A Numerical method for solving a class of fractional Sturm-Liouville eigenvalue problems

This article is devoted to both theoretical and numerical studies of eigenvalues of regular fractional 2α-order SturmLiouville problem where 2 < α ≤ 1. In this paper, we implement the reproducing kernel method RKM) to approximate the eigenvalues. To find the eigenvalues, we force the approximate solution produced by the RKM satisfy the boundary condition at x = 1. The fractional derivative is described in the Caputo sense. Numerical results demonstrate the accuracy of the present algorithm. In addition, we prove the existence of the eigenfunctions of the proposed problem. Uniformly convergence of the approximate eigenfunctions produced by the RKM to the exact eigenfunctions is proven.


Introduction
Fractional differential equations (FDEs) appear as generalizations to existing models with integer derivative and they also present new models for some physical problems [1].In recent years, great interests were devoted to the analytical and numerical treatments of fractional differential equations.In general, fractional differential equations don't have exact solutions in closed forms, and therefore, numerical methods such as, the variational iteration [4], the homotopy analysis method [5], and the Adomian decomposition method [3,7,8], have been implemented for several types of fractional differential equations.Also, the maximum principle and the method of lower and upper solutions have been extended to deal with FDEs and obtain analytical and numerical results [6].The Tau method, the Pseudo-spectral method, and the wavelet method based on the Legendre polynomials have been implemented for several types of FDEs [2].The fractional Sturm-Liouville eigenvalue problem was studied earlier [9] and [10].In [9], the existence of a solution to such boundary value problem was established.In [10], the aforementioned relation between eigenvalues and zeros of Mittag-Leffler function was shown.The Adomian decomposition method was established for estimating fractional second order eigenvalues [11,12].The Homotopy Analysis method has been used to numerically approximate the eigenvalues of the fractional Sturm-Liouville Problems [13].In [14], fractional differential transform method used to approximate the eigenvalues of Sturm-Liouville problems of fractional order.Fourier series were used in [15], the method of Haar wavelet operational matrix was used in [16] and [17].In [18]- [20], [21], extended some spectral properties of fractional Sturm-Liouville problem.Variational Methods and Inverse Laplace transform method applied in [23] and [24], respectively.In [25], it is presented a series method for solving higher eigenvalue Sturm-Liouville problems.Recently P. Antunes and R. Ferreira constructed numerical schemes using radial basis functions [26], B. Jin et al used Galerkin finite element method to solve fractional eigenvalue problems [27].In this paper, we discuss the following regular fractional Sturm-Liouville problem of the form subject to where a 0 , a 1 , a 2 , a 3 are constants, p(x), q(x), r(x) are continuous functions with p(x), q(x) > 0 for all x ∈ [0, 1], and D α is the Caputo fractional derivative.Historically, problem (1.1)-(1.2) had been studied theoretically for α = 1 by [28] and [29] who showed that it has an infinite sequence of eigenvalues {λ 0 , λ 1 , λ 2 , ...} with the following property η is a constant and each eigenvalue has multiplicity at most 3.However, the numerical treatments of such problems have always been far from trivial which, therefore, attracts several authors to initiate or apply different numerical methods to investigate their solution.For instance, Lesnic and Attili [33] used the Adomian decomposition method (ADM) whereas Greenberg and Marletta [30]- [32] developed their own code using Theta Matrices (SLEUTH).Syam and Siyyam [34] implemented the iterated variation method.The present work is motivated by approximating the eigenvalues of problem (1.1)-(1.2) using reproducing kernel method (RKM).This paper is organized as follows.In section 2, we present some preliminaries which we will use in this paper.A description of the RKM for discretization of the fractional 2α-order Sturm-Liouville problem (1.1)-(1.2) is presented in section 3. Several numerical examples and conclusions are discussed in Section 4. Conclusions and closing remarks are given in Section 5.

Preliminaries
In this section, we review the definition and some preliminary results of the Caputo fractional derivatives, as well as, the definition of the Rimann-Liouville fractional and their properties.
where Γ is the well-known Gamma function.
3 Analysis of RKM for a class of fractional second-order Sturm-Liouville Problems In this section, we discuss the numerical solution of the following class of fractional 2α-order Sturm-Liouville Problems using RKM: subject to ) where a 0 , a 1 , a 2 , a 3 are constants, p(x), q(x), and r(x) are continuous with p(x), r(x) > 0 for all x ∈ [0, 1].Assume that subject to where Applying the operator I α to both sides of Equation (3.4) or Applying the operator I α to both sides of Equation (3.6) yields where ) .
In order to solve problem (3.5) and (3.7), we construct kernel Hilbert space W 3 2 [0, 1] in which every function satisfy the boundary conditions (3.4).First, we define the reproducing kernel.

