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.