A Reliable Method for Solving Fractional Sturm – Liouville Problems

In this paper, a reliable method for solving fractional Sturm–Liouville problem based on the operational matrix method is presented. Some of our numerical examples are presented.


Introduction
The Sturm-Liouville theory plays an important role for the development of spectral methods and the theory of self-adjoint operators [1].Several applications on SLPs are studied as boundary-value problems [2].The Sturm-Liouville eigenvalue problem has played an important role in modeling many physical problems.The theory of the problem is well developed and many results have been obtained concerning the eigenvalues and corresponding eigenfunctions.It should be noted that, since finding analytical solutions for this problem is an extremely difficult task, several numerical algorithms have been developed to seek approximate solutions.However, to date, mostly integer-order differential operators in SLPs have been used, and such operators do not include any fractional differential operators.Fractional calculus is a theory which unifies and generalizes the notions of integer-order differentiation and integration to any real order [3][4][5].
Recently, the fractional Sturm-Liouville problems were formulated in [6,7].Authors in these papers considered several types of the fractional Sturm-Liouville equations and they investigated the eigenvalues and eigenfunctions properties of the fractional Sturm-Liouville operators.
Djrbashian [8] studied the existence of a solution to the fractional boundary value problem.In [9], authors discussed the aforementioned relation between eigenvalues and zeros of Mittag-Leffler function.In [10], they used the Homotopy Analysis method while, in [11], they used the fractional differential transform method to compute the eigenvalues.In [12], researchers used the Fourier series to solve this problem while, in [13,14], they used the method of Haar wavelet operational matrix.In [15][16][17][18][19], researchers extended the scope of some spectral properties of fractional Sturm-Liouville problems.Variational methods and Inverse Laplace transform method were applied in [20,21], respectively.Recently, in [22], authors constructed numerical schemes using radial basis functions while, in [23], they used Galerkin finite element method for such problems.Greenberg and Marletta [24,25] developed their own code using Theta Matrices (SLEUTH).In [26], researchers implemented the iterated variation method.
In this article, we present a numerical technique for solving class of FSLPs of the form subject to where c 0 , c 1 , c 2 and c 3 are constants such that det c 0 c 1 c 2 c 3 = 0, f (t), g(t), h(t) are continuous functions with f (t), g(t) > 0 for all t ∈ [0, 1], and D γ is the Caputo derivative.Next, we present some results related to the Caputo fractional derivative, as well as the definition of the fractional-order functions.
For y ∈ L 1 [0, 1] and γ ≥ 0: Let ∆ n be defined by The inner product on the set ∆ n is given by Theorem 1.The sequence of functions defined as follows are orthogonal: Assume the result of the theorem is true for i > 1.Then, for any j ∈ {0, 1, ..., i − 2}, we have

Operational Matrices of Fractional Integration
A set of l Block Pulse Functions (BPFs) in the interval [0, 1) are given by {b for i = 0, 1, .., l − 1.The following are some of the BPFs properties where , and Theorem 2. Let I γ be the Rimann-Liouville functional operator.Then, where Proof.For each i = 0, 1, ..., l − 1, we can write I γ b i as Multiply both sides by b r (t), for 0 ≤ r ≤ l − 1, then integrate both sides to get For more details, see [28,29].
. Then, there exists an M × l matrix Q γ such that where Proof.It is easy to see that y i (t) ∈ L 2 [0, 1), for each i = 0 : M − 1.Using Equations ( 10) and ( 11), we get for i = 0 : M − 1 and k = 0 : l − 1 which ends the proof.From now on, let M = l.Theorem 4. If 0 < γ < 1, then Q γ l×l is nonsingular matrix.

Operational Matrix of Fractional Integration
where Approximate the function y(t) by where Proof.Let H γ l be given by From Equations ( 13) and ( 17), we get and Combining Equations ( 18) and ( 19), we get Therefore,

Method of Solution
Using Equations ( 10) and ( 13), we get Thus, where = D γ y(0).Theorem 5 and Equations ( 10) and ( 13) imply that where ψ = y(0).Therefore, Hence, Using the boundary conditions in Equations ( 2) and ( 3), we get the following cases . Thus, We use the collocation points Substitute these values into Equation ( 23) and take the transpose of both sides to get a system of linear equations in terms of U of the form To have a nonzero solution to the system in Equation (24), G(µ) must be nonsingular.Thus, Therefore, we find the eigenvalues from Equation ( 25) and we find the corresponding eigenfunctions from Equation (21).

Numerical Results
We present two examples for l = 16.In this paper, we focus only on the eigenvalues.
Using the procedure described in the previous section, the generated eigenvalues are reported in Table 1.For γ = 1, the exact eigenvalues are well-known and they are given by It is worth mentioning that the eigenvalues of the problem in this example approach to n 2 π 2 when γ approaches to 1. Let For γ = 0.75, δ 1,2 = 5.7 × 10 −16 .Sample of these values for γ = 0.95 are given as Similarly, for γ = 0.99, This means the orthogonality relation holds.We notice that the eigenvalues satisfy the increasing property.

Example 2. Consider
Using the procedure described in the previous section, the generated eigenvalues are reported in Table 2.This means the orthogonality relation holds.We notice that the eigenvalues satisfy the property µ 1 ≤ µ 2 ≤ ....

Conclusions
In this article, a reliable method for solving fractional Sturm-Liouville problem based on the operational matrix method is presented.Two of our numerical examples are presented.From the previous discussion, we notice the following.

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From previous section, the orthogonality property 1 0 y i (t) y j (t) q(t) = 0, i = j holds.

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The proposed method can be generalized to other applications in Physics and Engineering.

Table 1 .
Eigenvalues for different values of γ.