Natale Manganaro

A reduction procedure for generalized Riemann problems with application to nonlinear transmission lines

Generalized simple wave solutions to quasilinear hyperbolic nonhomogeneous systems of PDEs are obtained through the differential constraint method. These solutions prove to be flexible enough to solve generalized Riemann problems where discontinuous initial data are involved. Within such a theoretical framework, the governing model of nonlinear transmission lines is investigated throughout.

Reduction Procedure and Generalized Simple Waves for Systems Written in Riemann Variables

In this article, the method of differential constraintsis applied for systems written in Riemann variables. Westudied generalized simple waves. This class of solutions can beobtained by integrating a system of ordinary differentialequations. Two models from continuum mechanics are studied:traffic flow and rate-type models.

Hodograph transformation and differential constraints for wave solutions to 2 ? 2 quasilinear hyperbolic nonhomogeneous systems

The differential constraint method is used to work out a reduction approach to determine solutions in a closed form to the highly nonlinear hodograph system arising from 2 × 2 hyperbolic nonhomogeneous models. These solutions inherit all of the features of the standard wave solutions obtainable via the classical hodograph transformation and in the meantime incorporate the dissipative effects induced on wave processes by the source-like term involved in the governing equations. Within such a theoretical framework the problem of integrating the standard linear hodograph system associated with 2 × 2 homogeneous models is also revisited and a number of results obtained elsewhere of relevant interest in wave problems are recovered as a particular case. Along the lines of the proposed reduction approach, different examples of 2 × 2 governing models are analysed thoroughly in order to highlight the flexibility of the provided solutions to describe hyperbolic dissipative wave processes.

The constant astigmatism equation. New exact solution

In this paper we present a new solution for the constant astigmatism equation. This solution is parameterized by an arbitrary function of a single variable.

Riemann problems and exact solutions to a traffic flow model

Within the theoretical framework of differential constraints method a nonhomogeneous model describing traffic flows is considered. Classes of exact solutions to the governing equations under interest are determined. Furthermore, Riemann problems and generalized Riemann problems which model situations of interest for traffic flows are solved.

Exact description of simple wave interactions in multicomponent chromatography

Nonlinear wave interaction processes for a quasilinear hyperbolic homogeneous system of first-order partial differential equations multicomponent chromatography are investigated. The wave analysis is worked out by extending to the present multicomponent case the leading ideas of a well-established method of approach that was developed for solving initial value-wave problems in terms of exact solutions to 2 × 2 hyperbolic homogeneous systems. These interaction processes may model different situations concerning the separation of a mixture into its chemical components. Several numerical plots are also given in order to illustrate the behavior of the exact wave solutions arising from the analysis that is accomplished.

Wave-Like Solutions for a Class of Parabolic Models

A reduction aproach is developed for determining exact solutions of anonlinear second order parabolic PDE. The method in point makes acomplementary use of the leading ideas of the theory of quasilinearhyperbolic systems of first order endowed by differential constraintsand of the techniques providing multiple wave-like solutions ofnonlinear PDEs. The searched solutions exhibit a inherent wave featuresand they are obtained by solving a consistent overdetermined system ofPDEs. Remarkably, in the process it is possible to define nonlinearmodel equations which allow special classes of initial or boundary valueproblems to be solved in a closed form. Within the present reductionapproach exact solutions and model material response functions areobtained for an equation of widespread application in many fields ofinterest.

Nonlinear wave interaction problems in the three-dimensional case

Three-dimensional nonlinear wave interactions have been analytically described. The procedure under interest can be applied to three-dimensional quasilinear systems of first order, whose hydrodynamic reductions are homogeneous semi-Hamiltonian hydrodynamic type systems (i.e. possess diagonal form and infinitely many conservation laws). The interaction of N waves was studied. In particular we prove that they behave like simple waves and they distort after the collision region. The amount of the distortion can be analytically computed.