A Class of Spline Functions for Solving 2-Order Linear Differential Equations with Boundary Conditions

Chengzhi Liu ,Xuli Han,Juncheng Li

doi:10.3390/a13090231

Abstract

In this paper, we exploit an numerical method for solving second order differential equations with boundary conditions. Based on the theory of the analytic solution, a series of spline functions are presented to find approximate solutions, and one of them is selected to approximate the solution automatically. Compared with the other methods, we only need to solve a tri-diagonal system, which is much easier to implement. This method has the advantages of high precision and less computational cost. The analysis of local truncation error is also discussed in this paper. At the end, some numerical examples are given to illustrate the effectiveness of the proposed method.

Highlights

Ordinary differential equations (ODEs) of second order are used throughout engineering, mathematics and science
We consider the numerical solutions for second order linear differential equations of the form y00 ( x ) + P( x )y = Q( x ), x ∈ [ a, b], (1)
In [9], Zahra proposed the exponential spline functions consisting of a polynomial part of degree three and an exponential part to find approximation of linear and nonlinear fourth order two-point boundary value problems

Summary

Introduction

Ordinary differential equations (ODEs) of second order are used throughout engineering, mathematics and science. Lodhi and Mishra exploited the numerical method to solve second order singularly perturbed two-point linear and nonlinear boundary value problems [8]. In [9], Zahra proposed the exponential spline functions consisting of a polynomial part of degree three and an exponential part to find approximation of linear and nonlinear fourth order two-point boundary value problems. By using B-spline, Rashidinia and Ghasemi developed a numerical method which can solve the general nonlinear two-point boundary value problems up to order 6 [12]. It is known to all that second order linear differential equations with constant coefficients can be solved analytically. We put forth effort on constructing a spline function based on analytic solutions of equations with constant coefficients

Preliminary

Definition of the Piecewise Spline Functions

Continuity Condition

Truncation Error Estimation

Numerical Examples

Conclusions