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

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 𝑛 to arbitrary order 𝛼, {𝑥^𝑛,𝜕^𝑛/𝜕𝑥^𝑛}→{𝑥^𝛼,𝜕^𝛼/𝜕𝑥^𝛼} 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, _𝑎𝐷_𝑡^−𝛼𝑓(𝑡)=1Γ(𝛼)^𝑡_𝑎𝑓(𝜏)(𝑡−𝜏)^𝛼−1(),𝑑𝜏𝑡>𝑎,𝛼>0_𝑎𝐷^𝛼_𝑡𝑓(𝑡)=1Γ()𝑑𝑘−𝛼𝑑𝑡^𝑘𝑡_𝑎𝑓(𝜏)(𝑡−𝜏)^{𝑘−𝛼−1}(),𝑑𝜏𝑘−1≤𝛼<𝑘 (1) 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 function 𝑓(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 ∞, 𝑓_𝜇(𝑧)=𝑓(𝑧)_𝜇=Γ𝜇+12𝜋𝑖_𝐶𝑓(𝑡)(𝑡−𝑧)^𝜇+1,𝑓𝑑𝑡𝜇≠−1,−2,…_−𝑛(𝑧)=lim_{𝜇→−𝑛}𝑓_𝜇(𝑧)𝑛∈ℤ⁺, (2) where 𝑡≠𝑧, −𝜋≤arg(𝑡−𝑧)≤𝜋for𝐶⁻,0≤arg(𝑡−𝑧)≤2𝜋for𝐶⁺, (3) then 𝑓_𝜇(𝑧)  (𝜇>0) is said to be the fractional derivative of 𝑓(𝑧) of order 𝜇 and 𝑓_𝜇(𝑧)  (𝜇<0) is said to be the fractional integral of 𝑓(𝑧) of order −𝜇, provided (in each case) that 𝑓_𝜇(𝑧)<∞𝜇∈ℝ. (4)

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 𝑓(𝑧) and 𝑔(𝑧) be analytic and single-valued functions. If 𝑓_𝜇 and 𝑔_𝜇 exist, then (𝑖)ℎ₁𝑓(𝑧)_𝜇=ℎ₁𝑓_𝜇(𝑧)(𝑖𝑖)ℎ₁𝑓(𝑧)+ℎ₂𝑔(𝑧)_𝜇=ℎ₁𝑓_𝜇(𝑧)+ℎ₂𝑔_𝜇(𝑧), (5) where ℎ₁ and ℎ₂ are constants and 𝜇∈ℝ; 𝑧∈ℂ.

Lemma 3 (Index Law). Let 𝑓(𝑧) be an analytic and single-valued function. If (𝑓_𝜌)_𝜇 and (𝑓_𝜇)_𝜌 exist, then 𝑓_𝜌(𝑧)_𝜇=𝑓_𝜌+𝜇(𝑧)=𝑓_𝜇(𝑧)_𝜌, (6) where 𝜌,𝜇∈ℝ and 𝑧∈ℂ, and |Γ(𝜌+𝜇+1)/Γ(𝜌+1)Γ(𝜇+1)|<∞.

Lemma 4 (Generalized Leibniz Rule). Let 𝑓(𝑧) and 𝑔(𝑧) be analytic and single-valued functions. If 𝑓_𝜇 and 𝑔_𝜇 exist, then 𝑓(𝑧)⋅𝑔(𝑧)_𝜇=^∞_𝑛=0Γ𝜇+1ΓΓ()𝑓𝜇−𝑛+1𝑛+1^𝜇−𝑛(𝑧)⋅𝑔_𝑛(𝑧), (7) where 𝜇∈ℝ; 𝑧∈ℂ and |Γ(𝜇+1)/Γ(𝜇−𝑛+1)Γ(𝑛+1)|<∞.

Property 1. For a constant 𝜆, 𝑒^𝜆𝑧_𝜇=𝜆^𝜇𝑒^𝜆𝑧𝜆≠0;𝜇∈ℝ;𝑧∈ℂ. (8)

Property 2. For a constant 𝜆, 𝑒^−𝜆𝑧_𝜇=𝑒^{−𝑖𝜋𝜇}𝜆^𝜇𝑒^−𝜆𝑧𝜆≠0;𝜇∈ℝ;𝑧∈ℂ. (9)

Property 3. For a constant 𝜆, 𝑧^𝜆_𝜇=𝑒^{−𝑖𝜋𝜇}Γ𝜇−𝜆Γ()𝑧−𝜆^𝜆−𝜇Γ𝜇∈ℝ;𝑧∈ℂ;𝜇−𝜆Γ().−𝜆<∞ (10)

