A numerical solution to nonlinear second order three-point boundary value problems in the reproducing kernel space

https://doi.org/10.1016/j.amc.2011.11.009Get rights and content

Abstract

In this paper, a new numerical algorithm is provided to solve nonlinear three-point boundary value problems in a very favorable reproducing kernel space which satisfies all boundary conditions. Its reproducing kernel function is discussed in detail. We also prove 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 second order three-point boundary value problems.

Introduction

Boundary value problems for nonlinear differential equations arise in the mathematical modeling of viscoelastic and inelastic flows, deformation of beams and plate deflection theory [1], [2], [3]. Multi-point nonlinear boundary value problems, which take into account the boundary data at intermediate points of the interval under consideration, have been addressed by many authors [4], [5], [6], [7], [8]. Despite of the large amount of works which are done on the theoretical aspects of these kind of equations (please see [4], [5], [6], [7], [9], [10], [11] and references therein), few works are available on the numerical analysis. Authors of [12] solved numerically multi-point boundary value problems by using the Sinc-collocation method. In [13], [14], [15], [16], [17], [18], [19], [20], finite difference methods have been proposed for the numerical solution of various nonlocal boundary value problem. Adomian decomposition method [21] and He’s variational iteration method [22] are employed for solving multi-point boundary value problems.

In [7], the existence and uniqueness of solutions of the following second-order three-point boundary value problems have been studied by the monotone iterative methodx(t)+f(t,x)=0,t(0,1),x(0)=0,x(1)=βx(η),where fC(I×R,R),I=[0,1],0<η<1,β>0.

In [8], the authors investigated the numerical solutions of singular second order three-point boundary value problems as follows:a(t)x(t)+b(t)x(t)+c(t)x(t)=f(t,x),t(0,1),x(0)=0,x(1)=αx(η)+γ,where a(0)=0 or a(1)=0 and η(0,1),α>0. Besides, the authors also imposed many restrictive conditions on a(t),b(t) and c(t).

In this work, a numerical method will be given for Eqs. (1.1), (1.2). It is worth mentioning that we only need to assume that a(t),b(t),c(t)L2[0,1] for Eq. (1.2). Without loss of generality, we may put γ=0. The proposed numerical method depends on a new reproducing kernel space. The construction of reproducing kernel space is innovative. Its reproducing kernel function has a lot of good properties that are beneficial to calculation, especially, the new reproducing kernel space satisfies all boundary conditions.

The rest of the paper is organized as follows. A new reproducing kernel space for solving problem Eq. (1.1) is constructed in Section 2; In Section 3, based on the reproducing kernel function and the minimum-point of function, the numerical algorithm is discussed. In Section 4, some numerical examples are shown to verify the effect of our method. Also a conclusion is given in Section 5.

Section snippets

A constructive method for the reproducing kernel space H[0,1]

In order to solve Eq. (1.1), a reproducing kernel space is defined byH[0,1]={x(t)|x(t)is absolutely continuous,x(0)=0,x(1)=βx(η),x(t)L2[0,1].}The inner product and norm of H[0,1] are defined byx(t),y(t)=x(0)y(0)+01x(t)y(t)dt,x(t)=x,x.

Lemma 2.1

The function space H[0,1] is a reproducing kernel space.

The proof can be found in [23]. It is very important to obtain the representation of reproducing kernel function, since it is the base of our algorithm. By Lemma 2.1, there exists a reproducing

Numerical algorithm

In this section, we shall explain how to obtain approximate solution of Eq. (1.1).

Let L:H[0,1]L2[0,1] and (Lx)(t)=x(t). Since operator L is bounded, we can rewrite Eq.(1.1) as(Lx)(t)=f(t,x).Choosing a countable dense subset S={t1,t2,}[0,1] and defining ψi(t) byψi(t)=defLtiRti(t)=2Rti(t)ti2,then {ψi(t)}i=1 is a complete system of the space H[0,1] (see [23]). Furthermore, we obtain an orthogonal basis {ψ˜i(t)}i=1 of H[0,1] by Gram–Schmidt process, such thatψ˜i(t)=k=1iβikψk(t).

Theorem 3.1

If x(t) is

Numerical examples

Here some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method are compared with analytical solution of each example and are found to be in good agreement with each other.

Example 1

Consider the following nonlinear second-order differential equation:x(t)=sinht-2-tx3(t),t(0,1),x(0)=0,x(1)=3x(35)with an exact solution x(t)=1125-t-t2+sinh1-3sinh352+sinht. Applying our method and taking the number of nodes n=10,30,50, the relative errors are shown

Conclusion

In this work, we construct a novel reproducing kernel space and give the way to express the reproducing kernel function. A numerical algorithm is presented based on the properties of reproducing kernel function and the minimum-point of function. Through numerical experiments, we demonstrated the efficiency and superiority of our proposed algorithm.

Acknowledgments

The authors appreciate the constructive comments and suggestions provided from the kind referees and editor. This work was supported by Youth Foundation of Heilongjiang Province under Grant QC2010036 and Fundamental Research Funds for the Central Universities under Grant No. HIT.NSRIF.2009050.

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