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

Authors: Resat Yilmazer; Okkes Ozturk
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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 n to arbitrary order α, {xn,∂n/∂xn}→{xα,∂α/∂xα} in a reasonable way. The first question was already raised by Leibniz (1646–1716): can we define a derivative of the order 1/2, 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,(1)aDt-αf(t)=1Γ(α)∫at‍f(τ)(t-τ)α-1dτ1Γ(α)∫at‍f(τ)τ(t>a,α>0),aDtαf(t)=1Γ(k-α)(ddt)k∫at‍f(τ)(t-τ)k-α-1dτ1Γ(k-α)(ddt)kk-α-1(k-1≤α<k),
where k∈N, N being the set of positive integers and Γ stands for Euler’s function gamma.Definition 1 (cf. [10–14, 17]).
If the functionf(z) is analytic (regular) inside and on C, where C={C-,C+}, C- is a contour along the cut joining the points z and -∞+iIm(z), which starts from the point at -∞, encircles the point z once counter-clockwise, and returns to the point at -∞,  and C+ is a contour along the cut joining the points z and ∞+iIm(z), which starts from the point at ∞, encircles the point z once counter-clockwise, and returns to the point at ∞,
(2)fμ(z)=(f(z))μ=Γ(μ+1)2πi∫C‍f(t)(t-z)μ+1dt(μ≠-1,-2,…),f-n(z)=limμ→-nfμ(z)(n∈ℤ+),
where t≠z,
(3)-π≤arg(t-z)≤πforC-,0≤arg(t-z)≤2πforC+,
then fμ(z)  (μ>0) is said to be the fractional derivative of f(z) of order μ and fμ(z)  (μ<0) is said to be the fractional integral of f(z) of order -μ, provided (in each case) that
(4)|fμ(z)|<∞(μ∈ℝ).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).
Letf(z) and g(z) be analytic and single-valued functions. If  fμ and gμ exist, then
(5)(i)(h1f(z))μ=h1fμ(z)(ii)(h1f(z)+h2g(z))μ=h1fμ(z)+h2gμ(z),
where h1 and h2 are constants and μ∈ℝ; z∈ℂ.Lemma 3 (Index Law).
Letf(z) be an analytic and single-valued function. If (fρ)μ and (fμ)ρ exist, then
(6)(fρ(z))μ=fρ+μ(z)=(fμ(z))ρ,
where ρ,μ∈ℝ and z∈ℂ, and |Γ(ρ+μ+1)/Γ(ρ+1)Γ(μ+1)|<∞.Lemma 4 (Generalized Leibniz Rule).
Letf(z) and g(z) be analytic and single-valued functions. If fμ and gμ exist, then
(7)(f(z)·g(z))μ=∑n=0∞Γ(μ+1)Γ(μ-n+1)Γ(n+1)fμ-n(z)·gn(z),
where μ∈ℝ; z∈ℂ and |Γ(μ+1)/Γ(μ-n+1)Γ(n+1)|<∞.Property 1.
For a constantλ,
(8)(eλz)μ=λμeλz(λ≠0;μ∈ℝ;z∈ℂ).Property 2.
For a constantλ,
(9)(e-λz)μ=e-iπμλμe-λz(λ≠0;μ∈ℝ;z∈ℂ).Property 3.
For a constantλ,
(10)(zλ)μ=e-iπμΓ(μ-λ)Γ(-λ)zλ-μ(μ∈ℝ;z∈ℂ;|Γ(μ-λ)Γ(-λ)|<∞).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]).
LetP(z;p) and Q(z;q) be polynomials in z of degrees p and q, respectively, defined by
(11)P(z;p)≔∑k=0pakzp-k=a0∏j=1p(z-zj)(a0≠0,p∈ℕ),Q(z;q):=∑k=0qbkzq-k(b0≠0,q∈ℕ).
Suppose also that f-ν≠0 exists for a given function f.
Then, the nonhomogeneous linear ordinary fractional differintegral equation(12)P(z;p)ϕμ(z)+[∑k=1p(νk)Pk(z;p)+∑k=1q(νk-1)Qk-1(z;q)]×ϕμ-k(z)+(νq)q!b0ϕμ-q-1(z)=f(z)000000000Qk-1(z;q)0000000(μ,ν∈ℝ,p,q∈ℕ)
has a particular solution of the form
(13)ϕ(z)=((f-ν(z)P(z;p)eH(z;p,q))-1e-H(z;p,q))ν-μ+10000000μ+1000(z∈ℂ∖{z1,…,zp}),
where, for convenience,
(14)H(z;p,q):=∫zQ(ζ;q)P(ζ;p)dζ(z∈ℂ∖{z1,…,zp}),
provided that the second member of (13) exists.
