Fractional Solutions of Bessel Equation with N-Method

This paper deals with the design fractional solution of Bessel equation. We obtain explicit solutions of the equation with the help of fractional calculus techniques. Using the N-fractional calculus operator N ν method, we derive the fractional solutions of the equation.


Introduction, Definitions, and Preliminaries
Fractional calculus has an important place in the field of math. Firstly, L'Hospital and Leibniz were interested in the topic in 1695, [1]. Fractional calculus is an area of applied mathematics that deals with derivatives and integrals of arbitrary orders and their applications in science, engineering, mathematics, economics, and other fields. The seeds of fractional derivatives were planted over 300 years ago. Since then many efficient mathematicians of their times, such as Riemann, have contributed to this field; all these references can be seen in [1][2][3][4][5]. The mathematics involved appeared very different applications of this field. Fractional calculus has been applied to almost every field of science. They are viscoelasticity, electrical engineering, electrochemistry, biology, biophysics and bioengineering, signal and image processing, mechanics, mechatronics, physics, and control theory. During the last decade, Samko et al. [4], Nishimoto [6][7][8][9][10][11][12], and Podlubny [3] have been helpful in introducing the field to engineering, science, economics and finance, and pure and applied field. Furthermore, there were many studies in this field [5,[13][14][15][16]. Various scientists have studied that concept. The progress in this field continues [2-4, 6-12, 17-20].
-Fractional calculus is a very interesting method because this method is applied to singular equation. Note that fractional solutions can be obtained for kinds of singular equation via this method [6][7][8][9][10][11][12]17]. In this paper, our aim is to apply the same way for singular Sturm-Liouville equation with Bessel potential and find fractional solutions of this equation. Furthermore, we give some applications and their graphs of fractional solutions of the equation. Now, consider the following the Bessel equation: where and are real numbers. By means of the substitution = √ (1) reduces to the form Bessel equation for having the analogous singularity is given in [21].
The differintegration operators and their generalizations [6-11, 17, 18] have been used to solve some classes of differential equations and fractional differential equations.
Two of the most commonly encountered tools in the theory and applications of fractional calculus are provided by the Riemann-Liouville operator ( ∈ C) and the Weyl operator ( ∈ C), which are defined by [17,19].

2
The Scientific World Journal provided that the defining integrals in (3) and (4) exist, N being the set of positive integers.

Lemma 2 (linearity property). If the functions ( ) and ( ) are single-valued and analytic in some domain Ω ⊆ C, then
for any constants ℎ 1 and ℎ 2 .

Lemma 3 (index law). If the function ( ) is single-valued and analytic in some domain
Lemma 4 (generalized Leibniz rule). If the functions ( ) and ( ) are single-valued and analytic in some domain Ω ⊆ C, then where ( ) is the ordinary derivative of ( ) of order ( ∈ N 0 := N ∪ {0}), being tacitly assumed (for simplicity) that ( ) is the polynomial part (if any) of the product ( ) ( ).
Property 1. For a constant , Property 2. For a constant , Property 3. For a constant , Now, let apply -fractional method to nonhomogeneous Bessel equation.
Putting (21) and (22) in (16), we obtained 2 With some rearrangement of the terms in (23), we have Here, we choose such that That is, (I) Let = + (1/2). From (21) and (24), we have Rewrite (28) in the form At this point, differentiating two times, and substituting from (29) and (31) in (30), we can express (30) as 2 + 1 (2 + 2 + 1) Choose such that That is, (I) (i): For instance, taking = − √ , we have from (29) and (32). Applying the operator ] to both members of (36), we find the following equality: The Scientific World Journal Using (3)-(12), we have Making use of the relations (38), rewrite (37) in the following form: Choose ] such that We then have we obtain the following equality from (41): This is an ordinary differential equation of the first order which has a particular solution, Making use of the reverse process to obtain , we finally obtain the solution (17) Choosing ] such that and replacing we then obtain from (48). A particular solution of (51) is given by Thus, we have (18) (19) and (20) different from (17) and (18), respectively, if ̸ = 0.

The Operator ] -Method to a Homogeneous Bessel Equation
Theorem 7. If ∈ ℘, just as in Theorem 5, then the homogeneous Bessel equation has solutions of the forms Proof. When = 0 in Section 2, we have for = − √ and = √ , instead of (43) and (51). Therefore, we get (54) for (58) and (55) (16), then we obtain the following equation: and its solution is By performing the necessary operations in (62), we get where Riemann Liouville operator is and using the definitions of Riemann Liouville operator again, we obtain the following solution: Now, let us show that the last equality is the solution of (61): Obviously, if (66) and (67) are put in (61), it is satisfied. The graph of the solution of (61) is given in Figures 1 and 2.
Application 2. If we substitute = −1 and = 0 in (53), then we obtain the following equation: and its solution is We prove that is the solution of (67). With the help of Riemann Liouville operator, Now, let us show that the last equality is the solution of (67), Obviously, if (71) and (72) are put into (68), it is satisfied. The graph of the solution of (68) is given in Figure 3.
The Scientific World Journal 7 Applying the operator ] to both members of (87), we then obtain We then have from (89). Next, writing We finally obtain the solution Let ] = − + (1/2). In the same way as in the procedure in (ii), replacing by -(ii. 1) and (ii. 2), we can obtain ( ) and ( V) .
for ̸ = 0, where is an arbitrary constant.

Conclusion
The -fractional calculus operator ] -method is applied to the nonhomogeneous and homogeneous Bessel equation. Explicit fractional solutions of Bessel equations are obtained. Furthermore, similar solutions were obtained for the modified same equation by using the method.