Abstract
Strang has discussed stability of difference equations whose solutions satisfy the special boundary condition u=O on and outside of the boundaries. We shall use Toeplitz matrices to generalize this theory to systems of equations in one space variable with arbitrary homogeneous boundary conditions. The discussion will be confined to problems in the quarter plane with constant coefficients. The results can be easily generalized to variable coefficient and two point boundary value problems, using Kreiss' method [1] and/or Strang's in [2] and [3]. We shall derive Kreiss' sufficient conditions for stability of dissipative hyperbolic systems with constant coefficients as a corollary to a more general result. In particular, the condition of dissipativity is replaced by a weaker condition. We treat the explicit case in the main part of this work and add the implicit case as an appendix in part 7. The main results are stated in XIX and XXVIII. Kreiss' Theorem is derived in XXII. We give nondissipative examples in XXIII and XXIX. We hope to extend this technique to include problems in several space variables in the near future.
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