Furthermore, the homogeneous linear ordinary fractional differintegral equation(15)P(z;p)ϕμ(z)+[∑k=1p(νk)Pk(z;p)+∑k=1q(νk-1)Qk-1(z;q)]×ϕμ-k(z)+(νq)q!b0ϕμ-q-1(z)=0000000000Qk-1(z;q)(μ,ν∈ℝ,p,q∈ℕ)
has solutions of the form
(16)ϕ(z)=K(e-H(z;p,q))ν-μ+1,
where K is an arbitrary constant and H(z;p,q) 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 givenα-dimensional fractional space with a Coulomb potential depending on a parameter.The Schrödinger equation to start with is given by(17)[-ℏ22mrα-1∂∂r(rα-1∂∂r)+ℓ22mr2-e2κrδ-2]φ(r,θ)=Eφ(r,θ),
where ℓ2 corresponds to the angular momentum operator given by
(18)ℓ2φ(r,θ)=[-ℏ2sinα-2∂∂θ(sinα-1∂∂θ)]φ(r,θ)=ℓ(ℓ+α-2)φ(r,θ),
where α is the dimension of a solid (1≤α≤3) and the radial distance r(0≤r≤∞) and related angle θ(0≤θ≤π) measured relative to an axis passing through the origin are two coordinates describing r in the α-dimensional (αD) space. The constant κ has the value of 1/4πϵ0 for δ=3 and is generally defined as [20]
(19)κ=Γ(δ/2)2πδ/2(δ-2)ϵ0(δ>2).
Looking for solutions of (17) in the form
(20)φ(r,θ)=R(r)Φ(θ),
we easily find that
(21)R′′(r)+α-1rR′(r)+[2mℏ2(E+e2κrδ-2)-ℓ(ℓ-α-2)r2]R(r)=0,Φ′′(θ)+(α-2)cotθΦ′(θ)+ℓ(ℓ-α-2)Φ(θ)=0.
The angular equation (18) has solutions in terms of Gegenbauer polynomials Cℓα/2-1(cosθ) as follows:
(22)Φℓ(θ)=Hℓ(α)Cℓα/2-1(cosθ)(ℓ=0,1,2,…,n-1),
where Hℓ is the normalization factor and is given by [21],
(23)Hℓ(α)={Γ(α2-1)[ℓ!(ℓ+α/2-1)23-απΓ(ℓ+α-2)]1/2(α≠2),1(2π)1/2(ℓ≠0)or12π1/2(ℓ=0)(α=2).
To solve the radial equation R(r), let us use the substitutions
(24)R(r)=rℓe-krϕ(r),
where k2=-2mE/ℏ2. We arrive at the following differential equation:
(25)zϕ′′(z)+[(2ℓ+α-1)-z]ϕ′(z)+[b23-δk4-δz3-δ-2ℓ+α-12]ϕ(z)=0,
where we use the substitutions
(26)z=2kr,b=me2κℏ2.
It is worthwhile to mention that for 3D(δ=3), we arrive at the special case as given in reference [21].Consider the differential equation(27)zd2ϕdz2+(τ-z)dϕdz+(σz3-δ-τ2)ϕ(z)=0,
where
(28)τ=2ℓ+α-1,σ=b23-δk4-δ.(i)
Letδ=2. For this δ (27) becomes the differential equation
(29)zd2ϕdz2+(τ-z)dϕdz+(σz-τ2)ϕ(z)=0.
For (29), we use the substitution
(30)ϕ(z)=ez/2z-τ/2w(z).
Thus, we have
(31)dϕdz=ez/2z-τ/2-1[zdwdz+12(z-τ)w(z)],(32)d2ϕdz2(z)=ez/2z-τ/2-2×{z2d2wdz2+z(z-τ)dwdz+14+14[(z-τ)2+2τ]w(z)d2wdz2}.
After substituting (30), (31), and (32) into (29) and doing some simplifications, we obtain at the differential equation
(33)z2d2wdz2+[2τ-τ24+(σ-14)z2]w(z)=0.
The transformation
(34)w(z)=z1/2φ(z)
has first and second derivative
(35)dwdz=z1/2[dφdz+12zφ(z)],(36)d2wdz2=z1/2[d2φdz2+1zdφdz-14z2φ(z)].
