Multistep Collocation Methods For Volterra Integral Equation With Proportional Delay

Integral equation has been widely used in natural science and engineering can be summed up in integral equation, many natural phenomena can be describe by integral equation, at the same time, many differential equations can be transformed into integral equations. Because of the widespread contact characteristics, integral equation theory has been rapid development. The development of many natural phenomena are associated with state of the past, so integral equation and integral equation with delay is often used to describe the law of the system changes. Due to the integral equation of the most difficult to calculate the exact solutions, all modern integral equation numerical analysis cause extensive research.Classic step configuration method is applied to delay integral equation has obtained a series of results. The purpose of this paper is to configure multistep collocation method used in the proportional delay integral equations, in order to get relative to the single step method configuration method higher accuracy. This paper are arranged as following. The first part briefly introduces delay integral equations based on collocation method of the research progress and main results, as well as development present situation, and put forward in this paper, the main train of thought. The second part under the uniform meshes, according to whether the delay item and the current interval will be overlap divided interval into three sections, get equation numerical formats and error equation in each interval, deduce Global convergence order, prove under m collocation point r step, the order can reach m+r, Under orthogonality condition, the order of iterated collocation solution can reach m+r+1. The third part verify the main result of this paper by numerical experiments.