Definition 3.1. Let A be a nonempty abstract set. A function K :
• (y(.), K(., x)) = y(x) for all x ∈ A and y ∈ H.
The second condition which possesses a reproducing kernel is called a reproducing kernel Hilbert space (RKHS).For more details, see [35].Let W 3  2 [0, 1] = {y(x) : y, y ′ , and y ′′ are absolutely continuous real-valued functions, The inner product in W 3 2 [0, 1] is defined as and the norm ∥y∥ where y, u where Proof.Using the integration by parts, we have ∂ iv K ∂t iv (x, 1) + y ′ (0) Since y(t) and K(x,t Thus, ∂ iv K ∂t iv (x, 1) + y ′ (0) where δ is the Dirac-delta function and ∂ iv K ∂t iv (x, 1) = 0. (3.18) Since the characteristic equation of ∂ vi K ∂t vi (x,t) = δ (x − t) is ξ 6 = 0 and its characteristic value is ξ = 0 with 6 multiplicity roots, we write K(η, y) as On the other hand, Integrating ∂ vi K ∂t vi (x,t) = δ (x − t) from x − ε to x + ε with respect to t and letting ε → 0 to get We solved system (3.16) using Mathematica to get Next, we study the space The inner product in W 1 2 [0, 1] is defined as where Theorem 3.2.The space W 1 2 [0, 1] is a reproducing kernel Hilbert space, i.e.; there exists R(x,t) ∈ W 1 2 [0, 1] such that for any y ∈ W 1 2 [0, 1] and each fixed x,t ∈ [0, 1], we have In this case, R(x,t) is given by Proof.Using integration by parts, we have Since R(x,t) is a reproducing kernel of W 1 2 [0, 1], we have Since the characteristic equation of Using the conditions (3.18)-(3.21),we get the following system of equations c 0 (x) − c 1 (x) = 0, (3.27) and the proof is completed.Now, we will present how to solve Problem (3.5) and (3.7) using the reproducing kernel method.Let where L * is the adjoint operator of L. Using Gram-Schmidt orthonormalization to generate orthonormal set of functions and α i j are the coefficients of Gram-Schmidt orthonormalization.
Proof.First, we want to prove that For each fixed y(x . Second, we prove that equation (3.25) holds true.Simple calculations implies that and the proof is completed.
Let the approximate solution of Problem (3.7) be given by In the next theorem, we want to show that {y N (x)} ∞ N=1 is uniformly convergent to y(x).Theorem 3.4.If y(x) and y N (x) are given as in (3.25) and (3.26), then {y N (x)} ∞ N=1 converges uniformly to y(x).Proof.For any x ∈ [0, 1], Thus, which implies that {y N (x)} ∞ N=1 converges uniformly to y(x).To find the τ 0 and τ 1 , we set if α = 1.In this case, y N (x) = 0 is a function of x and λ .To find the eigenvalues of Problem (3.1)-(3.3),we use the simple shooting method by setting y N (1) = 0.In the next section, we sketch the graph of y N (1).When the graph intersects the λ −axis, this means we have an eigenvalue to problem (3.1)-(3.3).

Numerical Results
In this section, we apply the RKM outlined in the previous section to solve numerically the following examples.Note that the maximum number of terms in the approximate series solution is taken as N = 12 for all examples considered in this paper.In this paper, we will focus only one the eigenvalues.
Using the procedure described in the previous section, we scan the function y N (1) for λ on the interval [0, χ] where y N (1) approaches to infinity when λ approaches to χ. Figure 1 shows the graph of the the function y N (1) against the parameter λ .When the graph intersects the λ −axis, this means we have an eigenvalue.Then, we use the FindRoot command in Mathematica to find this root.The available results for λ obtained by the present method are summarized in Table 1.
It is worth mentioning that the eigenvalues of the problem in this example approaches to n 2 π 2 when α approaches to 1.We noticed that the eigenvalue problem in Example (4.1) does not have any eigenvalue for α = 0.501.For this reason, we look for the numerical value of α * such that the eigenvalue problem of this example does not have any eigenvalue for 1 2 < α < α * .We noticed that α * = 0.7355.Let This means, the orthogonality relation holds.We notice that the eigenvalues satisfy the property where p(x) = 1, q(x) = 1 + x α , and r(x) = 0.
Following the same procedure described in Example (4.1), we scan the function y N (1) for λ on the interval [0, χ] where y N (1) approaches to infinity when λ approaches to χ.The available results for λ obtained by the present are summarized in Table 2.This means, the orthogonality relation holds.We notice that the eigenvalues satisfy the property where p(x) = q(x) = 1, and r(x) = 0.
Using the procedure described in Example (4.1), we scan the function y N (1) for λ on the interval [0, χ] where y N (1) approaches to infinity when λ approaches to χ.The available results for λ obtained by the present method are summarized in Table 3.It worth mention that, there are eigenvalues for all 1 2 < α ≤ 1.For example,the first eigenvalue for α = 0.5001 is 1.68861.Let δ i, j = ∫ 1 0 y i (x) y j (x) q(x)dx .This means, the orthogonality relation holds.We notice that the eigenvalues satisfy the property λ 1 ≤ λ 2 ≤ ....

Conclusion
In this paper, we study the eigenvalues of regular 2α-order fractional Sturm-Liouville problem for 1 2 < α ≤ 1.We used the RKM to approximate the eigenvalues.We present three examples.From these examples, we notice that our technique is very efficient for computing the eigenvalues of the fractional second order problems.We end this section by the following remarks.
• The results in this paper confirm that RKM is a powerful and can be used in different fields of sciences and engineering.

Table 1 :
Eigenvalues for different values of α.

Table 2 :
Eigenvalues for different values of α.

Table 3 :
Eigenvalues for different values of α.