Abstract
An iterative algorithm is proposed for solving the solution of a nonlinear fourth-order differential equation with integral boundary conditions. Its approximate solution is represented in the reproducing kernel space. It is proved that converges uniformly to the exact solution . Moreover, the derivatives of are also convergent to the derivatives of . Numerical results show that the method employed in the paper is valid.
1. 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 this paper, we study the nonlinear fourth-order differential equation with integral boundary conditions in the reproducing kernel space: where is a nonnegative constant. The existence of (1) was obtained in [4, 5]. However, the literature of numerical analysis contains little on the solution of (1).
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 [8–11]. In [8–11], two-point 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 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.
2. Preliminaries
2.1. The Reproducing Kernel Space
The inner product space is defined as follows: is a absolutely continuous real-valued function, , , .
The inner product and norm in are defined, respectively, by where .
Theorem 1. The space is a complete reproducing kernel space. That is, for each fixed , there exists , such that for any and . The reproducing kernel can be written as where, and , , which are related to and , will be given in the following proof.
Proof. (i) The proof of the completeness and reproducing property of 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 in .
Through several integration by parts for (2), we have
and since , it follows that
Then, we have
Note that the property of the reproducing kernel and , , and is the solution of the following generalized differential equation:
While , it is easy to know that is the solution of the following constant linear homogeneous differential equation with orders; that is,
with the following boundary conditions:
We know that (9) has the characteristic equation , and the eigenvalue is a root whose multiplicity is 10. Therefore, the general solution of (9) is
where Now, we are ready to calculate the coefficient , and , .
Since
we have
Since , it follows that
The above 10 equations in (12) and (15) provided 10 conditions for solving the coefficients , in (11). Noting that (10) and (16) provided 12 boundary conditions, we have 22 equations, that is, (12), (15), (10), and (16). It is easy to know that these 22 equations are linear equations with the variables , and , . Therefore, , and , could be calculated by many methods. As long as the coefficients , , , and , are known, the exact expression of the reproducing kernel function could be calculated from (11).
2.2. The Reproducing Kernel Space
The inner product space is defined by is a absolutely continuous real-value function, .
The inner product and norm in are given, respectively, by where . In [15], it has been proved that is also a complete reproducing kernel space and its reproducing kernel is
2.3. Introduction into a Linear Operator
Let and ; then (1) can be converted into the form as follows: where and as . Therefore, we have It is easy to prove that is a bounded linear operator.
Now, we construct an orthogonal function system. Let and , where is the conjugate operator of . In terms of the properties of reproducing kernel , one obtains
The normal orthogonal system of functions in can be derived from Gram-Schmidt orthogonalization process of : where are orthogonalization coefficients, , .
We collect two lemmas in [9] for future use.
Lemma 2. If is dense on , then is a complete system of and . The subscript by the operator indicates that the operator applies to the function of .
Lemma 3. If , then there exists , such that where .
Lemma 4. If , , , and is continuous with respect to for , , then
Proof. Since , , by Lemma 3, we know that is convergent uniformly to ; therefore, the proof is complete.
3. An Iterative Algorithm and Its Convergence
In this section, the exact solution of (1) is given in the reproducing kernel space .
Theorem 5. If is dense on and is the solution of (20), then satisfies the form
Proof. can be expanded to the Fourier series in terms of normal orthogonal basis in :
Remark 6. (i) If (20) is linear, that is, , then the analytical solution of (20) can be obtained directly by (26).
(ii) If (20) is nonlinear, that is, depends on , then the solution of (20) can be obtained by the following iterative method.
We construct an iterative sequence , putting
where
Next, we prove that in iterative formula (28) is convergent to the exact solution of (20).
Theorem 7. Suppose the following conditions are satisfied:(i) is bounded;(ii) is dense in ;(iii) for any .
Then, in iterative formula (28) converges to the exact solution of (20) in and
where are given by (29).
Proof. (i) First, we will prove the convergence of .
By (28), we have
From the orthogonality of , it follows that
From boundedness of , we have ; , .
Let ; in view of , it follows that
Considering the completeness of , there exists , 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
If , then
If , then
From (38) and (39), it is clear that ; furthermore, it is easy to see by induction that
Since is dense on interval , for any , there exists subsequence , such that , as . Hence, let in (40); by the convergence of and Lemma 4, we have
that is, is the solution of (20) and
where are given by (29).
From Lemma 3, we have Corollary 8.
Corollary 8. Assume that the conditions of Theorem 7 hold; then in (28) satisfies , , where is the solution of (20).
4. 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.
Example 1. Consider
The exact solution is , . The numerical results and the root-mean-square errors for the partial derivatives are displayed in Tables 1 and 2, respectively. The superimposed images of with and are displayed in Figure 1.
Example 2. Consider
The exact solution is , . Similar to Example 1, the numerical results are displayed in Tables 3 and 4 and Figure 2.
5. 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The research is supported by the Scientific Research Project of Heilongjiang Education Office (no. 12511155).