First Step to Spectral Theory with Generalized M Derivative and Applications

In this article, several fundamental spectral results are established for the Sturm–Liouville problem with discrete boundary conditions involving the generalized M-derivative. The paper is organized into four sections. The first section provides a brief historical background of the topic. The second section presents essential definitions and foundational theorems. In the third section, we investigate the uniqueness theorem for the generalized M-derivative Sturm–Liouville boundary value problem on a finite interval and offer two distinct methods for representing the solution. The final section offers a comprehensive evaluation of the study, including a detailed visual analysis using graphical illustrations.