Abstract
In this article, we develop the natural transform in terms of the M-derivative improving the basic notions for a new interesting version of the fractional transform. The properties and relations of certain functions for the natural transform of the M-derivative are introduced. The natural transform with the M-derivative is the more general version of the natural transform for the conformable operator. Furthermore, this method is successfully applied to find the general solutions of some fractional differential equations with M-derivative. We propose a significant spectral data with boundary conditions under M-Sturm-Liouville problem. We offer the representation of the solution for the M-Sturm-Liouville problem, depending on both the boundary and initial conditions. Our main aim is to extract the solution representation of the M-Sturm-Liouville problem by using the natural transform and observe the problem by supporting the spectral structure of the M-Sturm- Liouville problem with graphs. Finally, these results show that our new approach is simple, effective and accurate.