1 Introduction
In this paper, a new iterative reproducing kernel approach will be constructed for obtaining the numerical solution of nonlinear fractional threepoint boundary value problem,
(1) 
with following boundary conditions,
(2) 
Here, and are sufficiently smooth functions and fractional derivatives are taken in Caputo sense. Without loss of generality, we pay regard to , and . Because, , and boundary conditions can be easily reduced to , and .
Nonlinear fractional multipoint boundary value problems appear in a different area of applied mathematics and physics 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 and references therein. Many important studies have been concerned in engineering and applied science such as dynamical systems, fluid mechanics, control theory, oil industries, heat conduction can be wellturned by fractional differential equations 8 ; 9 ; 10 . Some applications, qualitative behaviors of solution and numerical methods to find approximate solution have been investigated for differential equation with fractional order 11 ; 12 ; 13 ; 14 .
More particularly, it is not easy to directly get exact solutions to most differential equations with fractional order. Hence, numerical techniques are utilised largely. Actually, in recent times many efficient and convenient methods have been developed such as the finite difference method 15 , finite element method 16 , homotopy perturbation method 17 , Haar wavelet methods 18 , Adomian decomposition method 19 , collocation methods 20 , homotopy analysis method 21 , differential transform method 22 , variational iteration method 23 , reproducing kernel space method 24 ; 25 and so on 26 ; 27 ; 28 .
In 1908, Zaremba firstly introduced reproducing kernel concept 29 . His resarches with regard to boundary value problems which includes Dirichlet condition. Reproducing kernel method (RKM) produces a solution in convergent series form for many differential, partial and integrodifferential equations. For more information, we refer to 30 ; 31 . Recently, this RKM is applied for different type of problem. For example, fractional order nonlocal boundary value problems 32 , Riccati differential equations 33 , forced Duffing equations with nonlocal boundary conditions 34 , Bratu equations with fractional order Caputo derivative 35 , timefractional Kawahara equation 36 , twopoint boundary value problem 37 , nonlinear fractional Volterra integrodifferential equations 38 .
Recently, Legendre reproducing kernel method is proposed for fractional twopoint 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 threepoint 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.
2 Preliminaries
In this section, several significant concepts, definitions, theorems, and
properties are provided which will be used in this research.
Definition 2.1 Let and . Then, the order left RiemannLiouville fractional integral operator is given
as 8 ; 12 ; 13 :
here is Gamma function, and .
Definition 2.2 Let and . Then, the
order left Caputo differential operator is given as 8 ; 12 ; 13 :
Definition 2.3 In order to construct polynomial type reproducing kernel, the first kind shifted Legendre polynomials are defined over the interval . For obtaining these polynomials the following iterative formula can be given:
The orthogonality requirement is
(3) 
here, weighted function is taken as,
(4) 
Legendre basis functions can be established so that this basis function system satisfy the homogeneous boundary conditions as:
(5) 
Eq. (5) has a advantageous feature for solving boundary value problems. Therefore, these basis functions for can be defined as;
(6) 
such that this system satisfy the conditions
(7) 
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 , and with its inner product be a Hilbert space of realvalued functions on . Then, the reproducing kernel of is iff


.
The last condition is known as reproducing property. Especially, for any , ,
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 .
Theorem 2.1 Let be an orthonormal basis of dimensional Hilbert space , then
(8) 
is reproducing kernel of 30 ; 31 .
Definition 2.5 Let polynomials space be
preHilbert space over with real coefficients and its degree
and inner product as:
(9) 
with described by Eq. (4), and the norm
(10) 
With the aid of definiton of Hilbert space, for any fixed , is a subspace of and ,
Theorem 2.2 Hilbert space is a reproducing
kernel space.
Proof. From Definition 2.5, it is quite apparent that functions space is a finitedimensional. It is well known that
all finitedimensional preHilbert space is a Hilbert space. Herewith, using
this consequence and Theorem 2.1, is a reproducing
kernel space.
For solving problem (1)(2), it is required to describe a
closed subspace of so that satisfy homogeneous boundary
conditions.
Definition 2.6 Let
One can easily demonstrate that is a reproducing kernel space using Eq. (6). From Theorem 2.1, the kernel function of can be written as
(11) 
Here, is complete system which is easily obtained from basis functions in Eq. (6) with the help of GramSchmidt orthonormalization process. Eq. (11) is very useful for implementation. In other words, and can readily recalculated by increasing .
3 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 . Then, we will construct an iterative process for nonlinear problem in third subsection.
3.1 Generation of reproducing kernel for threepoint boundary value problems
In this subsection, we shall generate a reproducing kernel Hilbert space in which every functions satisfies , and .
is defined as .
Obviously, reproducing kernel space is a
closed subspace of . The reproducing kernel of can be given with the following theorem.
Theorem 3.1 The reproducing kernel
of ,
(12) 
Proof. Frankly, not all elements of vanish at . This shows that 0. Hence, it can be easily seen that and therefore . For , clearly, , it follows that
Namely, is of reproducing kernel of . This completes the proof.
3.2 Representation of solution in Hilbert space
In this subsection, reproducing kernel method with Legendre polyomials is established for obtaining numerical solution of threepoint boundary value problem. For Eqs. (1)(2), the approximate solution shall be constructed in . Firstly, we will define linear operator as follow,
such that
The Eqs.(1)(2) can be stated as follows
(13) 
Easily can be shown that linear operator is bounded. We will obtain the
representation solution of Eq. (13) in the space. Let be the polynomial form of
reproducing kernel in space.
Theorem 3.2 Let be any distinct
points in open interval for Eqs. (1)(2), then
Proof. For any fixed , put
(14)  
It is quite obvious that . Therefore .
Here, shows the adjoint operator of . For any fixed and , .
Theorem 3.3 Let be any distinct
points in open interval for , then is complete in .
Proof. For every fixed , let
this result shows, for ,
(15)  
In Eq. (15), by use of inverse operator, it is decided that .
Thus, is complete in . 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 .
The orthonormal system of can be derived with the help of the GramSchmidt
orthogonalization process using ,
(16) 
here show the coefficients of orthogonalization.
Theorem 3.4 Suppose that is the exact solution of Eqs. (1)(2) and shows any distinct
points in open interval , in that case
(17) 
Proof. Since from Theorem 3.3 can be written
On the other part, using Eq. (14) and Eq. (16), we obtain which is the precise solution of Eq. (10) in as,
The proof is completed.
Theorem 3.5 If ,
then for , where is a constant.
Proof. We have for any , From the expression of , it pursue that
So,
Therefore, .
Theorem 3.6 and its derivatives
are respectively uniformly converge to and ().
Proof By using Theorem 3.5 for any we get
where are positive constants. Therefore, if in the meaning of the norm of as , and its
derivatives are
respectively uniformly converge to and its derivatives . This completes the proof.
If considered problem is linear, numerical solution can be directly get from (17). But, for nonlinear problem the following iterative procedure can be construct.
3.3 Construction of iterative procedure
In this subsection, we will use the following iterative sequence to overcome the nonlinearity of the problem, , inserting,
(18) 
here, orthogonal projection operator is defined as and shows the th iterative numerical solution of (18). Then, the following important theorem will be given for iterative
procedure.
Theorem 3.7 If is distinct points in
open interval , then
(19) 
Proof. Since , is the complete orthonormal system in ,
This completes the proof.
Taking and define the iterative sequence
(20) 
4 Numerical applications
In this section, some nonlinear threepoint boundary value problems are
considered to exemplify the accuracy and efficiency of proposed approach.
Numerical results which is achieved by LRKM are shown with tables.
Example 4.1 We consider the following fractional order nonlinear
threepoint boundary value problem with Caputo derivative:
(21) 
(22) 
Here, a known function such that the exact solution of this problem is .
By using proposed approach for Eqs. (21)(22), and choosing
nodal points as , the
approximate solution is computed by Eq. (20). For (21)(22), comparison of absolute errors for
different , values are demonstrated in Table 1 and Table 2
and comparison of exact solution and numerical solution for
and is given in Table 3.
Example 4.2 We take care of the following nonlinear threepoint boundary value problem with Caputo derivative
(23) 
(24) 
Here, a known function such that the exact solution of this problem is .
By using proposed approach for Eqs. (23)(24), and choosing
nodal points as , the
approximate solution is computed by Eq. (20). For (23)(24), comparison of absolute errors for
different , values are demonstrated in Table 4 and Table 5
and comparison of exact solution and numerical solution for
and is given in Table 6.
5 Conclusion
In this research, a novel numerical approach which is called LRKM has been proposed and successfully implemented to find the approximate solution of nonlinear threepoint 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 threepoint boundary value problems with fractional order.
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Tables
0.1  3.25E11  6.08E11  3.70E12  4.45E10  3.14E9 

