Solvability of fractional differential equations with applications of Morgan Voyce polynomials

Abstract

In this research, a new computational approach is presented to address multi-order fractional differential equations, including the renowned Bagley-Torvik and Painlevé equations. These equations are pivotal in scientific and engineering realms, like modelling the movement of a submerged plate restricted in a Newtonian fluid and gas in a fluid, along with simulating the coupled oscillations. We utilise the collocation approach employing a novel operational matrix derived for Morgan-Voyce polynomials via the Atangana-Baleanu fractional derivative. Initially, we introduce the fractional differential matrix to convert the problem and its constraints into a system of algebraic equations with unknown coefficients. These coefficients aid in finding numerical solutions for the given equations. To assess the efficiency of proposed method, various examples are simulated utilising the proposed approach and the outcomes are compared with existing results for different derivatives.