A reduction procedure for generalized Riemann problems with application to nonlinear transmission lines

Abstract

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.

Disturbance rejection in general heterodirectional 1-D linear hyperbolic systems using collocated sensing and control

“…Systems of this type have lately been subject to extensive research due to the vast amount of physical systems that can be modeled this way. Relevant systems are heat exchangers (Xu & Sallet, 2010), transmission lines (Curró, Fusco & Manganaro, 2011), road traffic (Amin, Hante & Bayen, 2008), oil wells (Landet, Pavlov & Aamo, 2013) and multiphase flow (Di Meglio, 2011;Diagne, Diagne & Tang, in press), to mention a few.…”

The constant astigmatism equation. New exact solution

“…Thus, we see that both transformations (x, t, u) → (x (1) , t (1) , u (1) ) and (x, t, u) → (x (−1) , t (−1) , u (−1) ) preserve the class of solutions (see Remark 2 in the previous Section) which cannot be associated with any solutions of the Sine-Gordon equation, i.e. all such iterated solutions are solutions of reduced equation (14).…”

“…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)…”

Observer design for a class of nonlinear system in cascade with counter-convecting transport dynamics

“…The first-order hyperbolic PDEs model a variety of physical systems. Specifically, 2×2 systems of first order hyperbolic linear PDEs model processes such as open channels [7], transmission lines [6], gas flow pipelines [10] or road traffic models [9]. They also have some resemblances with systems that model the gas-liquid flow in oil production pipes [15].…”