COMPUTING EIGENELEMENTS OF STURM – LIOUVILLE PROBLEMS OF FRACTIONAL ORDER VIA FRACTIONAL DIFFERENTIAL TRANSFORM METHOD

In this study, fractional differential transform method (FDTM), which is a semi analytical-numerical technique, is used for computing the eigenelements of the Sturm-Liouville problems of fractional order. The fractional derivatives are described in the Caputo sense. Three problems are solved by the present method. The calculated results are compared closely with the results obtained by some existing techniques in literature. It is observed that FDTM can be utilized as an alternative tool for the solution of this type of problems. Keywords-Fractional differential transform method, Caputo derivative, Sturm-Liouville prolems, Eigenelement

The organization of this paper is as follows: In Section 2, we briefly introduce the fractional differential transform method.In Section 3, the mentioned scheme in Section 3 is used to to approximate eigenelements of the Sturm-Liouville problems of fractional order.Also, the accuracy and efficiency of the scheme is investigated with three numerical illustrations in that section.Finally, Section 4 consists of some brief conclusions.

FRACTIONAL DIFFERENTIAL TRANSFORM METHOD
The fractional differentiation in Riemann-Liouville sense is defined by Let us expand the analytical and continuous function ) (x f in terms of a fractional power series as follows: where  is the order of fraction and In order to avoid fractional initial and boundary conditions, we define the fractional derivative in the Caputo sense.The relation between the Riemann-Liouville operator and Caputo operator is given by (3) in Eq. (2) and using Eq.(3), we obtain fractional derivative in the Caputo sense [23] as follows: . ) ( Since the initial conditions are implemented to the integer order derivatives, the transformation of the initial conditions are defined as follows: where n is the order of fractional differential equation considered.
By the following proposition we recall properties of the FDT from , where a is a constant.

If
, where

NUMERICAL EXAMPLES
In this section, we present, using the method outlined in the previous section, numerical results for three eigenvalue problems of fractional order.We are interested in the numerical computation of the eigenvalues and the corresponding normalized eigenfunctions for the considered problems.All the results are calculated by using the symbolic calculus software Mathematica.Example 1.Consider the regular fractional eigenvalue problem [21,24] ) Selecting the order of fraction as 2   and using the first, second and fifth properties given in Proposition 2.1, Eq. ( 6) is transformed to the following recurrence relation: The boundary condition in Eq. ( 7) given for 0  x are transformed by using Eq. ( 5) as follows: , is the missing boundary condition.The condition in Eq. ( 7) for 1  x is transformed by using Eq.(2) as follows: Here, N is the number of terms considered in Eq.(2).Eqs.(8) 1.The results obtained with the proposed method are in good agreement with the results obtained in Refs.[21] and [24].
Table 1.Numerical results with comparison to Refs.[21] and [24] The eigenvalues [21] 2.11027708 13.76538223 24.24328676 Ref. [24] 2.1103 13.7654 24.2433The proposed method 2.11027708 13.76538177 24.24328615The eigenfunctions corresponding to the obtained eigenvalues are shown in Fig. 1.The graphical results obtained with the proposed method are also in good agreement with the results obtained in Refs.[21] and [24].) with the following boundary conditions: and using the properties given in Proposition 2.1, Eq. ( 14) can be transformed as follows: The condition for 0  x in Eq. ( 15) can be transformed by using Eq. ( 5) as follows: (17) Also, the condition in Eq. ( 15) for 1  x are transformed by using Eq.(2) as follows: . 0 Repeating the same procedure used for the second example, we calculated terms by using Eqs.( 16)- (17).In this case, we solve .
The numerical results are presented in Table 2.The data in Table 2 shows good agreement with the results obtained in Refs.[21] and [24].
Table 2. Numerical results with comparison to Refs.[21] and [24] The eigenvalues 1  2  3  Ref. [21] 1.66091840 13.55041793 20.51445520 Ref. [24] 1.6609 13.5504 20.5145The proposed method 1.66091840 13.55041774 20.51445527 The eigenfunctions corresponding to the obtained eigenvalues and their derivatives are shown in Fig. 2. The graphical results are in good agreement with the results of Al-Mdallal [21] and Abbasbandy et al. [24].) where  is the unknown value of the fraction.The condition for 0  x in Eq. ( 22) can be transformed by using Eq.(2) as follows: .
The condition in Eq. ( 22) for 1  x are transformed by using Eq.(5) as follows: .
Using Eqs. ( 24) and ( 23 terms are used respectively.Numerical results with comparison to Ref. [22] is given in Table 3.The results are in good agreement with the results of Al-Mdallal [22].Table 3. Numerical results with comparison to Ref. [22] i (Ref.[22]) The eigenfunctions corresponding to the obtained eigenvalues are shown in Fig. 3.The graphical results are in good agreement with the results of Al-Mdallal [22].