A new reproducing kernel approach for nonlinear fractional three-point boundary value problems

In this article, a new reproducing kernel approach is developed for obtaining numerical solution of nonlinear three-point boundary value problems with fractional order. This approach is based on reproducing kernel which is constructed by shifted Legendre polynomials. In considered problem, fractional derivatives with respect to $\alpha$ and $\beta$ are defined in Caputo sense. This method has been applied to some examples which have exact solutions. In order to shows the robustness of the proposed method, some numerical results are given in tabulated forms.

Recently, Legendre reproducing kernel method is proposed for fractional two-point boundary value problem of Bratu Type Equations [39]. The main motivation of this paper is to extend the Legendre reproducing kernel approach for solving nonlinear three-point boundary value problem with Caputo derivative.
The remainder part of the paper is prepared as follows: some fundamental definitions of fractional calculus and the theory of reproducing kernel with Legendre basis functions are given in Section 2. The structure of solution with Legendre reproducing kernel is demonstrated in Section 3. In order to show the effectiveness of the proposed method, some numerical findings are reported in Section 4. Finally, the last section contains some conclusions.

Definition 2.3
In order to construct polynomial type reproducing kernel, the first kind shifted Legendre polynomials are defined over the interval [0, 1]. For obtaining these polynomials the following iterative formula can be given: . . .

The orthogonality requirement is
here, weighted function is taken as, Legendre basis functions can be established so that this basis function system satisfy the homogeneous boundary conditions as: Eq. (5) has a advantageous feature for solving boundary value problems. Therefore, these basis functions for j ≥ 2 can be defined as; such that this system satisfy the conditions It is worth noting that the basis functions given in Eq. (6) are complete system. For more information about orthogonal polynomials, please see [41,42,43].

Definition 2.4
Let Ω = ∅, and H with its inner product ·, · H be a Hilbert space of real-valued functions on Ω. Then, the reproducing kernel of H is R : The last condition is known as reproducing property. Especially, for any x, ξ ∈ Ω, If a Hilbert space satisfies the above two conditions then is called reproducing kernel Hilbert space. Uniqueness of the reproducing kernel can be shown by use of Riesz representation theorem [40].
Definition 2.5 Let W m ρ [0, 1] polynomials space be pre-Hilbert space over [0, 1] with real coefficients and its degree ≤ m and inner product as: with ρ [0,1] (ξ) described by Eq. (4), and the norm With the aid of definiton of L 2 Hilbert space, L 2 Proof. From Definition 2.5, it is quite apparent that W m ρ [0, 1] functions space is a finite-dimensional. It is well known that all finite-dimensional pre-Hilbert space is a Hilbert space. Herewith, using this consequence and Theorem 2.1, W m ρ [0, 1] is a reproducing kernel space.
For solving problem (1)-(2), it is required to describe a closed subspace of W m ρ [0, 1] so that satisfy homogeneous boundary conditions.
Here, h j (ξ) is complete system which is easily obtained from basis functions in Eq. (6) with the help of Gram-Schmidt orthonormalization process. Eq. (11) is very useful for implementation. In other words, R m x (ξ) and W m ρ [0, 1] can readily re-calculated by increasing m.

Main Results
In this section, some important results related to reproducing kernel method with shifted Legendre polynomials are presented. In the first subsection, generation of reproducing kernel which is satify threepoint boundary value problems is presented. In the second subsection, representation of solution is given θ W m ρ [0, 1]. Then, we will construct an iterative process for nonlinear problem in third subsection.

Generation of reproducing kernel for three-point boundary value problems
In this subsection, we shall generate a reproducing kernel Hilbert space θ W m ρ [0, 1] in which every functions satisfies z(0) = 0, z(θ) = 0 and z(1) = 0.
Proof. Frankly, not all elements of 0 W m ρ [0, 1] vanish at θ. This shows that R m θ (θ) = 0. Hence, it can be easily seen that θ R m The Eqs.(1)-(2) can be stated as follows Easily can be shown that linear operator L is bounded. We will obtain the representation solution of Eq.
It is quite obvious that Here, L * shows the adjoint operator of L. For any fixed m and ξ j ∈ (0, 1), ψ m j ∈ θ W m ρ [0, 1].
In Eq. (15), by use of inverse operator, it is decided that z ≡ 0. Thus, {ψ m j } m−2 j=0 is complete in θ W m ρ [0, 1]. This completes the proof. Theorem 3.3 indicates that in Legendre reproducing kernel approach, using a finite distinct points are enough. But, in traditional reproducing kernel method need to dense sqeuence on the interval. Namely, this new approach is vary from traditional method in [28,32,33,34,35,38].
here β m jk show the coefficients of orthogonalization.
If considered problem is linear, numerical solution can be directly get from (17). But, for nonlinear problem the following iterative procedure can be construct.

Numerical applications
In this section, some nonlinear three-point boundary value problems are considered to exemplify the accuracy and efficiency of proposed approach. Numerical results which is achieved by L-RKM are shown with tables.
Example 4.1 We consider the following fractional order nonlinear three-point boundary value problem with Caputo derivative: Here, f (ξ) a known function such that the exact solution of this problem is z(ξ) = ξ(ξ − 1 2 )(ξ − 1). By using proposed approach for Eqs.  Table 1 and Table 2 and comparison of exact solution and numerical solution for α = 1.75 and β = 0.75 is given in Table 3.

Conclusion
In this research, a novel numerical approach which is called L-RKM has been proposed and successfully implemented to find the approximate solution of nonlinear three-point boundary value problems with Caputo derivative. For nonlinear problem, a new iterative process is proposed. Numerical findings show that the present approach is efficient and convenient for solving three-point boundary value problems with fractional order.