Modified finite difference method for solving fractional delay differential equations

• Main Authors: Behrouz Parsa Moghaddam, Zeynab Salamat Mostaghim
• Format: Article
• Language: English
• Published: Sociedade Brasileira de Matemática 2017-01-01
• Series: Boletim da Sociedade Paranaense de Matemática
• Subjects: Finite difference method; Caputo derivative; Fractional delay differential equations; Boundary values problems
• Online Access: http://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/25081

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.