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The Scientific World Journal
Volume 2013 (2013), Article ID 685695, 8 pages
Fractional Solutions of Bessel Equation with -Method
Department of Mathematics, Firat University, 23119 Elazig, Turkey
Received 30 April 2013; Accepted 16 July 2013
Academic Editors: R. M. Guedes, Z. Mukandavire, and R. Wangkeeree
Copyright © 2013 Erdal Bas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 -fractional calculus operator method, we derive the fractional solutions of the equation.
1. 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, . 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 N. H. Abel, M. Caputo, L. Euler, J. Fourier, A. K. Grunwald, J. Hadamard, G. H. Hardy, O. Heaviside, H. J. Holmgren, P. S. Laplace, G. W. Leibniz, A. V. Letnikov, J. Liouville, and B. Riemann, have contributed to this field; all these references can be seen in [1–5]. 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. , Nishimoto [6–12], and Podlubny  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, 13–16]. 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 [6–12, 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 .
Definition 1 (cf. [6–10, 12, 20]). Let where is a curve along the cut joining two points and , is a curve along the cut joining two points and , is a domain surrounded by , and is a domain surrounded by . (Here contains the points over the curve .)
Moreover, let be a regular function in , where , Then is said to be the fractional derivative of of order and is said to be the fractional integral of of order , provided (in each case) that .
Lemma 2 (linearity property). If the functions and are single-valued and analytic in some domain , then for any constants and .
Lemma 3 (index law). If the function is single-valued and analytic in some domain , then
Lemma 4 (generalized Leibniz rule). If the functions and are single-valued and analytic in some domain , then where is the ordinary derivative of of order , 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.
2. The -Method Applied to Bessel Equation
Theorem 5. Let and . We consider the nonhomogeneous Bessel equation: and it has particular solutions of the forms where , , (an arbitrary given function), and are given constants.
With some rearrangement of the terms in (23), we have
Here, we choose such that That is,
Set Rewrite (28) in the form
Choose such that That is,
Applying the operator to both members of (36), we find the following equality:
Choose such that We then have from (39).
Next, writing 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) from (44), (42), (35), and (27).
Applying the operator to both members of (47), we have
(II) Let .
3. 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 for , where is an arbitrary constant.
Application 1. If we substitute , and in (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):
Application 2. If we substitute and 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.
4. Two Further Cases of Modified Bessel Equation
Theorem 10. In the similar way as in the previous sections, we can solve the following nonhomogeneous modified Bessel equation: which are obtained by replacing by ( instead of ) in (16); that is,
Choose as follows: That is
Choose as follows: That is,
Applying the operator to both members of (87), we then obtain
Choose such that We then have from (89).
(ii. 2) Similarly, in the case when , we obtain
Let . In the same way as in the procedure in , replacing by – (ii. 1) and (ii. 2), we can obtain and .
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.
The authors sincerely thank the reviewers for their valuable suggestions and useful comments.
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