Multi-Level General Fractional Derivatives: Fundamental Theorems and Applications
- 1Faculty of Mathematics, K. N. Toosi University of Technology, Tehran 16315-1618, Iran
- 2Institute of Research and Development of Processes IIDP, University of the Basque Country, Campus of Leioa, 48940 Leioa, Bizkaia, Spain
Abstract
General fractional derivatives (GFDs) generalize classical fractional operators by considering the specific kernel functions; however, extending this generalization to a multi-layered structure remains unexplored. Here, we propose a systematic analysis of the nth-level GFDs, which extend the framework developed in earlier studies. We begin by revisiting the first- and second-level GFDs, proceeding to explore the properties of general fractional integrals (GFIs) to provide a higher-level mathematical foundation. Subsequently, we examine illustrative examples and establish the first and second fundamental theorems of fractional calculus (FC) within the nth-level setting. The proof of the first fundamental theorem follows the pattern of the lower-level cases, while the second fundamental theorem is verified by deriving an inductive representation for the null-space of associated kernels. Finally, the paper concludes with an analysis of fractional differential equations (FDEs) employing GFDs.
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BibTeX
@article{ghasemi2026multilevel,
title = {{Multi-Level General Fractional Derivatives: Fundamental Theorems and Applications}},
author = {Mina Ghasemi and Hashem P. Masiha and Manuel De La De La Sen},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14213894},
url = {https://www.mdpi.com/2227-7390/14/21/3894},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Multi-Level General Fractional Derivatives: Fundamental Theorems and Applications
AU - Mina Ghasemi
AU - Hashem P. Masiha
AU - Manuel De La De La Sen
PY - 2026
DA - 2026/10/25
JO - Mathematics
DO - 10.3390/math14213894
UR - https://www.mdpi.com/2227-7390/14/21/3894
AB - General fractional derivatives (GFDs) generalize classical fractional operators by considering the specific kernel functions; however, extending this generalization to a multi-layered structure remains unexplored. Here, we propose a systematic analysis of the nth-level GFDs, which extend the framework developed in earlier studies. We begin by revisiting the first- and second-level GFDs, proceeding to explore the properties of general fractional integrals (GFIs) to provide a higher-level mathematical foundation. Subsequently, we examine illustrative examples and establish the first and second fundamental theorems of fractional calculus (FC) within the nth-level setting. The proof of the first fundamental theorem follows the pattern of the lower-level cases, while the second fundamental theorem is verified by deriving an inductive representation for the null-space of associated kernels. Finally, the paper concludes with an analysis of fractional differential equations (FDEs) employing GFDs.
SN - 2227-7390
ER -