The direct spectral problem via local derivative including truncated Mittag-Leffler function

Abstract

In this article, we propose some substantial spectral data for Sturm–Liouville problem with separated boundary conditions in frame of newly defined truncated M-derivative which contains truncated Mittag-Leffler function. The proposed such spectral data like representation of solution for Sturm-Liouville problem subjected to the both boundary and initial conditions, asymptotic formulas of the eigenfunctions and eigenvalues and their normalized counterparts are implemented in terms of underlying efficient limit-based local derivative. We establish all these mentioned data for two different representation of solution under different conditions. In order to obtain these crucial results, certain tools like integration by parts formula, Leibniz rule, variation of parameters method and so forth are offered in view of the truncated M-derivative. Moreover, uniqueness of the solution of the Sturm–Liouville equation is leadingly showed. Substantially, the main advantage of this potent derivative allowing arbitrary change of order is that it contains truncated Mittag–Leffler function enabling us to deal better with the behavior of the matter in hand and to consider it as a generalized version of other extensive local derivatives in literature owing to the additional parameter inside definition. Thereby the objective of this paper is pivotally dedicated to observe the behavior of the spectral structure of Sturm–Liouville problem by supporting with the simulation analysis.