Work

Multi-Level General Fractional Derivatives: Fundamental Theorems and Applications

Year: 2026

Type: article

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.

Source: Mathematics

DOI: 10.3390/math14213894

Authors: Mina Ghasemi, Hashem P. Masiha, Manuel De La Sen

Institutions: K. N. Toosi University of Technology, University of the Basque Country

Language: English

Topic: Fractional Differential Equations Solutions

Subfield: Modeling and Simulation

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,137

OpenAlex ID: W7223690521

FWCI:

Cites: 20

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Related to: 10