Modified finite difference method for solving fractional delay differential equations

Abstract

In this paper we present and discuss a new numerical scheme for solving fractional delay differential equations of the general
form:
$$D^{\beta}_{*}y(t)=f(t,y(t),y(t-\tau),D^{\alpha}_{*}y(t),D^{\alpha}_{*}y(t-\tau))$$
on $a\leq t\leq b$,$0<\alpha\leq1$,$1<\beta\leq2$ and under the following interval and boundary conditions:\\
$y(t)=\varphi(t) \qquad\qquad -\tau \leq t \leq a,$\\
$y(b)=\gamma$\\
where $D^{\beta}_{*}y(t)$,$D^{\alpha}_{*}y(t)$ and $D^{\alpha}_{*}y(t-\tau)$ are the standard Caputo fractional derivatives, $\varphi$ is the initial value and $\gamma$ is a smooth function.\\
We also provide this method for solving some scientific models. The obtained results show that the propose method is very
effective and convenient.