Numerical Solution via Operational Matrix for Solving Prabhakar Fractional Differential Equations

• Main Authors: Md Nasrudin, Farah Suraya, Chang Phang, Chang Phang
• Format: Article
• Language: English
• Published: Hindawi 2022
• Subjects: T Technology (General)
• Online Access: http://eprints.uthm.edu.my/7295/1/J14305_1eeb3fabff7d91e6857be8589c53a85f%5B1%5D.pdf; http://eprints.uthm.edu.my/7295/; https://doi.org/10.1155/2022/7220433

�e operational matrix method is one of the powerful tools for solving fractional di�erential equations. �is method uses the concept of replacing a symbol with another symbol, i.e., replacing symbol fractional derivative, Dα, with another symbol, which is an operational matrix, Pα. In [1], the authors had derived shifted Legendre operational matrix for solving fractional di�erential equations, de�ned in Caputo sense. �en, researchers started to apply the various types of poly�nomials to derive the operational matrix for solving various types of fractional calculus problems, including Genocchi operational matrix for fractional partial di�erential equations [2], Laguerre polynomials operational matrix for solving fractional di�erential equations with non-singular kernel [3], and Mu¨ntz–Legendre polynomial operational matrix for solving distributed order fractional di�erential equations [4] Recently, apart from the fractional di�erential equation de�ned in Caputo sense, this kind of operational matrix method had been extended to tackle another type of frac�tional derivative or operator, which includes the Caputo–Fabrizio operator [5] and Atangana–Baleanu de�rivative [6, 7]. In this research direction, the operational matrix method is either an operational matrix of derivative or operational matrix of integration based on certain polynomials. �e operational matrix method is possible to apply to another type of fractional derivatives if there is an analytical expression for xp (where p is integer positive) in the sense of certain fractional derivatives or operators. Hence, we extend this operational matrix to tackle operator de�ned by one parameter Mittag–Le�er function, i.e. Antagana–Baleunu derivative [6] to the operator that de- �ned by using three-parameter Mittag–Le�er function, so�called Prabhakar fractional integrals or derivative. In short, we aim to solve the following fractional di�erential equation defined in Prabhakar sense: