Application of residual correction method in calculating upper and lower approximate solutions of fifth-order boundary-value problems

This article attempts to obtain upper and lower approximate solutions of nonlinear fifth-order boundary-value problems by applying the sixth-degree B-spline residual correction method as put forth in the paper. As the first step, a sixth-degree B-spline function is used to discretize and convert a differential equation into mathematical programming problems of an inequation, and then the residual correction concept is applied to simplify such complex calculating problems into problems of equational iteration. The results from validation of the two examples indicate that the required iteration is less than 3 times in both cases. Therefore, such method can help to adequately identify the range of the error of mean approximate solutions in relation to exact solutions, in addition to its effectiveness in obtaining the upper and lower approximate solutions of a fifth-order differential equations accurately and quickly. Hence, it can avoid random addition of grid points for the purpose of increased numerical accuracy.