A new Bell function approach to solve linear fractional differential equations

In this study, a new collocation approach based on the Bell polynomials is presented to solve linear fractional differential equations. The aim of this study is to find the approximate solutions of linear fractional differential equations in the truncated Bell series. Firstly, fractional Bell functions are defined by generalizing the Bell polynomials with fractional powers. The generalized Bell functions are expressed in matrix form. Secondly, by using the Caputo derivative, the derivatives of the assumed solution form are transformed to the matrix forms. The matrix forms of the desired solution and its derivatives are substituted in the equation. By using equally spaced collocation points, the linear fractional equation is reduced to a system of linear algebraic equations. The obtained system is written in the compact form. The matrix forms of the conditions in the problem are formed by using the matrix form of the desired solution. Hence, a new linear algebraic system is obtained by means of the previous system and the system of the conditions. The obtained last system is solved and thus its solutions give the coefficients of the desired solution. The error and convergence analysis of the method is studied. Lastly, to show the applicability and efficiency of the technique, the method is applied to numerical examples.(c) 2022 IMACS. Published by Elsevier B.V. All rights reserved.