Boundary shape function method for nonlinear BVP, automatically satisfying prescribed multipoint boundary conditions

Chein-Shan Liu,Chih-Wen Chang

doi:10.1186/s13661-020-01436-y

Abstract

It is difficult to exactly and automatically satisfy nonseparable multipoint boundary conditions by numerical methods. With this in mind, we develop a novel algorithm to find solution for a second-order nonlinear boundary value problem (BVP), which automatically satisfies the multipoint boundary conditions prescribed. A novel concept of boundary shape function (BSF) is introduced, whose existence is proven, and it can satisfy the multipoint boundary conditions a priori. In the BSF, there exists a free function, from which we can develop an iterative algorithm by letting the BSF be the solution of the BVP and the free function be another variable. Hence, the multipoint nonlinear BVP is properly transformed to an initial value problem for the new variable, whose initial conditions are given arbitrarily. The BSF method (BSFM) can find very accurate solution through a few iterations.

Highlights

Many engineering problems can be modeled by ordinary differential equations (ODEs)
The multipoint boundary value problem (BVP) arise when the states of an ODE system are measured at many points, which are important in many areas of engineering applications
According to the new idea of boundary shape function, we have developed an iterative numerical algorithm used in solutions of the second-order nonlinear multipoint BVPs

Summary

Introduction

Many engineering problems can be modeled by ordinary differential equations (ODEs). When they are subjected to prescribed boundary conditions, we encounter the boundary value problems (BVPs), which manifest themselves in many applications, for instance, engineering, control theory, and optimization. Zhou and Xu [10] studied the three-point BVPs for systems of nonlinear second-order ODEs and shown the existence and multiplicity of positive solutions of the above problem by applying the fixed point index theory in cones. The current IVP being obtained exactly by using the variable transformation from u(x) to y(x) is different from the IVP that appeared in the shooting method in two aspects: the governing equation is Eq (16) instead of Eq (1), and the initial conditions y(0) and y (0) are given arbitrarily, not unknown values. The accuracy is very good, which is much better than that computed in [19]

Example 7 Let us consider the following BVP in a large spatial range:

Conclusions