tailieunhanh - EIGENSTRUCTURE OF NONSELFADJOINT COMPLEX DISCRETE VECTOR STURM-LIOUVILLE PROBLEMS ´ RAFAEL J.

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) − Auxx (x,t) = 0, 0 0, t 0, t 0, () () () () A1. | EIGENSTRUCTURE OF NONSELFADJOINT COMPLEX DISCRETE VECTOR STURM-LIOUVILLE PROBLEMS _ z _ 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 - Auxx x t 0 0 x 1 t 0 A1u 0 t B1ux 0 t 0 t 0 A2U 1 t B2Ux 1 t 0 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 e Cmxm. We divide the domain 0 1 x 0 00 into equal rectangles of sides Ax h and At l introduce coordinates of a typical mesh point p kh jl and represent u kh jl U k j . Approximating the partial derivatives appearing in by the forward difference approximations U k j 1 - U k j l U k 1 j - U k j h U k 1 j - 2U k j U k - 1 j h2 Ut k j Ux k j Uxx k j Copyright 2005 Hindawi Publishing Corporation Advances inDifferenceEquations 2005 1 2005 15-29 DOI 16 Nonselfadjoint discrete vector Sturm-Liouville problems takes the form U k j 1 - U k j U k 1 j - 2U k j U k - 1 j --------l----------- A--------------- V------------------ where h 1 N 1 k N - 1 j 0. Let r l h2 and we can write the last equation in the form rA U k 1 j U k - 1 j I - 2rA U k j - U k j 1 0 1 k N - 1 j 0 where I is the identity matrix in Cmxm. Boundary and initial conditions - take the form A1U 0 j NB1 U 1 j - U 0 j 0 j 0 A2U N j NB2 U N j - U N - 1 j 0 j 0 U k 0 F kh 0 k N. Once we discretized problem - we seek solutions of the boundary problem - of the form separation of variables U k j G j H k G j e Cmxm H k e Cm. Substituting U k j given by in .