Finally, substituting (34) and (36) into (33) and doing simplifications we arrived at the equation
(37)z2d2φdz2+zdφdz+[(4σ-12z)2-(τ-12)2]φ(z)=0.(ii)
Letδ=4. For this δ (27) becomes the following differential equation:
(38)zd2ϕdz2+(τ-z)dϕdz+(σz-τ2)ϕ(z)=0.
For (38), we use the substitution
(39)ϕ(z)=ez/2z-τ/2u(z).
Therefore, we obtain
(40)dϕdz=ez/2z-τ/2-1[zdudz+12(z-τ)u(z)],d2ϕdz2=ez/2z-τ/2-2×{z2d2udz2+z(z-τ)dudz++14[(z-τ)2+2τ]u(z)d2udz2}.
After substituting (39) and (40) into (38) and doing simplifications, we arrived at the equation
(41)4z2d2udz2+[4σ-τ2+2τ-z2]u(z)=0.
Similarly, for (41), we use the transformation
(42)u(z)=z1/2φ(z).
Thus, we have
(43)dudz=z1/2[dφdz+12zφ(z)],(44)d2udz2=z1/2[d2φdz2+1zdφdz-14z2φ(z)].
Finally, substitute (42) and (44) into (41) and do simplifications to obtain the equation
(45)z2d2φdz2+zdφdz-[(z2)2+((τ-1)2-4σ2)2]φ(z)=0.Our aim is to obtain explicit solutions of (37) and (45), by means of (27), according to different δ.


## 3. Applications of Theorem5 to a Class of Ordinary Second-Order Equations

In order to apply Theorem5 to the following class of ordinary homogeneous differential equations:
(46)Az2d2φdz2+(Bz+C)dφdz+(Dz2+Ez+F)φ(z)=0(z∈ℂ∖{0}),
If the first two lines and last lines of (47) substitute into (46) (A,D≠0), (37) and (45) is obtained, respectively. 						(47)A=B=1,D=4σ-14,C=E=0,F=-(τ-12)2(A=B=1,D=-14,C=E=0,F=σ-(τ-12)2).
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 functionf satisfies the constraint (4) and f-ν≠0, then the nonhomogeneous linear ordinary differential equation
(48)Az2d2φdz2+Bzdφdz+(Dz2+Ez+F)φ(z)=f(z)(A≠0,D≠0)
has a particular solution in the form
(49)φ(z)=zρeλz[(A-1z-ν-1+(2Aρ+B)/Ae2λzzρeλz×(z-ρ-1e-λzf(z))-ν)-1zρeλz×zν-(2Aρ+B)/Ae-2λz]ν-1zρeλzzρeλz(A≠0,D≠0,z∈ℂ∖{0}),
where ρ and λ are given by
(50)ρ=A-B±(A-B)2-4AF2A,λ=±iDA,(51)ν=(2Aρ+B)λ+E2Aλ,
provided that the second member of (49) exists.Furthermore, the homogeneous linear ordinary differential equation(52)Az2d2φdz2+Bzdφdz+(Dz2+Ez+F)φ(z)=0
has solutions of the form
(53)φ(z)=Kzρeλz(zν-(2Aρ+B)/Ae-2λz)ν-1(A≠0,D≠0,z∈ℂ∖{0}),
where K is an arbitrary constant, ρ and λ are given by (50), and ν is given by (51), provided that the second member of (53) exists.Theorem 7.
Under the hypotheses of Theorem6, the homogeneous linear ordinary differential equation
(54)z2d2φdz2+zdφdz+[(4σ-12z)2-(τ-12)2]φ(z)=0
has a particular solituon in the form
(55)φ(z)=Nzν-1/2eλz(z-νe-2λz)ν-1(ν∈ℝ,z∈ℂ∖{0}),
where N is an arbitrary constant and  ρ and λ are given by
(56)ρ=±(τ-1)2,λ=±i4σ-12,ν=2ρ+12,
provided that the second member of (55) exists.Theorem 8.
Under the hypotheses of Theorem6, the homogeneous linear ordinary differential equation
(57)z2d2φdz2+zdφdz-[(z2)2+((τ-1)2-4σ2)2]φ(z)=0
has a particular solution of the form
(58)φ(z)=Hzν-1/2eλz(z-νe-2λz)ν-1(ν∈ℝ,z∈ℂ∖{0}),
where H is an arbitrary constant and  ρ and λ are given by
(59)ρ=±(τ-12)2-σ,λ=∓12,ν=2ρ+12,
provided that the second member of (58) exists.



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