Numerical Solution of Nonlinear System of Ordinary Differential Equations by the Newton-Taylor Polynomial and Extrapolation with Application from a Corona Virus Model

Abstract

In this paper, we consider a nonlinear non autonomous system of differential equations. We linearize this system by the Newton's method and obtain a sequence of linear systems of ODE. We are going to solve this system on [0,Nl] , for some positive integer N and a positive real l>0 . For this purpose, in the first step we solve the problem on [0,l]. By knowing the solution on [0,l], we solve the problem on [l,2l] and obtain the solution on [0,2l]. We continue this procedure until [0,Nl]. In each partial interval [(k-1)l,kl], first of all, we solve the problem by the extrapolation method and obtain an initial guess for the Newton-Taylor polynomial solutions. These procedures cause that the errors don’t propagate. The sequence of linear systems in Newton's method are solved by a famous method called Taylor polynomial solutions, which have a good accuracy for linear systems of ODE. Finally, we give a mathematical model of the novel corona virus disease and illustrate accuracy and applicability of the method by some examples from this model and compare them by similar work, that simulate the numerical solutions.