Mid-knot cubic non-polynomial spline for a system of second-order boundary value problems

Qinxu Ding,Patricia J Y Wong

doi:10.1186/s13661-018-1075-y

Abstract

In this paper, a mid-knot cubic non-polynomial spline is applied to obtain the numerical solution of a system of second-order boundary value problems. The numerical method is proved to be uniquely solvable and it is of second-order accuracy. We further include three examples to illustrate the accuracy of our method and to compare with other methods in the literature.

Highlights

IntroductionDing and Wong Boundary Value Problems (2018) 2018:156 at mid-knots of a uniform mesh, a second-order interpolation is used to obtain the numerical solutions at the knots
We consider a system of second-order boundary value problems of the type⎧ ⎪⎪⎨f (x), a ≤ x ≤ c, y (x) = ⎪⎪⎩gf ((xx)),y(x) + f (x) + r, c ≤ x ≤ d, d ≤ x ≤ b, (1.1)y(a) = a, y(b) = b, with continuity conditions of y and y at c and d
Thereafter, Al-Said et al [5] show that cubic spline method gives numerical solutions that are more accurate than that computed by quintic spline and finite difference techniques

Summary

Introduction

Ding and Wong Boundary Value Problems (2018) 2018:156 at mid-knots of a uniform mesh, a second-order interpolation is used to obtain the numerical solutions at the knots. Out of all these work, only [26] gives the numerical solutions of the third-order boundary value problem at mid-knots of a uniform mesh while the rest obtains the numerical solutions at the knots. Motivated by the above work especially those involving the use of non-polynomial splines, in this paper we shall develop a cubic non-polynomial spline scheme at mid-knots of a uniform mesh for the problem (1.1). Three examples will be presented to illustrate the numerical efficiency and the better performance over other methods in the literature

Mid-knot cubic non-polynomial spline method

Methods

Conclusion