On Abu-Shady-Kaabar fractional Chebyshev differential equation of the first kind

Francisco  Martínez; Mohammed K. A.  Kaabar

doi:10.5269/bspm.68585

Francisco Martínez Department of Applied Mathematics and Statistics, Technological University of Cartagena, Cartagena 30203, Spain https://orcid.org/0000-0002-3733-1239
Prof. Mohammed K. A. Kaabar Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, Kuala Lumpur 50603, Malaysia

Abstract

The newly proposed Abu-Shady-Kaabar (A-S-K) fractional derivative, initiated as generalized form of fractional derivative, is studied in this paper. New results are obtained via this newly proposed definition are investigated. The derivability and integrability of the sum function of fractional power series are studied in this context. Likewise, the homogeneous sequential linear A-S-K fractional differential equation of order 2α solutions’ existence around an ordinary point is analysed. The series solutions of the first kind A-S-K fractional Chebyshev differential equation, briefly named in this work as A-S-K Chebyshev-I for simplicity, are obtained. Some interesting properties of the A-S-K Chebyshev-I polynomials are also included. The novelty of this work lies in introducing the A-S-K derivative, which generalizes fractional calculus by integrating a new definition of the fractional derivative through an incremental ratio based on a power law. The A-S-K derivative provides a framework that is differentiable, integrable, and capable of handling complex fractional systems effectively.

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