Riemann invariants-like solutions for a class of rate-type materials

Fusco, D.; Manganaro, N.

doi:10.1007/bf01183939

Cited by 11 publications

(5 citation statements)

References 11 publications

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“…In each particular case, the method of differential constraints utilizes specific features of a corresponding nonlinear system (see Refs. [13], [15], [12], [4], [5], [1], [2], [3]). First, for further convenience we change the sign u → −u in the Constant Astigmatism equation ( 1)…”

Section: Reduction Proceduresmentioning

confidence: 99%

The Constant Astigmatism Equation. New Exact Solution

Manganaro,

Pavlov

2013

Preprint

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Section: Reduction Proceduresmentioning

confidence: 99%

The Constant Astigmatism Equation. New Exact Solution

Manganaro,

Pavlov

2013

Preprint

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“…In most cases the qualitative analyses based upon these techniques show that the effects due to the source-like term B produce a time damping in the process of nonlinear distortion of the wave amplitude whose dynamics is ruled by the differential operator involved in the left hand side of system (1) [4]. Exact wave-like solutions for models belonging to the class (1) were obtained by means of different reduction methods [24], [18], [16], [5], [6], [7], [9], [17], [3], [15], [8]. Whithin such a theoretical framework, of relevant interest in modeling nonlinear wave propagation is developing appropriate approaches for obtaining solutions of (1) in a closed form which generalize the standard simple wave solution ( [12], [20]) of the homogeneous and autonomous system…”

Section: General Appearancementioning

confidence: 99%

A reduction approach for determining generalized simple waves

Fusco

Manganaro

2007

Z. angew. Math. Phys.

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A reduction approach is developed in order to construct generalized simple wave solutions to quasilinear nonhomogenous hyperbolic systems of first order PDEs. The solutions sought must possess a special ansatz which permits time-evolution of the profile of a simple wave due to a source-like term. These solutions involve a free function which can be used to fit classes of initial or boundary value problems. By means of the proposed approach two governing models of interest in a variety of applications are investigated. Model constitutive laws consistent with the full reduction process are obtained and the occurence of singularities at a finite time for the resulting solutions is analysed. Furthermore a comparison is made between the results obtained within the present theoretical framework and the standard simple wave solutions of the corresponding homogeneous (source free) governing models.Several nonlinear wave processes are modeled by quasilinear systems of first order PDEs of the formHere x and t are space and time coordinates, respectively; U ∈ R N is a column vector denoting the dependent field variables; A(U) and B(U) are matrix coefficients. Hereafter, a subscript denotes a derivative with respect to the indicated variable. The system (1) is assumed to be hyperbolic, namely the N -th order matrix A is required to admit N real eigenvalues to which there correspond N right and left eigenvectors spanning the Euclidean space E N . The vector B on the left-hand side of (1) usually represents a "source-like" term and it can model a number of dissipative or dispersive mechanisms which are present in the nonlinear wave dynamics of interest. For the hyperbolic nonhomogeneous system (1) well established approaches permit to study the evolution of the acceleration waves [1], [23] or of the asymptotic

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“…This seems to be relevant in the mathematical modelling of the behaviour of some metals and polymers [3,10,12]. Several papers have been devoted to the investigation of system (4.1-4.3) either within the framework of Lie-group analysis [1,11,14,15] or within reduction procedures [19,26,27]. The eigenvalues (characteristic wave speeds) of the matrix 0 0-1 -10 0 -</ > 0 0. where p and q must be determined by satisfying conditions (2.8-2.10).…”

Section: Rate-type Materialsmentioning

confidence: 99%

“…In a framework more general than that of group-theoretic methods, the key question becomes which kind of constraint equations are admissible by a given system of PDEs. In many cases of interest [17,18,19,20,22,28] the assumption made a priori about the form of the constraints is essentially based on special requirements of the solution that is sought and/or on the possibility of developing an algorithmic method leading to a closed integration of the overdetermined set of differential equations at hand.…”

Section: Introductionmentioning

confidence: 99%

A method for finding exact solutions to hyperbolic systems of first-order PDEs

Fusco

1996

IMA Journal of Applied Mathematics

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Differential constraints are used as a means of developing a systematic method for finding exact solutions to quasilinear nonautonomous hyperbolic systems of first-order partial differential equations (PDEs) involving two independent variables. The leading assumption of the hyperbolicity of the basic system together with a strict compatibility argument permits characterization of the most general class of quasilinear first-order constraint equations, which can be appended to the governing mathematical model under consideration. Furthermore, the exact solutions can be found by integrating the resulting overdetermined and consistent system of PDEs. When the number of auxiliary constraint equations is equal to that of the field equations, the solutions provided by the 'nonclassical' group method for PDEs are recovered; for a very special form of the appended constraint equations simple wave solutions to hyperbolic systems of PDEs are also obtained. Within the present theoretical framework a model describing rate-type media is considered and several classes of initial-and/or boundary-value problems are solved.

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Riemann invariants-like solutions for a class of rate-type materials

Cited by 11 publications

References 11 publications

The Constant Astigmatism Equation. New Exact Solution

The Constant Astigmatism Equation. New Exact Solution

A reduction approach for determining generalized simple waves

A method for finding exact solutions to hyperbolic systems of first-order PDEs

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