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

Abstract 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.

[1]  Fazhan Geng.  Iterative reproducing kernel method for a beam equation with third-order nonlinear boundary conditions , 2012 .

[2]  To Fu Ma,et al.  Iterative solutions for a beam equation with nonlinear boundary conditions of third order , 2004, Appl. Math. Comput..

[3]  K. Parand,et al.  An iterative reproducing kernel method in Hilbert space for the multi-point boundary value problems , 2018, J. Comput. Appl. Math..

[4]  M. Grossinho,et al.  The dual variational principle and equilibria for a beam resting on a discontinuous nonlinear elastic foundation , 2000 .

[5]  Saeid Abbasbandy,et al.  Some error estimates for the reproducing kernel Hilbert spaces method , 2016, J. Comput. Appl. Math..

[6]  P. Amster,et al.  A shooting method for a nonlinear beam equation , 2008 .

[7]  S. Abbasbandy,et al.  Picard-Reproducing Kernel Hilbert Space Method for Solving Generalized Singular Nonlinear Lane-Emden Type Equations , 2015 .

[8]  Ravi P. Agarwal,et al.  On fourth order boundary value problems arising in beam analysis , 1989, Differential and Integral Equations.

[9]  Chiping Zhang,et al.  Numerical solution of nonlinear three‐point boundary value problem on the positive half‐line , 2012 .

[10]  Minggen Cui,et al.  A numerical solution to nonlinear second order three-point boundary value problems in the reproducing kernel space , 2012, Appl. Math. Comput..

[11]  Fazhan Geng,et al.  A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems , 2009, Appl. Math. Comput..

[12]  M. L. Pelicer,et al.  Monotone positive solutions for a fourth order equation with nonlinear boundary conditions , 2009 .

[13]  Xiaoqin Zhang,et al.  Existence and uniqueness of monotone positive solutions for an elastic beam equation with nonlinear boundary conditions , 2012, Comput. Math. Appl..

[14]  Minggen Cui,et al.  Nonlinear Numerical Analysis in Reproducing Kernel Space , 2009 .

[15]  Fazhan Geng,et al.  SOLVING SINGULAR NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEMS IN THE REPRODUCING KERNEL SPACE , 2008 .

[16]  Ping Li,et al.  Numerical Algorithm for the Third-Order Partial Differential Equation with Three-Point Boundary Value Problem , 2014 .

[17]  Saeid Abbasbandy,et al.  A shooting reproducing kernel Hilbert space method for multiple solutions of nonlinear boundary value problems , 2015, J. Comput. Appl. Math..

[18]  Chun-li Li,et al.  The exact solution for solving a class nonlinear operator equations in the reproducing kernel space , 2003, Appl. Math. Comput..