Solving a Class of Singular Fifth-Order Boundary Value Problems Using Reproducing Kernel Hilbert Space Method

Yulan Wang,Fugui Tan,Shuai Lu,Mingjing Du,Hao Yu

doi:10.1155/2013/925192

Abstract

We use the reproducing kernel Hilbert space method to solve the fifth-order boundary value problems. The exact solution to the fifth-order boundary value problems is obtained in reproducing kernel space. The approximate solution is given by using an iterative method and the finite section method. The present method reveals to be more effective and convenient compared with the other methods.

Highlights

The reproducing kernel Hilbert space method has been shown [1,2,3,4,5,6,7] to solve effectively, and accurately a large class of linear and nonlinear, ordinary, partial differential equations
Let gz(x) = (AsRx(s))(z); ‖gz(x)‖2 = (As(AtRs(t))(z))(z), where H[a, b] denotes any reproducing kernel space of functions over [a, b], the symbol As indicates that the operator A applies to functions of the variable s, and the symbol (AsRx(s))(z) indicates that the operator A applies to function Rx(s) of the variable s and s = z
The numerical results demonstrate that the new method is quite accurate and efficient for singular problems of fifth-order ordinary differential equations

Summary

Introduction

The reproducing kernel Hilbert space method has been shown [1,2,3,4,5,6,7] to solve effectively, and accurately a large class of linear and nonlinear, ordinary, partial differential equations. In [8], we give a new reproducing kernel Hilbert space for solving singular linear fourth-order boundary value problems with mixed boundary conditions. We use the new reproducing kernel Hilbert function space method to solve the nonlinear fifthorder boundary value problems. Let us consider the following class of singular fifth-order mixed boundary value problems: u(5). We suppose that the linear conditions can always be homogenized; after homogenization of these conditions, we put these conditions into the reproducing kernel space W26[0, 1] constructed .

Reproducing Kernel Hilbert Space

Analytical Solution

Numerical Solution

Numerical Experiment

Conclusions and Remarks