Domenico Fusco

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 to linear canonical forms and generation of conservation laws for a class of quasilinear hyperbolic systems

Abstract We consider a quasilinear hyperbolic homogeneous system of two first order equations involving two dependent and two independent variables. For the associated hodograph equations we investigate the reducibility to canonical forms allowing for an explicit integration. Such a kind of requirement, in problems of physical interest, provides a suitable method for characterizing possible material model laws. The theoretical approach shown herein can be relevant for studying the existence of conservation laws to non-homogeneous first order systems and also for describing the evolution of weak shock waves.

Group analysis and constitutive laws for fluid filled elastic tubes

Abstract The group properties are investigated for the non-linear mathematical model describing thin walled elastic tubes filled with incompressible fluid. The analysis developed for determining the generators of the group provides a mathematical approach for characterizing classes of constitutive laws for internal area, outflow and viscous retarding force. Thus integration of a system of (non-linear) ordinary differential equations gives rise to several classes of invariant solutions for non-linear arterial flow problems. Some features of these solutions are also discussed.

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.

Complete exceptionality, constitutive laws and symmetric form for a non-linear inelastic rod

Abstract In this paper we consider quasilinear constitutive laws characterizing ‘Maxwellian Materials’ in the case of a one-dimensional inelastic rod. By using the complete exceptionality condition, we are able to determine the most general class of quasilinear constitutive relations by which a weak discontinuity wave propagating along characteristics cannot evolve into a non- linear shock. The last part of the paper is devoted to point out some properties connected with the special class of constitutive relations where the material functions do not depend on the strain. In such a case by means of a field variables transformation, we are able to reduce the fundamental system of equations to a symmetric hyperbolic system where the coefficient of the field spatial derivative is a constant matrix. This special symmetric form is very useful to study shocks.

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.

Wave features and infinitesimal group analysis for a second order quasilinear equation in conservative form

Abstract In this paper we consider a quasilinear second order equation in conservative form. The complete exceptionality condition is used as a vehicle for characterizing classes of constitutive laws by which a weak discontinuity wave cannot evolve into a non-linear shock. As these classes of response functions involve arbitrary functions of the dependent variable, the invariance condition is required further in order to determine completely the functional form of the constitutive laws for the model of interest. The last part of the paper is concerned with relevant examples of second order equations arising from different physical frameworks for which the theory developed herein holds.

A reduction method to quasilinear hyperbolic systems of multicomponent field PDEs with application to wave interaction

Abstract A reduction method is worked out for determining a class of exact solutions with inherent wave features to quasilinear hyperbolic homogeneous systems of N>2 first-order autonomous PDEs. A crucial point of the present approach is that in the process the original set of field equations induces the hyperbolicity of an auxiliary 2×2 subsystem and connection between the respective characteristic velocities can be established. The integration of this auxiliary subsystem via the hodograph method and through the use of the Riemann invariants provides the searched solutions to the full governing system. These solutions also represent invariant solutions associated with groups of translation of space/time coordinates and involving arbitrary functions that can be used for studying non-linear wave interaction. Within such a theoretical framework the two-dimensional motion of an adiabatic fluid is considered. For appropriate model pressure–entropy–density laws, we determine a solution to the governing system of equations which describes in the 2+1 space two non-linear waves which were initiated as plane waves, interact strongly on colliding but emerge with unaffected profile from the interaction region. These model material laws include the classical pressure–entropy–density law which is usually adopted for a polytropic fluid.

Nerve pulse transmission: Recovery variable and rate-type effects

Abstract The classical models of Hodgkin-Huxley or FitzHugh-Nagumo type include recovery variables governed by simple relaxation equations. In this paper an attempt is made to use a rate type equation for describing the dynamics of the recovery variable. The basic dynamics of nerve pulse transmission are governed by an evolution equation while the voltage and the recovery variable are related each other by a rate-type equation which has been widely used in the theory of viscoelasticity for describing similar effects. The mathematical model is discussed and an evolution equation within the context of the wave hierarchy is obtained. The travelling wave solutions are investigated numerically by means of the Runge-Kutta method for different choices of the involved parameters.