Numerical Solutions for the Three-Point Boundary Value Problem of Nonlinear Fractional Differential Equations

and Applied Analysis 3 Definition 2.1 see 39 . The Riemann-Liouville fractional derivative of order α is defined as D 0 u x 1 Γ n − s ( d dx )n ∫x 0 u t x − t α−n 1 dt. 2.1 Γ · is the gamma function and n α 1, α denotes the integerd part of number α. Definition 2.2 see 40 . Let W be a Hilbert function space on a set X. W is called a reproducing kernel space if and only if for any x ∈ X, there exists a unique function Kx y ∈ W , such that 〈f,Kx〉 u x for any u ∈ W . Meanwhile, K x, y . Kx y is called a reproducing kernel. We now define a new reproducing kernel space W which includes the nonlocal boundary value conditions and give an explicit representation formula for calculation of the reproducing kernel. Definition 2.3. W . W 0, 1 {u x | u 4 x is an absolutely continuous real value function in 0, 1 , u 5 x ∈ L2 0, 1 , u 0 u′ 0 u′′ 0 0, u′′ 1 − βu′′ η 0, 0 < η < 1, 0 < βηα−3 < 1}. The inner product is given by 〈u x , v x 〉 u′′ηv′′η u 4 0 v 4 0 ∫1 0 u 5 x v 5 x dx. 2.2 Theorem 2.4. W is a reproducing kernel space, that is, there exists a function K x, y ∈ W , for any fixed y ∈ 0, 1 and any u x ∈ W , such that u y 〈u x , K x, y 〉. Moreover, the reproducing kernel can be denoted by K ( x, y ) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ k1 ( x, y ) a1 a2x a3x · · · a10x, y < x < η, k2 ( x, y ) b1 b2x b3x · · · b10x, y < η < x, k3 ( x, y ) c1 c2x c3x · · · c10x, η < y < x, k4 ( x, y ) d1 d2x d3x · · · d10x, η < x < y, k5 ( x, y ) e1 e2x e3x · · · e10x, x < η < y, k6 ( x, y ) f1 f2x f3x · · · f10x, x < y < η. 2.3 Proof. For any u x ∈ W , we only need to prove that there exists a K x, y ∈ W for any fixed y ∈ 0, 1 and any u x ∈W , K x, y must satisfy 〈 u x , K ( x, y )〉 u ( y ) . 2.4 4 Abstract and Applied Analysis


Introduction
Fractional differential equations have gained considerable importance due to their frequent applications in various fields of science and engineering including physics 1-4 , bioengineering 5-7 , hydrology 8-10 , solid mechanics 11 , chaos 12-14 , control theory 15 , and finance [16][17][18] . It has been found that fractional derivatives provide an excellent instrument for th e description of memory and hereditary properties of different substances 19 . With these features, the fractional-order models become more practical and realistic than the classical integerd-order models, in which such effects are not taken into account.
Finding exact solutions in closed forms for most differential equations of fractional order is a difficult task. As a result, a number of methods have been proposed and applied successfully to approximate fractional differential equations, such as Adomian decomposition method 20 , variational iteration method 21 , homotopy analysis method 22 , implicit and explicit difference method 23 , and collocation method 24 . Especially Momani and Odibat have applied He's variational iteration method to fractional differential equations [25][26][27] . Meanwhile, various fractional order differential equation have been solved very recently including fractional advection-dispersion equations 28, 29 , reactiondiffusion system with fractional derivatives 30 , fractional partial differential equations fluid mechanics 31 , and fractional-order two-point boundary value problem 32 . In contrast to the initial value and two-point boundary value problems, not much attention has been paid to the multipoint fractional boundary value problem MFBVP . Ahmed and Wang 33 considered existence and uniqueness of solutions for a four-point nonlocal boundary value problem of nonlinear impulsive differential equations of fractional order. Rehman and Khan 34 also studied existence and uniqueness of solutions for a class of multipoint boundary value problems for fractional differential equations. However, the research on the numerical treatment for MFBVP has proceeded very slowly in the recent years. This motivates us to investigate computationally efficient numerical techniques for solving the MFBVP.
In the present work, we are concerned with the numerical solution of the following three-point boundary value problem of fractional differential equation 35 in a reproducing kernel space: where D α 0 is the Riemann-Liouville fractional derivative and 0 < η < 1 satisfies 0 < βη α−3 < 1, while f : 0, 1 × 0, ∞ → 0, ∞ is continuous.
Recently, the reproducing kernel space method RKSM has been used for obtaining approximate solutions of differential and integral equations 36-38 . However, due to the multipoint boundary value conditions in 1.1 , especially for fractional differential equations, it is difficult to find the corresponding reproducing kernel space by applying traditional RKSM.
The aim of this work is to extend the RKSM to derive the numerical solutions of 1.1 . One important improvement is that we successfully construct a novel reproducing kernel space so as to overcome difficulties with the nonlocal multipoint boundary value conditions. By using the new reproducing kernel functions, we present an efficient numerical algorithm to solve problem 1.1 . The emphasis of the result is that uniformly convergence of the approximate solution, error estimation, and complexity analysis of our algorithm are studied.
The organization of this paper is as follows. In Section 2, we present some important definitions and preparations used in this paper. In Section 3, we construct and develop algorithms for solving nonlinear fractional differential equation with three-point boundary value conditions. In Section 4 the proposed methods are applied to several examples. Also a conclusion is given in Section 5.

