Numerical Investigation of Two Fractional Operators for Time Fractional Delay Differential Equation

Abstract

This article compared two high-order numerical schemes for convection-diffusion delay differential equation via two fractional operators with singular kernels. The objective is to present two effective schemes that give (3-alpha)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(3-\alpha )$$\end{document} and second order of accuracy in the time direction when alpha is an element of(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,1)$$\end{document} using Caputo and Modified Atangana-Baleanu Caputo derivatives, respectively. We also implemented a trigonometric spline technique in the space direction, giving second order of accuracy. Moreover, meticulous analysis shows these numerical schemes to be unconditionally stable and convergent. The efficiency and reliability of these schemes are illustrated by numerical experiments. The tabulated results obtained from test examples have also shown the comparison of these operators.