Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4623044 | Journal of Mathematical Analysis and Applications | 2007 | 7 Pages |
Abstract
We consider a second order hyperbolic system of the type(1)Lu=uttâBuxx=f(x,t),(x,t)âTm, where matrix B is a nonsingular constant matrix with positive eigenvalues, (x,t)âR2 and u,fâRn. The set Tm is defined to be(2)Tm={(x,t)|0⩽t⩽1/m,|x|⩽1âmt}, where m=min{μk} and μk2 is any eigenvalue of the matrix B. We will show that, under the condition u(x,0)=0, |x|⩽1, a symmetric Green's function GnÃn can be constructed [K. Kreith, A selfadjoint problem for the wave equation in higher dimensions, Comput. Math. Appl. 21 (5) (1991) 12-132] so that(3)u(x,t)=â«â«TmGnÃn(x,t;ξ,Ï)f(ξ,Ï)dξdÏ for any function fâL2(Tm). This will imply that the operator L in (1) over the set L2(Tm) of functions given by Eq. (3) and u(x,0)=0, |x|⩽1, is selfadjoint. We also note that the same result holds for u in (1), under the condition that ut(x,0)=0, |x|⩽1. We further note that when B has only one eigenvalue μ2, the function u in Eq. (3) satisfies a boundary condition similar to that of Kalmenov [T. Kalmenov, On the spectrum of a selfadjoint problem for the wave equation, Akad. Nauk. Kazakh SSR Vestnik 1 (1983) 63-66] on the characteristic boundaries of Tμ.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Nezam Iraniparast,