Theory on Linear L-Fractional Differential Equations and a New Mittag–Leffler-Type Function

• Published in: Fractal and fractional Vol. 8; no. 7; p. 411
• Main Author: Jornet, Marc
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.07.2024
• Subjects: Analysis; Banach spaces; Calculus; Continuity (mathematics); Derivatives; Differential calculus; Differential equations; Distribution (Probability theory); Exponential functions; Fields (mathematics); Fractional calculus; Integrals; Leibniz and Caputo fractional operators; linear and sequential linear equations; Linear equations; Linear systems; Mathematical functions; Mittag–Leffler function; non-integer-order differential equation; Operators (mathematics); Ordinary differential equations; Picard’s iterations; Power series; Smoothness
• ISSN: 2504-3110; 2504-3110
• DOI: 10.3390/fractalfract8070411

The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a differential form associated to the system. We develop a theory of this fractional derivative as follows. We prove a fundamental theorem of calculus. We deal with linear systems of autonomous homogeneous parts, which correspond to Caputo linear equations of non-autonomous homogeneous parts. The associated L-fractional integral operator, which is closely related to the beta function and the beta probability distribution, and the estimates for its norm in the Banach space of continuous functions play a key role in the development. The explicit solution is built by means of Picard’s iterations from a Mittag–Leffler-type function that mimics the standard exponential function. In the second part of the paper, we address autonomous linear equations of sequential type. We start with sequential order two and then move to arbitrary order by dealing with a power series. The classical theory of linear ordinary differential equations with constant coefficients is generalized, and we establish an analog of the method of undetermined coefficients. The last part of the paper is concerned with sequential linear equations of analytic coefficients and order two.