In 1773 Laplace obtained two fundamental semi-invariants, called Laplace invariants, for scalar linear hyperbolic partial differential equations PDEs in two independent variables. He utilized this in his integration theory for such equations. Recently, Tsaousi and Sophocleous studied semiinvariants for systems of two linear hyperbolic PDEs in two independent variables. Separately, by splitting ...

The relationship between the local temperature and the local heat flux has been established for the homogeneous hyperbolic heat equation. This relationship has been written in the form of a convolution integral involving the modified Bessel functions. The scale analysis of the hyperbolic energy equation has been performed and the dimensionless criterion for the mode of energy transport, similar...

This paper presents an optimal control method for a class of distributed-parameter systems governed by first order, quasilinear hyperbolic partial differential equations that arise in many physical systems. Such systems are characterized by time delays since information is transported from one state to another by wave propagation. A general closed-loop hyperbolic transport model is controlled b...

Rapid processes of heat transfer are not described by the standard heat conduction equation. To take into account a finite velocity of heat transfer, we use the hyperbolic model of heat conduction, which is connected with the relaxation of heat fluxes. In this case, the mathematical model is based on a hyperbolic equation of second order or a system of equations for the temperature and heat flu...

We show how partner symmetries of the elliptic and hyperbolic complex Monge-Ampère equations (CMA and HCMA) provide a lift of non-invariant solutions of threeand two-dimensional reduced equations, i.e., a lift of invariant solutions of the original CMA and HCMA equations, to non-invariant solutions of the latter four-dimensional equations. The lift is applied to non-invariant solutions of the t...

We consider an optimal control problem with indeenite cost for an abstract model which covers in particular hyperbolic and hyperbolic-like systems in a general bounded domain. Necessary and suucient conditions are given for the synthesis of the optimal control, which is given in terms of the Riccati operator arising from a nonstandard Riccati equation. The theory also extends a nite-dimensional...

We construct the traveling wave solutions of the 1 1 -dimensional modified Benjamin-BonaMahony equation, the 2 1 -dimensional typical breaking soliton equation, the 1 1 -dimensional classical Boussinesq equations, and the 2 1 -dimensional Broer-Kaup-Kuperschmidt equations by using an extended G′/G -expansion method, where G satisfies the second-order linear ordinary differential equation. By us...

In this paper we consider numerical approximations of hyperbolic conservation laws in the one-dimensional scalar case, by studying Godunov and van Leer’s methods. Before to present the numerical treatment of hyperbolic conservation laws, a theoretical introduction is given together with the definition of the Riemann problem. Next the numerical schemes are discussed. We also present numerical ex...

We study the asymptotic behavior of a one-dimensional, dynamical model of solid-solid elastic transitions in which the phase is determined by an order parameter. The system is composed of two coupled evolution equations, the mechanical equation of elasticity which is hyperbolic and a parabolic equation in the order parameter. Due to the strong coupling and the lack of smoothing in the hyperboli...

In this paper, the Chebyshev spectral collocation method(CSCM) for one-dimensional linear hyperbolic telegraph equation is presented. Chebyshev spectral collocation method have become very useful in providing highly accurate solutions to partial differential equations. A straightforward implementation of these methods involves the use of spectral differentiation matrices. Firstly, we transform ...