Mariano Torrisi

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.

Self-adjointness and conservation laws of a generalized Burgers equation

A (2 + 1)-dimensional generalized Burgers equation is considered. Having written this equation as a system of two dependent variables, we show that it is quasi self-adjoint and find a nontrivial additional conservation law.

Quasi self-adjoint nonlinear wave equations

In this paper we generalize the classification of self-adjoint second-order linear partial differential equation to a family of nonlinear wave equations with two independent variables. We find a class of quasi self-adjoint nonlinear equations which includes the self-adjoint linear equations as a particular case. The property of a differential equation to be quasi self-adjoint is important, e.g. for constructing conservation laws associated with symmetries of the differential equation.

EQUIVALENCE TRANSFORMATIONS AND SYMMETRIES FOR A HEAT CONDUCTION MODEL

Abstract In the framework of Extended Thermodynamics, we consider the unidimensional case of a non-linear system of partial differential equations describing the heat conduction in a rigid body and removing the well known paradox of thermal pulses propagating with infinite speed. Because of the strong non-linearity of the model we look for an invariant classification via equivalence transformations. Some general results concerned with the equivalence transformations for 2 × 2 quasilinear systems of partial differential equations are shown. Special classes of exact solutions for the system considered are obtained and discussed.

Symmetry classification and optimal systems of a non-linear wave equation

Abstract In this paper the complete Lie group classification of a non-linear wave equation is obtained. Optimal systems and reduced equations are achieved in the case of a hyperelastic homogeneous bar with variable cross section.

Application of weak equivalence transformations to a group analysis of a drift-diffusion model

A group analysis of a class of drift-diffusion systems is performed. In account of the presence of arbitrary constitutive functions, we look for Lie symmetries starting from the weak equivalence transformations. Applications to the transport of charges in semiconductors are presented and a special class of solutions is given for particular doping profiles.

Second-order differential invariants of a family of diffusion equations

An equivalence transformation algebra for a class of nonlinear diffusion equations is found. After having obtained the second-order differential invariants with respect to , we get some results which allow us to linearize a subclass of the equations considered.

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.

Group properties and invariant solutions for infinitesimal transformations of a non-linear wave equation

Abstract This paper concerns the infinitesimal group analysis for a second order non-linear wave equation involving non-homogeneous processes. In the first part we characterize the most general expression for the generator of the Lie group. Then we calculate the invariant surfaces in some special cases, obtaining the corresponding ordinary differential equations whose integration allow us to get classes of solutions for the original equation.

On some differential invariants for a family of diffusion equations

The equivalence transformation algebra LE and some of its differential invariants for the class of equations ut = (h(u)ux)x + f(x, u, ux) (h ? 0) are obtained. Using these invariants, we characterize subclasses which can be mapped by means of an equivalence transformation into the well-studied family of equations vt = (vkvx)x.