Two New Quadratic Scheme for Fractional Differential Equation with World Population Growth Model

In this paper, we proposed two quadratic convergence numerical technique which are very accurate and fast for finding the approximate solution of an initial value problem (IVP) of the linear as well as non-linear fractional differential equations (FDEs) of arbitrary order ρ,where 0 < ρ ≤ 1. Here, our fractional derivative are taken in the Caputo sense. In this suggested work, we provides the numerical solution which is comparatively faster and accurate than the Euler method (EM), Improved Euler method (IEM), and many other linear, quadratic convergence methods. Also, here we have been found the numerical solution without the help of any kind of linearisation, perturbations or any other assumptions. Illustrated example shows the numerical comparisons in sense of efficiency and accuracy between the our proposed scheme and the exact solution, Euler method and the Improved Euler method of the IVP of FDEs. Our both the proposed scheme Ralston’s method and Ralston’s method of minimum local error bound has quadratic convergence which is more accurate and faster than the Euler method and Improved Euler method, while Improved Euler has also quadratic convergence rate.