A New Generalized Definition of Fractal–Fractional Derivative with Some Applications

• Published in: Mathematical and computational applications Vol. 29; no. 3; p. 31
• Main Authors: Martínez, Francisco, Kaabar, Mohammed K. A.
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.06.2024
• Subjects: Calculus; Derivatives; Differential calculus; Differential equations; Fractals; fractal–fractional derivative in Caputo sense; fractal–fractional differentiation; fractal–fractional integration; fractal–fractional ordinary differential equations; Fractional calculus; Geometry; Integral calculus; Laws, regulations and rules; Ordinary differential equations
• ISSN: 2297-8747; 1300-686X; 2297-8747
• DOI: 10.3390/mca29030031

In this study, a new generalized fractal–fractional (FF) derivative is proposed. By applying this definition to some elementary functions, we show its compatibility with the results of the FF derivative in the Caputo sense with the power law. The main elements of classical differential calculus are introduced in terms of this new derivative. Thus, we establish and demonstrate the basic operations with derivatives, chain rule, mean value theorems with their immediate applications and inverse function’s derivative. We complete the theory of generalized FF calculus by proposing a notion of integration and presenting two important results of integral calculus: the fundamental theorem and Barrow’s rule. Finally, we analytically solve interesting FF ordinary differential equations by applying our proposed definition.