Low-cost numerical algorithm to find the series solution of nonlinear fractional differential equations with delay

This paper presents a low-cost numerical algorithm to find the series solution of the nonlinear fractional differential equation containing the delay term D∗αy(x)=f(x,y(x−τ)) subject to the initial conditions y(k)(0)=yn(k)(k=0,1,…,m−1), where α and τ are positive real constants, m=[α], and D∗αis Caputo’s fractional derivative operator. In the proposed method, first the above equation is transformed to the so-called Volterra integral equation and then the trapezoidal and Simpson’s rules are used to find the explicit series solution of the above mentioned equation. One main advantage of the proposed method comparing to other similar algorithms is that it can be applied without the need to the so-called predictor term which makes the proposed algorithm very effective and low-cost. A numerical example is presented to confirm the effectiveness of the proposed method