The Construction of a New Reproducing Kernel Space
We give some basic definitions and theories which are used further in this paper.
Γ · is the gamma function and n α 1, α denotes the integerd part of number α. We now define a new reproducing kernel space W which includes the nonlocal boundary value conditions and give an explicit representation formula for calculation of the reproducing kernel.
x is an absolutely continuous real value function in 0, 1 , The inner product is given by Theorem 2.4. W is a reproducing kernel space, that is, there exists a function K x, y ∈ W, for any fixed y ∈ 0, 1 and any u x ∈ W, such that u y u x , K x, y . Moreover, the reproducing kernel can be denoted by k 5 x, y e 1 e 2 x e 3 x 2 · · · e 10 x 9 , x < η < y, x < y < η.

2.3
Proof. For any u x ∈ W, we only need to prove that there exists a K x, y ∈ W for any fixed y ∈ 0, 1 and any u x ∈ W, K x, y must satisfy

2.5
We have the following equality using the integration by parts: Substituting 2.6 in 2.5 , we get

2.7
Therefore, by 2.4 K x, y is the solution of the following generalized differential equation:

2.8
Abstract and Applied Analysis 5 where δ denotes δ function. For x / y, it is known that K x, y is the solution of the following linear homogeneous differential equation: with the boundary value conditions: x 0 0,

2.10
We find that 2.9 has characteristic equation λ 10 0, and the eigenvalue λ 0 is a root whose multiplicity is 10. Applying the feature of functions in W, the general solution of 2.9 is given by the universal representation as 2.3 , in which every function k i x, y i 1, 2, 3, . . . , 6 has the situation in Figure 1. Next we will calculate 60 coefficients in 2.3 . By integrating repeatedly ∂ 10 K x, y /∂t 10 −δ x − y from y − ε to y ε with respect to x, we have and the following 20 equations are inferred as ε → 0 x y − ∂ 9 k 6 x, y ∂x 9 x y −1, x y − ∂ 9 k 4 x, y ∂x 9 x y −1, Abstract and Applied Analysis By 2.10 , one can obtain 14 equations. In view of boundary value conditions, the following 8 equations can be obtained:

2.14
Hence, the unknown coefficients of K x, y are governed by solving the 60 independent equations by 2.10 -2.14 .
Abstract and Applied Analysis 7

Description of the Proposed Numerical Method
The method consists of two steps. In the first step, a normal orthogonal basis is established in the reproducing kernel space W, and in the second step, it is used to successively obtain the approximate solution of 1.1 . Let us consider these steps in detail.

