Abstract
Fractional derivative models with an Abdon-Baleanu-Caputo (ABC) fractional deriva-tive with a non-singular Mittag-Leffler kernel in the Liouville-Caputo (LC) sense are investigated using a spectral collocation method based on Legendre approximations. This method reduces the first two models studied to a system of algebraic equations based on the properties of Legendre polynomials, which can then be solved using standard methods, such as Newton iteration. The remaining models are reduced to a system of ordinary differential equations, which are reduced, using finite differences, to a system of algebraic equations, again using properties of Legendre poly-nomials. This is the first time that this method has been used to solve equations in the ABC-sense. We illustrate through numerical results the effectiveness and accuracy of this method based on eval-uating the absolute and the residual error functions. These results clearly demonstrate the superior performance of the method. (C) 2019 The Author. Published by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).