Open Access Research

On sampling theories and discontinuous Dirac systems with eigenparameter in the boundary conditions

Mohammed M Tharwat

Author affiliations

Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

Citation and License

Boundary Value Problems 2013, 2013:65  doi:10.1186/1687-2770-2013-65

Published: 29 March 2013

Abstract

The sampling theory says that a function may be determined by its sampled values at some certain points provided the function satisfies some certain conditions. In this paper we consider a Dirac system which contains an eigenparameter appearing linearly in one condition in addition to an internal point of discontinuity. We closely follow the analysis derived by Annaby and Tharwat (J. Appl. Math. Comput. 2010, doi:10.1007/s12190-010-0404-9) to establish the needed relations for the derivations of the sampling theorems including the construction of Green’s matrix as well as the eigen-vector-function expansion theorem. We derive sampling representations for transforms whose kernels are either solutions or Green’s matrix of the problem. In the special case, when our problem is continuous, the obtained results coincide with the corresponding results in Annaby and Tharwat (J. Appl. Math. Comput. 2010, doi:10.1007/s12190-010-0404-9).

MSC: 34L16, 94A20, 65L15.

Keywords:
Dirac systems; transmission conditions; eigenvalue parameter in the boundary conditions; discontinuous boundary value problems