A novel numerical method to solve fractional ordinary differential equations with proportional Caputo derivatives

Abstract

In this paper, we develop a novel numerical scheme, namely 'NPCM-PCDE,' to integrate fractional ordinary differential equations with proportional Caputo derivatives of the type (pc)D(alpha)u(t) = f(1)(t, u(t)), t >= 0, 0 < alpha < 1 involving a non-linear operator f(1). A new method is developed using a natural discretization of the proportional Caputo derivative and the decomposition method to decompose the non-linear operator f(1). The error and stability analyses for the proposed method are provided. Some illustrated examples are given to compare the solution curves graphically with the exact solution and to prove the utility and efficiency of the method. The proposed NPCM-PCDE is found to be efficient, easy to implement, convergent, and stable.