Improved Reproducing Kernel Methods For Singular Initial Value Problem

Singular differential equations arise in a variety of applied mathematics and physics such as atomic structures, gas dynamics, chemical reactions and studies of nuclear physics. So, this kind of problem has attracted much attention and has been studied by many authors. And it is difficult to acquire analytic expression of this class of equations so that the study of right numerical methods has been an important research topic in theory and application. Therefore, it is very significant how to solve these equations has practical significance.This article introduces the reproducing kernel method, the Runge-Kutta method and the extrapolation method, and handles the second-order singular nonlinear differential equations with initial value conditions by the three methods. The main contents and results are divided into the following sections.Firstly, the research achievement for singular differential equation with initial value conditions is introduced and the development of the reproducing kernel is described in detail.Secondly, the definitions and the basic properties of the reproducing kernel are introduced, the basic theory of the extrapolation method is described in detail.Thirdly, combining extrapolation method with the reproducing kernel method,a numerical method for the solutions of the second-order singular nonlinear differential equations with initial value conditions is posed in the reproducing kernel space. The computational formula and the error estimate of this method are given,and the error estimates are proved. The numerical examples give the comparative analysis of this method.Finally, the Runge-Kutta method and the traditional reproducing kernel method are introduced. Combining the two methods, a new numerical method for the solutions of the second-order singular nonlinear differential equations with initial value conditions is posed. The numerical examples give the comparative analysis of this method.