Orthonormal Eigenfunction Expansions for Iterative Boundary Value Problems on Time Scales

Abstract

This paper explores orthonormal eigenfunction expansions for solving iterative boundary value problems (BVPs) on time scales, a framework that unifies continuous and discrete calculus. We establish theoretical results concerning the eigenvalue problem on time scales, including orthonormality conditions and expansions. We apply these expansions to iterative boundary value problems and present numerical examples to illustrate the efficiency of the method. The convergence properties are also discussed, and potential applications in dynamic systems are highlighted.