Rapid Convergence for Telegraph Systems with Periodic Boundary Conditions

Received Jul 4, 2017; Revised Oct 7, 2017; Accepted Oct 19, 2017

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1. Introduction

It is well known that the telegraph equation can be used to model many problems such as fluid mechanics, thermodynamics, and elastic mechanics; examples can be found in [1–4]. Recently, the maximum principle for the telegraph equation has been studied by many authors. Ortega and Robles-Perez [5] established a maximum principle for periodic solutions of the telegraph equation (1)utt-uxx+cut+λu=ht,x, t,x∈R2with doubly periodic boundary condition. Mawhin et al. [6] gave a maximum principle for bounded solutions of the telegraph equation which are 2π-periodic with respect to x and bounded over R with respect to t. Li [7] built a maximum principle for time-periodic solutions of the telegraph equation (2)utt-uxx+cut+at,xu=ht,x, t,x∈R2,which are bounded on R for x and 2π-periodic for t. Wang and An [8] gave the maximum principle for the linear telegraph system (3)utt-uxx+c1ut+a11t,xu+a12t,xv=f1t,x,vtt-vxx+c2vt+a21t,xu+a22t,xv=f2t,x,ut+2π,x=ut,x+2π=ut,x,vt+2π,x=vt,x