Antonino Valenti

Preliminary group classification of equations vtt=f(x,vx)vxx+g(x,vx)

A classification is given of equations vtt=f(v,vx)vxx+g(x,vx) admitting an extension by one of the principal Lie algebra of the equation under consideration. The paper is one of few applications of a new algebraic approach to the problem of group classification: the method of preliminary group classification. The result of the work is a wide class of equations summarized in Table II.

A group analysis approach for a nonlinear differential system arising in diffusion phenomena

We consider a class of second‐order partial differential equations which arises in diffusion phenomena and, following a new approach, we look for a Lie invariance classification via equivalence transformations. A class of exact invariant solutions containing an arbitrary function is obtained.

A non-local thermodynamic analysis of second sound propagation in crystalline dielectrics

A thermodynamic model of second sound propagation in rigid solids like dielectric crystals is proposed: this is achieved within !he framework of extended irreversible thermodynamics. The independent variables are the temperature, the heat flux vector plus a supplementary variable that is identified as the flux of the heat flux; to include non-local effects, the constitutive equations are assumed to depend on the gradients of the temperature and the heat flux vector. After establishing the evolution equations governing the behaviour of the basic variables in the course of space and time, the entropy production is calculated and a generalized Gibbs equation is derived. The present model is shown to be rather general as it encompasses the particular models of Cattaneo and Guyer-Krumhansl. Onsager-like reciprocal relations are also displayed and discussed. Working within the lowest-order approximation, a general wave equation for the temperature is derived. This relation is a third-order hyperbolic differential equation with respect to time, allowing for propagation of waves at finite velocity. A dispersion relation between the wavevector and the frequency is established and the corresponding phase velocity is calculated.

Exact solutions for a nonlinear model of dissipative media

Group classification of a mathematical model describing one-dimensional motion in nonlinear dissipative medium is performed. Reduced equations through the optimal system of subalgebras are obtained and some exact solutions are shown.

Heat pulse propagation by second sound in dielectric crystals

A non-linear thermodynamic model describing heat pulse propagation in dielectric crystals at low temperature is proposed. This work is a generalization of those of Cattaneo and of Guyer and Krumhansl, and is complementary to an earlier paper (Lebon G, Torrisi M and Valenti A 1995 J. Phys.: Condens. Matter 7 1461), which was mainly devoted to a linear approach. The model is based on extended irreversible thermodynamics, and uses as field variables the temperature, the heat flux, and the flux of the heat flux. Unlike the simple phonon gas model, the present formalism is compatible with the experimental observation that second sound is temperature dependent. In this paper, explicit expressions for the internal energy and velocity of propagation of weak discontinuities are determined. By making use of experimental data for NaF and Bi, the values of the relevant parameters have been evaluated.

Approximate symmetries in nonlinear viscoelastic media

Approximate symmetries of a mathematical model describing one-dimensional motion in a medium with a small nonlinear viscosity are studied. In a physical application, the approximate solution is calculated making use of the approximate generator of the first-order approximate symmetry.MSC:35J25, 32A37, 43A15, 35A58, 42B20.

On the Linearization of Semilinear Wave Equations

We consider the class of wave equations utt−uxx=f(u, ut, ux). By using the differential invariants, with respect to the equivalence transformation algebra of this class, we characterize subclasses of linearizable equations. Wide classes of general solutions for some nonlinear forms of f(u, ut, ux) are found.

On Equivalence Transformations Applied to a Non-Linear Wave Equation

The equivalence algebra for the non-linear wave equations \( {u_{{xx}}} = {u_{{tt}}} = f(u,{u_t},{u_x}) \) is obtained. Some algorithms are performed in order to extend the principal Lie algebra.

Symmetries and reduction techniques for dissipative models

Symmetries and reduction techniques are applied to a mathematical model describing one-dimensional motions in nonlinear dissipative media. Reduced equations through the optimal system of subalgebras are performed and some exact invariant solutions are shown in a physical application.

Extended thermodynamics revisited : renormalized flux variables and second sound in rigid solids

Propagation of heat waves in rigid bodies is investigated. The originality of the approach is that it rests on a revisited version of extended irreversible thermodynamics. In comparison with earlier developments, two innovations are proposed. First, we depart from the linear approach, best illustrated by Cattaneos relation, to explore the non-linear regime. Second, the extra variables are no longer the usual dissipative fluxes, but renormalized expressions of the fluxes, in order to include the specific material properties of the systems under study. The present model is particularly well suited for studying heat transport at low temperatures in dielectric crystals.