Báo Cáo Báo cáo hóa học EIGENSTRUCTURE OF NONSELFADJOINT COMPLEX DISCRETE VECTOR STURM-LIOUVILLE PROBLEMS ´

EIGENSTRUCTURE OF NONSELFADJOINT COMPLEX DISCRETE VECTOR STURM-LIOUVILLE PROBLEMS ´ RAFAEL J. VILLANUEVA AND LUCAS JODAR Received 15 March 2004 and in revised form 5 June 2004

We present a study of complex discrete vector Sturm-Liouville problems, where coefficients of the difference equation are complex numbers and the strongly coupled boundary conditions are nonselfadjoint. Moreover, eigenstructure, orthogonality, and eigenfunctions expansion are studied. Finally, an example is given.

1. Introduction and motivation Consider the parabolic coupled partial differential system with coupled boundary value conditions ut (x,t) ư Auxx (x,t) = 0, 0 0, t 0, t 0, (1.1) (1.2) (1.3) (1.4) A1 u(0,t) + B1 ux (0,t) = 0, A2 u(1,t) + B2 ux (1,t) = 0, u(x,0) = F(x), 0 ≤ x ≤ 1, where u = (u1 ,u2 , .,um )T , F(x) are vectors in Cm , and A,A1 ,A2 ,B1 ,B2 ∈ Cm×m . We divide the domain [0,1] × [0, ∞[ into equal rectangles of sides ∆x = h and ∆t = l, introduce