Solving nonlinear third-order three-point boundary value problems by boundary shape functions methods

Ji Lin,Yuhui Zhang,Chein-Shan Liu

doi:10.1186/s13662-021-03288-x

Abstract

For nonlinear third-order three-point boundary value problems (BVPs), we develop two algorithms to find solutions, which automatically satisfy the specified three-point boundary conditions. We construct a boundary shape function (BSF), which is designed to automatically satisfy the boundary conditions and can be employed to develop new algorithms by assigning two different roles of free function in the BSF. In the first algorithm, we let the free functions be complete functions and the BSFs be the new bases of the solution, which not only satisfy the boundary conditions automatically, but also can be used to find solution by a collocation technique. In the second algorithm, we let the BSF be the solution of the BVP and the free function be another new variable, such that we can transform the BVP to a corresponding initial value problem for the new variable, whose initial conditions are given arbitrarily and terminal values are determined by iterations; hence, we can quickly find very accurate solution of nonlinear third-order three-point BVP through a few iterations. Numerical examples confirm the performance of the new algorithms.

Highlights

Boundary value problems (BVPs) have a lot of applications, like engineering technique, control theory and optimization, the boundary layer of fluid mechanics, aero-elasticity, sandwich beam analysis and beam deflection theory, electromagnetic waves, theory of thin film and incompressible flows
In the paper we propose new numerical methods for solving the nonlinear thirdorder three-point BVPs, designing the algorithms to automatically satisfy the three-point boundary conditions, which are based on a novel concept of boundary shape function
The non-separated three-point nonlinear BVP is difficult to be treated by the numerical method, the accuracy of the problem we considered is very good

Summary

Introduction

Boundary value problems (BVPs) have a lot of applications, like engineering technique, control theory and optimization, the boundary layer of fluid mechanics, aero-elasticity, sandwich beam analysis and beam deflection theory, electromagnetic waves, theory of thin film and incompressible flows. In the paper we propose new numerical methods for solving the nonlinear thirdorder three-point BVPs, designing the algorithms to automatically satisfy the three-point boundary conditions, which are based on a novel concept of boundary shape function. 4, we develop the first numerical algorithm based on the collocation technique and the trial functions, which are generated from the boundary shape functions by asking the bases of solution to satisfy the three-point boundary conditions automatically. Taking advantage of the new concept of boundary shape function, it is easy to develop the second iterative algorithm to solve the third-order nonlinear BVPs with three-point boundary conditions, where two numerical examples are tested. 7, we extend the idea of boundary shape function to the nonlinear third-order BVP with general non-separated three-point boundary conditions. Will be considered later, where L1, L2, and L3 are linear operators

The boundary shape function

Conclusions