Numerical and Analytical Investigations of Fractional Self-Adjoint Equations and Fractional Sturm-Liouville Problems via Modified Conformable Operator

This paper introduces a new modified conformable operator and explores its properties in detail. The motivation for studying this operator lies in its potential applications in fractional calculus and differential equations. We analyze the self-adjoint modified conformable equation by discussing the existence and uniqueness solution in the two cases, homogeneous and non-homogeneous, and establish its connection to specific modified conformable initial value problems. Furthermore, we investigate the modified conformable Sturm-Liouville problem by determining its eigenvalues and corresponding eigenfunctions. Key theoretical results related to orthogonality and linear dependence are presented. To validate the theoretical findings, we provide numerical methods and illustrative examples, demonstrating the applicability of our approach. These results contribute to a deeper understanding of modified conformable operators and their role in mathematical physics and engineering.