximation methods have been studied. Problems arising in structural engineering The problems taken under consideration are multi-scale delamination, reliability of building materials and the possibly chaotic oscillations of a fixed-end beam. The first two types of problem are studied in collaboration with engineers from the Department of Structural Engineering. Computer graphics A newly developed field still under evolution is one dealing with activity research on synthesis between geometric and computational aspects. Namely, we study and apply geometric proprieties of a class of spline functions with shape parameter, aimed at obtaining free shapes in direct and reverse engineering applications.
Departments
Professors
Full Professors
Franca Calị
Laura Gotusso
Rodolfo Talamo
Associate Professors
Marco Frontini
Elena Marchetti
Raffaella Pavani