Modified finite difference method for solving fractional delay differential equations - Behrouz Parsa Moghaddam (Boletim da Sociedade Paranaense de Matemática, 2017)
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.