An iterative kernel based method for fourth order nonlinear equation with nonlinear boundary condition

作者：Azarnavid Babak^*; Parand Kourosh; Abba**andy Saeid

来源：Communications in Nonlinear Science & Numerical Simulation, 2018, 59: 544-552.

DOI：10.1016/j.cnsns.2017.12.002

• Reproducing kernel Hilbert space
• Shooting technique
• Nonlinear boundary condition
• Error estimate

This article discusses an iterative reproducing kernel method with respect to its effectiveness and capability of solving a fourth-order boundary value problem with nonlinear boundary conditions modeling beams on elastic foundations. Since there is no method of obtaining reproducing kernel which satisfies nonlinear boundary conditions, the standard reproducing kernel methods cannot be used directly to solve boundary value problems with nonlinear boundary conditions as there is no knowledge about the existence and uniqueness of the solution. The aim of this paper is, therefore, to construct an iterative method by the use of a combination of reproducing kernel Hilbert space method and a shooting-like technique to solve the mentioned problems. Error estimation for reproducing kernel Hilbert space methods for nonlinear boundary value problems have yet to be discussed in the literature. In this paper, we present error estimation for the reproducing kernel method to solve nonlinear boundary value problems probably for the first time. Some numerical results are given out to demonstrate the applicability of the method.

• 出版日期2018-6