An accurate numerical algorithm based on the generalized Narayana polynomials to solve a class of Caputo-Fabrizio and Liouville-Caputo fractional-order delay differential equations

Abstract

A spectral collocation-based approximate algorithm is adapted for numerical evaluation of a class of fractional-order delay differential equations. The involved fractional operators are defined as the Caputo-Fabrizio and Liouville-Caputo derivatives. A novel family of polynomials called Narayana polynomials and their generalization forms are utilized in our collocation procedure. We study the convergent analysis of the Narayana polynomials in a weighted $L_2$ norm and obtain an upper bound for their series expansion form.
The performance of the present matrix collocation is justified through solving three test examples using both fractional operators and various fractional orders. The outcomes are compared with two existing numerical approaches, i.e., the modified operational matrix method (MOMM) and the operational matrix of integration relied on Taylor bases (OMTB). A comparison study reveals that our results have more accuracy than these two methods and thus the presented matrix algorithm is superior in terms of efficacy and applicability. schemes.