Piecewise Picard iteration method for solving nonlinear fractional differential equation with proportional delays

Abstract

In this paper, a numerical method for solving a class of nonlinear fractional differential equation with proportional delays is proposed. In order to overcome the strongly nonlinear case, we propose the piecewise Picard iteration method(PPIM). The convergence proof and error estimations of the Picard and the PPIM are obtained. Meanwhile, a sufficient condition for the stability of the PPIM is also given. Some numerical examples confirm the validity of the PPIM. It’s worth noting that the PPIM is quite effective for solving linear, weakly nonlinear and some strongly nonlinear fractional differential equations with proportional delays.