A computational study of multi-term nonlinear fractional delay differential equations with collocation approach

Abstract

In this work, a shifted Legendre orthogonal polynomial based on the collocation method is implemented for obtaining computational solutions of fractional order multi-delay differential equations (FOMDDEs). The proposed numerical technique is applied to both linear and non-linear FOMDDEs. These FOMDDEs encompass multi-term integer and fractional order derivatives in both delayed and non-delayed components, with the fractional derivative regarded in the Caputo sense. The proposed method transforms the entire problem into a system of linear or non-linear algebraic equations, which is then solved numerically. Error analysis and approximation bounds are also discussed. To validate the implemented numerical scheme, various examples from the literature are compared, demonstrating the accuracy of the method even when using only a few terms of the Legendre polynomials. The numerical results show that the shifted Legendre collocation technique (SLCT) is both simple and efficient for solving a wide range of FOMDDEs.