0.2  5.45E11  8.75E11  4.93E12  8.70E10  5.51E9 
0.3  6.97E11  9.00E11  4.34E12  1.32E9  7.36E9 
0.4  8.17E11  7.81E11  2.55E12  1.85E9  8.96E9 
0.5  0  0  0  0  0 
0.6  1.10E10  5.14E11  2.03E12  3.36E9  1.24E8 
0.7  1.34E10  5.65E11  3.56E12  4.44E9  1.49E8 
0.8  1.70E10  8.70E11  3.73E12  5.80E9  1.81E8 
0.9  2.21E10  1.53E11  1.89E12  7.49E9  2.25E8 
0.1  3.78E17  1.33E16  2.49E19  4.86E15  7.48E14 

0.2  5.27E17  1.94E16  3.36E19  1.13E14  1.36E13 
0.3  5.10E17  2.03E16  3.80E19  1.99E14  1.91E13 
0.4  3.91E17  1.81E16  5.20E19  3.09E14  2.43E13 
0.5  0  0  0  0  0 
0.6  1.05E17  1.36E16  1.56E18  6.18E14  3.60E13 
0.7  6.65E18  1.57E16  2.74E18  8.25E14  4.36E13 
0.8  1.81E17  2.34E16  4.50E18  1.07E14  5.30E13 
0.9  5.14E17  3.91E16  7.02E18  1.36E14  6.48E13 
Exact Sol.  Approximate Sol.  Absolute Error  

0.0  0.000000000000000000000  0.000000000000000000000  0 
0.1  0.036000000000000000000  0.036000000000000000018  1.80E20 
0.2  0.048000000000000000000  0.048000000000000000044  4.40E20 
0.3  0.042000000000000000000  0.042000000000000000061  6.10E20 
0.4  0.024000000000000000000  0.024000000000000000071  7.10E20 
0.5  0.000000000000000000000  0.000000000000000000000  0 
0.6  0.024000000000000000000  0.023999999999999999899  1.01E19 
0.7  0.042000000000000000000  0.041999999999999999819  1.81E19 
0.8  0.048000000000000000000  0.047999999999999999694  3.06E19 
0.9  0.036000000000000000000  0.035999999999999999491  5.09E19 
1.0  0.000000000000000000000  0.000000000000000000000  0 
0.1  5.00E11  4.11E15  2.78E13  6.10E12  1.40E12 

0.2  8.39E11  3.45E15  1.87E13  1.17E11  2.42E11 
0.3  1.06E10  1.24E15  1.87E13  1.45E11  7.08E11 
0.4  1.23E10  9.21E15  7.60E13  1.20E11  1.32E10 
0.5  1.39E10  1.97E14  1.44E12  1.75E12  2.02E10 
0.6  0  0  0  0  0 
0.7  1.88E10  4.52E14  2.81E12  5.17E11  3.45E10 
0.8  2.31E10  5.88E14  3.32E12  9.98E11  4.05E10 
0.9  2.93E10  7.18E14  3.60E12  1.65E10  4.49E10 
0.1  3.49E13  1.93E18  4.48E16  2.68E14  4.75E15 

0.2  5.87E13  8.32E19  2.99E16  5.58E14  7.68E14 
0.3  7.47E13  2.90E18  3.06E16  7.18E14  2.25E13 
0.4  8.65E13  8.81E18  1.23E15  6.00E14  4.21E13 
0.5  9.76E13  1.65E17  2.34E15  5.54E15  6.45E13 
0.6  0  0  0  0  0 
0.7  1.31E12  3.55E17  4.55E15  2.91E13  1.09E12 
0.8  1.61E12  4.60E17  5.37E15  5.64E13  1.29E12 
0.9  2.05E12  5.65E17  5.83E15  9.39E13  1.43E12 
Exact Sol.  Approximate Sol.  Absolute Error  

0.0  0.000000000000000000000  0.000000000000000000000  0 
0.1  0.045000000000000000000  0.045000000000480782793  4.80E13 
0.2  0.064000000000000000000  0.064000000000580045412  5.80E13 
0.3  0.063000000000000000000  0.063000000000488602930  4.88E13 
0.4  0.048000000000000000000  0.048000000000398645450  3.98E13 
0.5  0.025000000000000000000  0.025000000000468872800  4.68E13 
0.6  0.000000000000000000000  0.000000000000000000000  0 
0.7  0.021000000000000000000  0.020999999998651962040  1.34E12 
0.8  0.032000000000000000000  0.031999999998006864020  1.99E12 
0.9  0.027000000000000000000  0.026999999997598991320  2.40E12 
1.0  0.000000000000000000000  0.000000000000000000000  0 
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