A Normal Orthogonal Basis in W
Define a bounded linear operator T : W 0, 1 → L 1 0, 1 satisfying The proof of existence and uniqueness of solution for 1.1 has been studied in 35 . Therefore T is reversible. Now, 1.1 is turned into the following operator equation in W: Choosing a countable dense subset {x i } ∞ i 1 on 0, 1 , for the reproducing kernel K x, y of W, we define a complete system in W as Then the orthogonal system of W is derived from Gram-Schmidt orthogonalization process, namely, Next, the complexity estimation of the orthogonal basis is discussed. We know that the orthogonal basis is obtained by orthogonalization of complete system ψ i x . The algorithm with time complexity may be analyzed as following.
Step 1 Computing ψ i x , ψ k x . In fact, according to the properties of reproducing kernel and bounded linear operator, we have Thus, we only need to calculate the specific function value Tψ i x k instead of the usual integral. Denote the computing time of the specific function value by T . It needs n n 1 /2 T to compute those ψ i x , ψ k x k 1, 2, . . . , i, i 1, 2, . . . , n.

Abstract and Applied Analysis
Step 2. Orthogonalization can be obtained by the following cycle.

The Approximate Solution of 3.2 in W
Proof. According to the orthogonal basis

10
Abstract and Applied Analysis Proof. Firstly, because of denseness, for any x ∈ 0, 1 and n ∈ N, we take x i ∈ {x 1 , x 2 , . . .}, i ≤ n such that |x − x i | < 1/n. Then due to the reproducing property and the property of projector P n , it follows that Tu n x i u n · , TK x i , · P n u · , ψ i · u · , P n ψ i · u · , ψ i · T u · , K x i , · Tu x i .

3.15
This implies that

3.16
By the mean value theorem, we have Finally, the following conclusion follows from above:

3.18
Thus, according to u n − u → 0, |x − x i | < 1/n and the boundedness of ∂/∂y TK y, · , we get |u n x − u x | o 1/n .

Numerical Experiments
In this section, we give some computational results of three numerical experiments with methods based on preceding sections, to support our theoretical discussion.
Example 4.1. Consider the following three-point boundary value problem of nonlinear fractional differential equation 35 :
Step 1. By using the representation formula for calculation of reproducing kernel in Section 2, the concrete expression of K x, y for the three-point boundary value conditions in 4.1 is given as k 4 x, y k 3 y, x ; k 5 x, y k 2 y, x ; k 6 x, y k 1 y, x .

4.2
Step 2. According to the numerical algorithm in Section 3, we get By 3.4 we get the orthogonalization coefficients β ik and ψ i x . Then the approximate solution u n x can be obtained iteratively by 3.12 .
The obtained numerical results are displayed in Table 1. Furthermore, the graph of u x and u n x for n 128 is plotted in Figure 2. It can be shown that the numerical solutions agree with exact solution by means of the proposed method.
Due to the multipoint boundary value conditions, to our knowledge, there is no the same example as 1.1 in the literature about numerical method. For the purpose of comparison, we compare the approximate solution of our method, together with the approximate solution by Adomian Decomposition AD method given in 41 .

Example 4.2.
In this example, we consider the following nonlinear fourth-order fractional integrodifferential equation of the form:

4.4
In this case, a similar numerical method can be proposed as 1.1 . In brief, the conditions u 0 u 0 u 0 0, u 1 βu η in the reproducing kernel space W are replaced by y 0 y 0 1, y 1 y 1 e. There are some slight changes in the process of derivation. The obtained numerical results for α 3.25 are displayed in Table 2.      According to the numerical scheme above, one can obtain the approximation u n x of u x for n 128. The numerical results are displayed in Table 3 and Figure 3, which show that the numerical solutions agree with exact solution.

Conclusion
In this paper, the RKSM is applied to derive approximate analytical solution of nonlinear fractional-order differential equations with three-point boundary value conditions. We have constructed a novel reproducing kernel space and give the way to express the reproducing kernel function, while traditional RKSM still can not be mentioned. The explicit series solution is obtained using the orthogonal basis established in the new reproducing kernel space. The numerical results given in the previous section demonstrate the better accuracy of our algorithms. Moreover, the numerical algorithms introduced in this paper can be well suited for handling general linear and nonlinear fractional-order differential equations with multipoint boundary conditions. We note that the corresponding analytical and numerical solutions are obtained using Mathematica 7.0.