Numerical Integration Methods For A Class Of Implicit Neutral Integro-differential Equations With Singular Kernel

The models of integro-differential equations appear widely in power systems, the heat conduction systems, biological medicine and air elastic mechanics and so on.Although in recent years, more and more scholars have studied the characteristics of integro-differential equations, there is not too much research for integro-differential equations with kernel function, especially for the numerical methods for the integro-differential equations with singular kernel. For the integro-differential equations with singular kernel, it is much more difficult to deal with the numerical solutions as the singularity of their kernel.This paper studies the numerical integrating methods for a class of implicit neutral integro-differential equations with singular kernel. The corresponding numerical format is given, as well as the proof of its convergence. Firstly, this paper introduce the research situation of the integro-differential equations and their applications in reality. Secondly,the numerical format which uses the numerical integrating method to solve a class of linear implicit neutral integro-differential equations with singular kernel is given, and the convergence is proven using the Lagrange theorem and Bernoulli inequality theorem.Thirdly, applying these methods to a class of nonlinear implicit neutral integro-differential equations with singular kernel, present the corresponding numerical format and convergence analysis. In addition, the implicit method is also given for nonlinear implicit neutral integro-differential equations. Furthermore, a comparison between the numerical integrating method and the graded mesh method for solving linear integro-differential equations is made, and the numerical experiments are shown to verify the theoretical results in this paper. Finally, make a summary about this paper, and conduct a discussion about future research.