Abstract and Applied Analysis

Volume 2013 (2013), Article ID 715258, 6 pages

http://dx.doi.org/10.1155/2013/715258

Research Article

Explicit Solutions of Singular Differential Equation by Means of Fractional Calculus Operators

Resat Yilmazer and Okkes Ozturk

Department of Mathematics, Firat University, 23119 Elazig, Turkey

Received 8 July 2013; Revised 6 September 2013; Accepted 10 September 2013

Academic Editor: Juan J. Trujillo

Copyright © 2013 Resat Yilmazer and Okkes Ozturk. 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.

Recently, several authors demonstrated the usefulness of fractional calculus operators in the derivation of particular solutions of a considerably large number of linear ordinary and partial differential equations of the second and higher orders. By means of fractional calculus techniques, we find explicit solutions of second-order linear ordinary differential equations.

1. Introduction, Definitions, and Preliminaries

The widely investigated subject of fractional calculus (i.e., calculus of derivatives and integrals of any arbitrary real or complex order) has gained considerable importance and popularity during the past three decades or so, due chiefly to its demonstrated applications in numerous seemingly diverse fields of science and engineering (see, for details, [1–6]). The fractional calculus provides a set of axioms and methods to extend the coordinate and corresponding derivative definitions from integer to arbitrary order , in a reasonable way. The first question was already raised by Leibniz (1646–1716): can we define a derivative of the order , that is, so that a double action of that derivative gives the ordinary one? We can mention that the fractional differential equations are playing an important role in fluid dynamics, traffic model with fractional derivative, measurement of viscoelastic material properties, modeling of viscoplasticity, control theory, economy, nuclear magnetic resonance, geometric mechanics, mechanics, optics, signal processing, and so on.

The differintegration operators and their generalizations [7–16] have been used to solve some classes of differential equations and fractional differential equations.

Some of most obvious formulations based on the fundamental definitions of Riemann-Liouville fractional integration and fractional differentiation are, respectively,

where , being the set of positive integers and stands for Euler’s function gamma.

Definition 1 (cf. [10–14, 17]). If the function is analytic (regular) inside and on , where , is a contour along the cut joining the points and , which starts from the point at , encircles the point once counter-clockwise, and returns to the point at ,  and is a contour along the cut joining the points and , which starts from the point at , encircles the point once counter-clockwise, and returns to the point at ,

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

We find it to be worthwhile to recall here the following useful lemmas and properties associated with the fractional differintegration which is defined above (cf., e.g., [10–14, 18]).

Lemma 2 (Linearity). Let and be analytic and single-valued functions. If and exist, then

where and are constants and ; .

Lemma 3 (Index Law). Let be an analytic and single-valued function. If and exist, then

where and , and .

Lemma 4 (Generalized Leibniz Rule). Let and be analytic and single-valued functions. If and exist, then

where ; and .

Property 1. For a constant ,

Property 2. For a constant ,

Property 3. For a constant ,

Some of the most recent contributions on the subject of particular solutions of linear ordinary and partial fractional differintegral equations are those given by Tu et al. [19] who presented unification and generalization of a significantly large number of widely scattered results on this subject. We begin by recalling here one of the main results of Tu et al. [19], involving a family of linear ordinary fractional differintegral equations, as Theorem 5 below.

Theorem 5 (Tu et al. [19, p. 295, Theorem 1; p. 296, Theorem 2]). Let and be polynomials in of degrees and , respectively, defined by

Suppose also that exists for a given function .

Then, the nonhomogeneous linear ordinary fractional differintegral equation

has a particular solution of the form

where, for convenience,

provided that the second member of (13) exists.

Furthermore, the homogeneous linear ordinary fractional differintegral equation

has solutions of the form

where is an arbitrary constant and is given by (14), provided that the second member of (16) exists.

2. Schrödinger Equation

In this stud, the main aim is to investigate the Schrödinger equation in a given -dimensional fractional space with a Coulomb potential depending on a parameter.

The Schrödinger equation to start with is given by

where corresponds to the angular momentum operator given by

where is the dimension of a solid and the radial distance and related angle measured relative to an axis passing through the origin are two coordinates describing in the -dimensional space. The constant has the value of for and is generally defined as [20]

Looking for solutions of (17) in the form

we easily find that

The angular equation (18) has solutions in terms of Gegenbauer polynomials as follows:

where is the normalization factor and is given by [21],

To solve the radial equation , let us use the substitutions

where . We arrive at the following differential equation:

where we use the substitutions

It is worthwhile to mention that for , we arrive at the special case as given in reference [21].

Consider the differential equation

where

Let . For this (27) becomes the differential equation

For (29), we use the substitution

Thus, we have

After substituting (30), (31), and (32) into (29) and doing some simplifications, we obtain at the differential equation

The transformation

has first and second derivative

Finally, substituting (34) and (36) into (33) and doing simplifications we arrived at the equation

Let . For this (27) becomes the following differential equation:

For (38), we use the substitution

Therefore, we obtain

After substituting (39) and (40) into (38) and doing simplifications, we arrived at the equation

Similarly, for (41), we use the transformation

Thus, we have

Finally, substitute (42) and (44) into (41) and do simplifications to obtain the equation

Our aim is to obtain explicit solutions of (37) and (45), by means of (27), according to different .

3. Applications of Theorem 5 to a Class of Ordinary Second-Order Equations

In order to apply Theorem 5 to the following class of ordinary homogeneous differential equations:

If the first two lines and last lines of (47) substitute into (46) (), (37) and (45) is obtained, respectively.

Indeed, by applying Theorem 5 in order to find explicit solutions of the homogeneous differential equation (46), Lin et al. [22] deduced the following result.

Theorem 6 (see [22, Theorem 3, p. 39]). If the given function satisfies the constraint (4) and , then the nonhomogeneous linear ordinary differential equation

has a particular solution in the form

where and are given by

provided that the second member of (49) exists.

Furthermore, the homogeneous linear ordinary differential equation

has solutions of the form

where is an arbitrary constant, and are given by (50), and is given by (51), provided that the second member of (53) exists.

Theorem 7. Under the hypotheses of Theorem 6, the homogeneous linear ordinary differential equation

has a particular solituon in the form

where is an arbitrary constant and and are given by

provided that the second member of (55) exists.

Theorem 8. Under the hypotheses of Theorem 6, the homogeneous linear ordinary differential equation

has a particular solution of the form

where is an arbitrary constant and and are given by

provided that the second member of (58) exists.

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