The Solution of Nonlinear Fourth-Order Differential Equation with Integral Boundary Conditions

An iterative algorithm is proposed for solving the solution of a nonlinear fourth-order differential equation with integral boundary conditions. Its approximate solutionu n (x) is represented in the reproducing kernel space. It is proved thatu n (x) converges uniformly to the exact solution u(x). Moreover, the derivatives of u n (x) are also convergent to the derivatives of u(x). Numerical results show that the method employed in the paper is valid.


Introduction
Integral boundary conditions for evolution problems have various applications in chemical engineering, thermoelasticity, underground water flow, and population dynamics.In fact, boundary value problems (BVPs) involving integral boundary conditions have received considerable attention.For BVPs with integral boundary conditions and comments on their importance, we refer the reader to the papers by Gallardo [1], Karakostas and Tsamatos [2], Lomtatidze and Malaguti [3], and the references therein.
In the reproducing kernel spaces, many classical problems such as population models and complex dynamics have been solved [6,7].For more details of the reproducing kernel spaces, we refer the readers to [8][9][10][11].In [8][9][10][11], twopoint BVPs were solved in the reproducing kernel space which satisfied two-point boundary conditions.In this paper, however, we can solve integral boundary problems in the reproducing kernel space which satisfies integral boundary conditions.Reference [12] investigated the existence and multiplicity of symmetric positive solutions for a class of Laplacian fourth-order differential equations with integral boundary conditions.The arguments were based upon a specially constructed cone and the fixed point theory for cones.In [13], the authors are concerned with a new algorithm for giving the analytical and approximate solutions of a class of fourth-order in the new reproducing kernel space.Theorem on the completeness of the system of eigenvector and associated vectors of the operator were proved.Based upon a specially constructed cone and the fixed point theory in a cone, [14] established various results on the existence and nonexistence of symmetric positive solutions to fourth-order boundary value problems with integral boundary conditions.
In the paper, the representation of the exact and approximate solutions of (1) in the reproducing kernel space is given.The advantages of this method are as follows.First, the conditions for determining solution in (1) can be imposed on the reproducing kernel space and therefore the reproducing kernel satisfying the conditions for determining solution can be calculated.We will use the kernel to solve problems.Second, the iterative sequence   () of approximate solutions converges in  4 to the solution ().
This paper is organized as follows.Several reproducing spaces and a linear operator are introduced in Section 2. Section 3 provides the main results; the exact and approximate solutions of (1) and an iterative method are developed for the kind of problems in the reproducing kernel space.We verify that the approximate solution converges to the exact solution uniformly.Some numerical experiments are illustrated in Section 4. Finally, Section 5 is the conclusions.
Proof.(i) The proof of the completeness and reproducing property of  5 [0, 1] is similar to the proof of Theorem 1.3.1 in [15].(ii) Now, let us find out the expression form of the reproducing kernel function  5 (, ) in  5 [0, 1].

An Iterative Algorithm and Its Convergence
In this section, the exact solution of ( 1) is given in the reproducing kernel space  5 [0, 1].
(ii) If ( 20) is nonlinear, that is,  depends on , then the solution of (20) can be obtained by the following iterative method.
Next, we prove that   () in iterative formula ( 28) is convergent to the exact solution of (20).Theorem 7. Suppose the following conditions are satisfied: Then,   () in iterative formula (28) converges to the exact solution () of (20) in  5 [0, 1] and where   are given by (29).
Proof.(i) First, we will prove the convergence of   ().By (28), we have From the orthogonality of From boundedness of ‖  ‖  5 , we have Considering the completeness of  5 [0, 1], there exists () ∈  5 [0, 1], such that   () (ii) Second, we will prove that () is the solution of (20).By Lemma 3 and (i) of Theorem 7, we know that   () converge uniformly to ().It follows that, on taking limits in (28), we have Since it follows that that is, () is the solution of (20) and where   are given by (29).

Numerical Experiment
In this section, the method in the paper will be applied to some numerical examples.All computations are performed in Mathematica 5.0.

Conclusions
In summary, we use an iterative method to find the approximate solution of the nonlinear fourth-order equation with integral boundary conditions in the reproducing kernel space.Using this method, we obtain the sequence which is proved to converge to the exact solution uniformly.
Numerical results show that the method employed in the paper is valid.It is worthy to note that the new method can be used as a very accurate algorithm for solving linear and nonlinear integral boundary problems.

Table 2 :
The root-mean-square errors for the partial derivatives for Example 1.

Table 3 :
The numerical results for Example 2.

Table 4 :
The root-mean-square errors for the partial derivatives for Example 2.