On Some Spectral Properties of Discrete Sturm-Liouville Problem

Abstract

Time scale theory helps us to combine differential equations with difference equations. Especially in models such as biology, medicine, and economics, since the independent variable is handled discrete, it requires us to analyze in discrete clusters. In these cases, the difference equations defined in $\mathbb{Z}$ are considered. Boundary value problems (BVP's) are used to solve and model problems in many physical areas. In this study, we examined spectral features of the discrete Sturm-Liouville problem. We have given some examples to make the subject understandable. The discrete Sturm-Liouville problem is solved by using the discrete Laplace transform. In the classical case, the discrete Laplace transform is preferred because it is a very useful method in differential equations and it is thought that the discrete Laplace transform will show similar properties. The other method obtained for the solution of this problem is the solutions obtained according to the states of the characteristic equation and $\lambda$ parameter. In this solution, discrete Wronskian and Cramer methods are used.