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

Abstract

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.

Introduction

Let us begin with a review of the efforts to solve boundary-value problems (BVPs) for differential equations in the past. In early days, Scott and Watts [1] attempted to calculate linear boundary-value problems through superposition coupled with an orthonormalization procedure and a variable-step Runge–Kuttta–Fehlberg integration scheme. Watson [2] and Watson and Scott [3] further introduced the Chow–Yorke algorithm to solve related problems and verify convergence and effectiveness of these problems. Agarwal [4] carried out researches on existence and uniqueness of solutions to equations of higher-order; Davies et al. [5] made attempts to solve problems concerning viscoelastic flows by using Galerkin methods; Siddiqi and Twizell [6] and Caglar et al. [7] tried to find solutions of such problems by using the sixth-degree B-spline, and concluded that the numerical values obtained through this method and their differential values below the fifth-order are both continuous and very accurate. Recently, Wazwaz [8] and Syam and Atili [9] made attempts to solve these problems through the Adomain decomposition method together with a modified technique, and after having compared this approach with other numerical methods, came to the conclusion that such numerical method can raise computational efficiency effectively.

When we look back to the researches conducted in the past, we may find out that most of the numerical methods, when used to solve higher-order boundary-value problems, always base validation of the error range of calculation results on the known exact solutions of these problems. In practical applications, however, the exact solutions are often unknown to us. Under such circumstances, more grid points or approximation functions are requisite to determine reliability of these results and their error range roughly, which requires a lot of time and energy from us. In light of the above-mentioned situations, this paper suggests a new method “sixth-degree B-spline residual correction method” to calculate maximum and minimum approximate solutions of fifth-order boundary-value problems for differential equations. The steps are as follows: first establish the residual expressions of a differential equation based on the maximum principle for differential equations; then discretize and convert the residual relation into optimized constraint problems for inequations through sixth-degree B-spline; and finally adopt the iterative technique for residual correction and the concept of virtual time (if necessary) to translate originally complex inequational constraint problems into simple problems involving equational iteration. The solutions obtained in such a way cannot only be defined as the upper or lower solutions of the exact solution, but also the value of error between the exact solution and the upper or lower solution can be considered as the maximum possible error of these approximate solutions.