On the solutions of boundary value problems

We investigate the nonlinear boundary value problems by reproducing kernel Hilbert space technique in this paper. We construct some reproducing kernel Hilbert spaces. We deﬁne a bounded linear operator to obtain the solutions of the problems. We demonstrate our numerical results by some tables. We com-pare our numerical results with some results exist in the literature to present the eﬃciency of the proposed method.


Introduction
We investigate the following boundary value problems by reproducing kernel method.
p(x)y ′ ′ = f (x, y) (1) subject to the boundary values Reproducing kernel space is a special Hilbert space. Many problems have been investigated by reproducing kernel Hilbert space method in the literature.
Safari et al. [1] have investigated the rainfallrunoff modeling through regression in the reproducing kernel Hilbert space algorithm. Najafi et al. [2] have worked on the combining fractional differential transform method and reproducing kernel Hilbert space method to solve fuzzy impulsive fractional differential equations. Sahihi et al. [3] have searched the system of second-order boundary value problems using a new algorithm based on the reproducing kernel Hilbert space. Agud et al. [4] have investigated the weighted p-regular kernels for reproducing kernel Hilbert spaces. Mundayadan et al. [5] have studied on the linear dynamics in the reproducing kernel Hilbert spaces. Arqub et al. [6] have constructed the modulation of reproducing kernel technique successfully. Emamjome et al. [7] have presented the reproducing kernel pseudospectral technique in details. Foroutan et al. [8] have investigated this technique for the nonlinear three-point boundary value problems. Akgül et al. [9] have worked on the representation for the reproducing kernel Hilbert space method for a nonlinear system. Allahviranloo et al. [14] have investigated the reproducing kernel method to solve parabolic partial differential equations with nonlocal conditions. For more details see [15][16][17][18][19][20][21][22][23].

Reproducing kernel Hilbert spaces
We define the reproducing kernel Hilbert spaces and find some reproducing kernel functions in these spaces in this section.
Definition 1. Let (H, ·, · ) be a Hilbert space of real functions defined on a nonempty set E. A function K : E × E → R is called a reproducing kernel for H if and only if (a) K(·, z) ∈ H for each fixed z ∈ E, (b) ϕ, K(·, z) = ϕ(z) for all z ∈ E and all ϕ ∈ H.
We will refer to such a Hilbert space H for which there exists a reproducing kernel function K as a reproducing kernel Hilbert space.
Condition (b) is called "the reproducing property" of the kernel K because the value of an arbitrary function ϕ ∈ H at an arbitrary point z ∈ E is reproduced by the inner product of ϕ with K(·, z). 1], and as the inner product and the norm in W 1 2 [0, 1] respectively. Reproducing kernel function T x (y) of W 1 2 [0, 1] is presented as: Proof. First, let us suppose Solving the coefficients, we get the reproducing kernel function as: We consider the solution of Eq. (1) We should homogenize the conditions. Put then we can obtain homogeneous boundary-value conditions of problem (1) With the boundary conditions: where Theorem 2. The operator L is a bounded linear operator.
Proof. Firstly, we present Lu 2 Moreover, by reproducing property we have: Then, we get Also, from

Solutions of the problems
Obviously, defined operator in (9) as L : Theorem 3. If y(x) is the exact solution of (10), then Proof. By the (12) and uniqueness of solution of (10) we obtain: Finite terms of (13) concludes the approximate solution: where Then the approximate solution y n (x) converges to the exact solution of (13) in o W 3 2 and we have Proof. First, we prove the convergence of y n (x). From (15), we have Also, orthonormality of and from boundedness of i.e., {A i } ∈ l 2 (i = 1, 2, ...).
So, y(x) is the exact solution of (10) given by where A i are given by (16).
Moreover a sequence y n − y o W 3 2 is monotonically decreasing in n.

Proof.
By (13) and (14), we acquire Therefore, we obtain In addition

Numerical examples
We consider the following problems by reproducing kernel Hilbert space method in this section. We computed our results by MAPLE. We showed our results by tables. Example 1. We investigate: We have the exact solution of the above problem as: . We searched the boundary value problem (20) by the proposed method and gave corresponding error-norms by Table 1.
Example 2. We solved the following problem for the second example in the reproducing kernel Hilbert space.
In Table 2, we computed absolute errors for (21).

Conclusions
In this work, we gave a new application of the reproducing kernel Hilbert space method. We obtained very useful reproducing kernel functions in the reproducing kernel Hilbert spaces. We proved the accuracy of the method. We compared the reproducing kernel Hilbert space method with the techniques existed in the literature. We concluded that the proposed technique is very effective for solving nonlinear two-point boundary value problems.