A numerical solution to nonlinear multi-point boundary value problems in the reproducing kernel space

Abstract

In this paper, a new numerical algorithm is provided to solve nonlinear multi-point boundary value problems in a very favorable reproducing kernel space, which satisfies all complex boundary conditions. Its reproducing kernel function is discussed in detail. The theorem proves that the approximate solution and its first-and second-order derivatives all converge uniformly. The numerical experiments show that the algorithm is quite accurate and efficient for solving nonlinear multi-point boundary value problems.

Reproducing Kernel Functions

“…From (14) and (19), the unknown coefficients c i y ðÞand d i y ðÞ i ¼ 1; 2; …; 6 ðÞ can be obtained. Thus, R y is given by…”

A new algorithm based on differential transform method for solving multi-point boundary value problems

“…Also, the same authors [31] proposed a numerical method for solving nonlinear singular fourth-order four-point BVPs by combining the HPM and RKM. Some kinds of the second-order multi-point BVPs have been solved numerically by using the RKM in [19,20,32,33,35]. Also, this method is applied to deal with a class of linear non-local BVPs [34,57].…”

Theoretical and Numerical Aspects of a Third-order Three-point Nonhomogeneous Boundary Value Problem

“…Multi-point boundary value problems there has been attention of several studies mainly focused on the existence of solutions with qualitative and quantitative aspects, we recommend [2,3,4,5,6,7,8,9,10,11,12,14,15] and the references therein. It is well known that the Krasnoselskii's fixed point theorem, Avery-Peterson and Leggett-Williams theorems are massively used in this line.…”