Advanced Theoretical and Applied Studies of Fractional Differential Equations 2013View this Special Issue
Research Article | Open Access
Explicit Solutions of Singular Differential Equation by Means of Fractional Calculus Operators
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.
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
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.  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. , 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  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 , 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 .
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
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.  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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