Use residual correction method to calculate the upper and lower solutions of initial value problem

This article attempts to establish solutions’ monotonic relation with residuals of initial value problem through the concept of maximum principle for differential equations, then discretizes the equation by the simple spline approximation for initial value problem, converts the once complex inequation constraint mathematical programming problem into a simple problem of equation iteration by applying the new “residual correction method for initial value problem” put forth by this article, and obtains the upper and lower solutions of initial value problem. Results from numerical validation indicate this method is accurate, simple and fast, the approximate solutions obtained can correctly analyze error range and the new method put forth hereby helps increase the accuracy of mean approximate solutions.