<abstract xmlns="http://www.w3.org/1999/xhtml">
<p>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.</p>
</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.

Representation of the solution of the M-Sturm-Liouville problem with natural transform

International Journal of Mathematics and Computer in Engineering

This work is licensed under the Creative Commons Attribution 4.0 International License.

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...

Representation of the solution of the M-Sturm-Liouville problem with natural transform

Publié en ligne: 13 sept. 2023

Pages: 243 - 252

Reçu: 09 juin 2023

Accepté: 17 août 2023

DOI: https://doi.org/10.2478/ijmce-2023-0020

Mots clés
M-derivative, Sturm-Liouville problem, natural transform

© 2023 Erdal Bas et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Representation of the solution of the M-Sturm-Liouville problem with natural transform

Publié en ligne: 13 sept. 2023

Pages: 243 - 252

Reçu: 09 juin 2023

Accepté: 17 août 2023

DOI: https://doi.org/10.2478/ijmce-2023-0020

Mots clésM-derivative, Sturm-Liouville problem, natural transform

© 2023 Erdal Bas et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Mots clés
M-derivative, Sturm-Liouville problem, natural transform