# On mixed steps-collocation schemes for nonlinear fractional delay differential equations

## TL;DR

This paper introduces a new numerical approach combining the method of steps and shifted Legendre collocation for solving nonlinear fractional delay differential equations, with proven convergence and demonstrated accuracy.

## Contribution

It presents a novel formula for fractional derivatives of shifted Legendre polynomials and develops efficient schemes for nonlinear fractional delay equations.

## Key findings

- Schemes effectively solve nonlinear fractional delay differential equations.
- Convergence analysis confirms the reliability of the methods.
- Numerical examples demonstrate high accuracy and efficiency.

## Abstract

This research deals with the numerical solution of non-linear fractional differential equations with delay using the method of steps and shifted Legendre (Chebyshev) collocation method. This article aims to present a new formula for the fractional derivatives (in the Caputo sense) of shifted Legendre polynomials. With the help of this tool and previous work of the authors, efficient numerical schemes for solving nonlinear continuous fractional delay differential equations are proposed. The proposed schemes transform the nonlinear fractional delay differential equations to a non-delay one by employing the method of steps. Then, the approximate solution is expanded in terms of Legendre (Chebyshev) basis. Furthermore, the convergence analysis of the proposed schemes is provided. Several practical model examples are considered to illustrate the efficiency and accuracy of the proposed schemes.