Piecewice Newton Method For Two Classes Of Nonlinear Equations

Most phenomena in our world are essentially nonlinear and can be described by nonlinear ordinary or partial differential equations. Thus solving nonlinear problems is of great importance for gaining insight into real-world, especially for some engineering problems and physical problems. Usually, it is difficult to obtain the analytic solution for this kind of equations or it is no need to get the analytic solution. So finding the numerical solutions becomes very important and has practical value. In this paper, we have given the numerical solutions of two classes of nonlinear equations.The Newton method is an extremely effective method. Researchers have achieved great success in this respect. In this paper, a new and effective algorithm method named by the piecewise Newton method is proposed and introduced. The method is an improvement of the Newton method for finding the numerical solutions of two classes of nonlinear equations. The basic idea of the proposed methods is that we divide the interval [0,T] into some subintervals equally and apply Newton iterations on every subinterval. The proof of convergence is obtained and the error estimation is also given.In the first part of this paper, the piecewise Newton method is presented for solving nonlinear oscillator equations. It is worth nothing that our method is still effective when the equations are strongly nonlinear problems. Moreover, the previously proposed methods are only for weakly nonlinear problems. In addition, the piecewise Newton method is more effective than the Newton method for long interval. Finally, four examples are given to show the effectiveness of th e method.In the second part of this paper, the piecewise Newton method is presented for solving nonlinear weakly singular Volterra integral equations of the second kind. Our method has a second-order convergence rate. Besides, it is more effective than Newton method. In fact, our method still has a great numerical result when Newton method is divergent. Some numerical examples are given to demonstrate the validity and applicability of the method.