A novel method to determine the time series of a second-order chemical reaction in the unstable region of the Euler scheme

摘要

A novel method is presented in this paper, to obtain the equilibrium point and the time series of an autonomous first-order differential equation, in a region where the Euler and higher order Runge–Kutta methods fail. The method is illustrated for the chemical reaction that follows second-order kinetics both in the forward and backward direction. The method uses Euler scheme and determines the equilibrium point from the phase plot. Here there is no iteration. The equilibrium point is then used to construct the time series. This is determined from the difference equation. Here, instead of determining the value of xn for a particular tn, the time tn is determined to reach a particular value of xn. This is done iteratively. For an autonomous differential equation, the errors in time are accumulated linearly. The results wherever possible are compared with the Runge–Kutta fourth-order scheme. Agreement is good.