The Scientific World Journal

Volume 2013 (2013), Article ID 685695, 8 pages

http://dx.doi.org/10.1155/2013/685695

Research Article

Fractional Solutions of Bessel Equation with -Method

Erdal Bas, Resat Yilmazer, and Etibar Panakhov

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.

Abstract.

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, [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 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. [4], Nishimoto [6–12], 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, 13–16]. 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 [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    and the Weyl operator   , which are defined by [17, 19].

Consider

provided that the defining integrals in (3) and (4) exist, being the set of positive integers.

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 .

Finally, let the fractional calculus operator (Nishimoto’s operator) be defined by (cf. [6–10])

with

We find it to be worthwhile to recall here the following useful lemmas and properties associated with the fractional differintegration which was defined earlier (cf. e.g., [6–10, 12]).

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.

Remark 6. The cases of (19) and (20) coincide with those (17) and (18).

Proof. Set

Thus

Putting (21) and (22) in (16), we obtained

With some rearrangement of the terms in (23), we have

Here, we choose such that

That is,

(I) Let . From (21) and (24), we have

Set

Rewrite (28) in the form

At this point, differentiating two times,

and substituting from (29) and (31) in (30), we can express (30) as

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:

Using (3)–(12), we have

Making use of the relations (38), rewrite (37) in the following form:

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).

Inversely, (44) satisfies (43); then

satisfies (41). Therefore, (17) satisfies (16) because we have (27), (35), (44), and (45).

(I) (ii): In the case when , we have

from (29) and (32).

Applying the operator to both members of (47), we have

Choosing such that

and replacing

we then obtain

from (48). A particular solution of (51) is given by

Thus, we have (18) from (52), (50), (46), and (27).

(II) Let .

With the help of the similar method in , replacing by in (I) (i) and (I) (ii), we have other solutions (19) and (20) different from (17) and (18), respectively, if .

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.

Remark 8. In the case when (56) and (57) coincide with (54) and (55).

Proof. When in Section 2, we have

for and , instead of (43) and (51).

Therefore, we get (54) for (58) and (55) for (59).

And, for , replacing by in (58) and (59), we have (56) and (57).

Theorem 9. Let and just as in Theorem 5. Then the nonhomogeneous modified Sturm-Liouville equation (16) is satisfied by the fractional differintegrated functions

Proof. It is clear by Theorems 5 and 7.

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):

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.

Figure 1: Rey.

Figure 2: Imy.

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.

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,

(i) Therefore, the solutions for (74) are given by replacing by in (17), (18), (19), and (20) as follows:

(ii) In the same way, for the solutions for (75), substituting the relations (21), and (22) into (75), we have

Choose as follows:

That is

Let . From (21) and (77), we have

Next, set (29); then (81) is rewritten in the form

Substituting the relations (29) and (31) into (82), we have

Choose as follows:

That is,

(ii. 1) In the case when , we have

from (29) and (83).

Applying the operator to both members of (87), we then obtain

Using (4), (10), and (11), we have

Choose such that

We then have

from (89).

Next, writing

we obtain

from (91). This is an ordinary differential equation of the first order which has a particular solution

We finally obtain the solution

from (94), (92), (86) and (80).

(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 .

Theorem 11. In the homogeneous case for (74) with , using the solutions (54), (55), (56), and (57) and replacing by , we obtain

for , where is an arbitrary constant.

5. 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.

Acknowledgment

The authors sincerely thank the reviewers for their valuable suggestions and useful comments.

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