Diamond-Type Dirac Dynamic System in Mathematical Physics

Abstract

In order to merge continuous and discrete analyses, a number of dynamic derivative equations have been put out in the process of developing a time-scale calculus. The investigations that incorporated combined dynamic derivatives have led to the proposal of improved approximation expressions for computational application. One such expression is the diamond alpha (& diamond;alpha) derivative, which is defined as a linear combination of delta and nabla derivatives. Several dynamic equations and inequalities, as well as hybrid dynamic behavior-which does not occur in the real line or on discrete time scales-are analyzed using this combined concept. In this study, we consider a & diamond;alpha Dirac system under boundary conditions on a uniform time scale. We examined some basic spectral properties of the problem we are considering, such as the simplicity, the reality of eigenvalues, orthogonality of eigenfunctions, and self adjointness of the operator. Finally, we construct an expression for the eigenfunction of the & diamond;alpha Dirac boundary value problem (BVP) on a uniform time scale.