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

Abstract

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.

Pulselike Exact Solutions for a Model Describing Nerve Fibers

“…In [15] it was proved that the only auxiliary equations of form (2.5) that are compatible with the hyperbolic system (2.1) are those for which the vector coefficients C i belong to a given subspace of the left eigenvectors of the matrix A, so that, in the present case, C i = l i for i = 1 2 3. Thus, taking…”

“…Our aim here is to develop for the model (2.1)-(2.2) the reduction procedure considered in [15] to which we refer for details. The approach consists of appending a set of auxiliary partial differential equations (constraints) to the governing system of PDEs.…”

“…The approach consists of appending a set of auxiliary partial differential equations (constraints) to the governing system of PDEs. Then, the solutions themselves will be found by solving the entire system of differential equations, consisting of the original system and of the prescribed constraint equations [15,18,19].…”

“…These solutions are obtained by the mathematical approach worked out in [15] for quasilinear hyperbolic systems of PDEs along the lines of the reduction methods based on the introduction of differential constraints [16,17].…”

A reduction approach for determining generalized simple waves

“…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…”