A generalization of truncated M-fractional derivative and applications to fractional differential equations

• Published in: Applied mathematics and nonlinear sciences Vol. 5; no. 1; pp. 171 - 188
• Main Authors: İlhan, Esin, Kıymaz, İ. Onur
• Format: Journal Article
• Language: English
• Published: Beirut Sciendo 01.01.2020; De Gruyter Poland
• Subjects: 26A33; 33E20; 34A08; alternative fractional derivative; conformable fractional derivative; M-series; Truncated M-fractional derivative
• ISSN: 2444-8656; 2444-8656
• DOI: 10.2478/amns.2020.1.00016

In this paper, our aim is to generalize the truncated M-fractional derivative which was recently introduced [Sousa and de Oliveira, A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties, Inter. of Jour. Analy. and Appl., 16 (1), 83–96, 2018]. To do that, we used generalized M-series, which has a more general form than Mittag-Leffler and hypergeometric functions. We called this generalization as truncated ℳ-series fractional derivative. This new derivative generalizes several fractional derivatives and satisfies important properties of the integer-order derivatives. Finally, we obtain the analytical solutions of some ℳ-series fractional differential equations.