Rita Tracinà

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.

Basic Theorems in the Equilibrium Theory of Thermoelasticity with Microtemperatures

In this paper we consider the linear equilibrium theory of thermoelasticity with microtemperatures and some basic results of the classical theories of elasticity and thermoelasticity are generalized. The Greens formulae in the considered theory are obtained. The uniqueness theorems of the internal and external basic boundary value problems (BVPs) are proved. The representation of Galerkin type solution is obtained and the completeness of this solution is established. The formulae of integral representations of regular vector and regular (classical) solutions are obtained. The basic properties of thermoelastopotentials and singular integral operators are presented. Finally, the existence theorems for the internal and external basic BVPs are proved by means of the potential method and the theory of singular integral equations.

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.

Invariants of a family of nonlinear wave equations

Abstract In this work, by using an infinitesimal technique, we find invariants of a family of nonlinear wave equations. The differential invariants are applied to the linearization problem.

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.

Representations of Solutions in the Theory of Thermoelasticity with Microtemperatures for Microstretch Solids

Here, the linear theory of thermoelasticity with microtemperatures for isotropic microstretch solids is considered. First, the representation of Galerkin type solution of equations of motion is obtained. Then, the representation theorem of Galerkin type for the system of equations of steady vibrations is presented. Finally, the representation formula for general solution of the system of homogeneous equations of steady vibrations in terms of 10 metaharmonic functions is established.

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.