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 𝑃(𝑧;𝑝) and 𝑄(𝑧;𝑞) be polynomials in 𝑧 of degrees 𝑝 and 𝑞, respectively, defined by 𝑃𝑧;𝑝∶=^𝑝_𝑘=0𝑎_𝑘𝑧^𝑝−𝑘=𝑎₀^𝑝_𝑗=1𝑧−𝑧_𝑗𝑎₀,𝑄≠0,𝑝∈ℕ𝑧;𝑞∶=^𝑞_𝑘=0𝑏_𝑘𝑧^𝑞−𝑘𝑏₀≠0,𝑞∈ℕ. (11) Suppose also that 𝑓_−𝜈≠0 exists for a given function 𝑓.
Then, the nonhomogeneous linear ordinary fractional differintegral equation 𝑃𝑧;𝑝𝜙_𝜇(𝑧)+^𝑝_𝑘=1𝜈𝑘𝑃_𝑘𝑧;𝑝+^𝑞_𝑘=1𝜈𝑄𝑘−1^𝑘−1𝑧;𝑞×𝜙_𝜇−𝑘(𝑧)+𝜈𝑞𝑞!𝑏₀𝜙_{𝜇−𝑞−1}(𝑧)=𝑓(𝑧)𝜇,𝜈∈ℝ,𝑝,𝑞∈ℕ (12) has a particular solution of the form 𝜙(𝑧)=𝑓_−𝜈(𝑧)𝑒𝑃(𝑧;𝑝)^{𝐻(𝑧;𝑝,𝑞)}₋₁𝑒^{−𝐻(𝑧;𝑝,𝑞)}_{𝜈−𝜇+1}𝑧∈ℂ⧵𝑧₁,…,𝑧_𝑝, (13) where, for convenience, 𝐻𝑧;𝑝,𝑞∶=^𝑧𝑄𝜁;𝑞𝑃𝑧𝜁;𝑝𝑑𝜁𝑧∈ℂ⧵₁,…,𝑧_𝑝, (14) provided that the second member of (13) exists.
Furthermore, the homogeneous linear ordinary fractional differintegral equation 𝑃𝑧;𝑝𝜙_𝜇(𝑧)+^𝑝_𝑘=1𝜈𝑘𝑃_𝑘𝑧;𝑝+^𝑞_𝑘=1𝜈𝑄𝑘−1^𝑘−1𝑧;𝑞×𝜙_𝜇−𝑘(𝑧)+𝜈𝑞𝑞!𝑏₀𝜙_{𝜇−𝑞−1}(𝑧)=0𝜇,𝜈∈ℝ,𝑝,𝑞∈ℕ (15) has solutions of the form 𝜙(𝑧)=𝐾𝑒^{−𝐻(𝑧;𝑝,𝑞)}_{𝜈−𝜇+1}, (16) 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 given 𝛼-dimensional fractional space with a Coulomb potential depending on a parameter.

The Schrödinger equation to start with is given by −ℏ²2𝑚𝑟^𝛼−1𝜕𝑟𝜕𝑟^𝛼−1𝜕+ℓ𝜕𝑟2𝑚²𝑟²−𝑒²𝜅𝑟^𝛿−2𝜑()(),𝑟,𝜃=𝐸𝜑𝑟,𝜃 (17) where ℓ² corresponds to the angular momentum operator given by ℓ²𝜑()=−ℏ𝑟,𝜃²sin^𝛼−2𝜕𝜕𝜃sin^𝛼−1𝜕𝜑()()𝜑(),𝜕𝜃𝑟,𝜃=ℓℓ+𝛼−2𝑟,𝜃 (18) 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] Γ()𝜅=𝛿/22𝜋^𝛿/2(𝛿−2)𝜖₀(𝛿>2). (19) Looking for solutions of (17) in the form 𝜑(𝑟,𝜃)=𝑅(𝑟)Φ(𝜃), (20) we easily find that 𝑅^′′(𝑟)+𝛼−1𝑟𝑅^′(𝑟)+2𝑚ℏ²𝐸+𝑒²𝜅𝑟^𝛿−2−ℓ(ℓ−𝛼−2)𝑟²𝑅(𝑟)Φ=0,_′′(𝜃)+(𝛼−2)cot𝜃Φ^′(𝜃)+ℓ(ℓ−𝛼−2)Φ(𝜃)=0. (21) The angular equation (18) has solutions in terms of Gegenbauer polynomials 𝐶_ℓ^𝛼/2−1(cos𝜃) as follows: Φ_ℓ(𝜃)=𝐻_ℓ(𝛼)𝐶_ℓ^𝛼/2−1()(),cos𝜃ℓ=0,1,2,…,𝑛−1 (22) where 𝐻_ℓ is the normalization factor and is given by [21], 𝐻_ℓ(𝛼)=Γ𝛼2()−1ℓ!ℓ+𝛼/2−12^3−𝛼𝜋Γ(ℓ+𝛼−2)^1/2(),1𝛼≠2(2𝜋)^1/2()1ℓ≠0or2𝜋^1/2(ℓ=0)(𝛼=2). (23) To solve the radial equation 𝑅(𝑟), let us use the substitutions 𝑅(𝑟)=𝑟^ℓ𝑒^−𝑘𝑟𝜙(𝑟), (24) where 𝑘²=−2𝑚𝐸/ℏ². We arrive at the following differential equation: 𝑧𝜙^′′(𝑧)+[(2ℓ+𝛼−1)−𝑧]𝜙^′(𝑧)+𝑏2^3−𝛿𝑘^4−𝛿𝑧^3−𝛿−2ℓ+𝛼−12𝜙(𝑧)=0, (25) where we use the substitutions 𝑧=2𝑘𝑟,𝑏=𝑚𝑒²𝜅ℏ². (26) 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 𝑧𝑑²𝜙𝑑𝑧²+(𝜏−𝑧)𝑑𝜙+𝑑𝑧𝜎𝑧^3−𝛿−𝜏2𝜙(𝑧)=0, (27) where 𝑏𝜏=2ℓ+𝛼−1,𝜎=2^3−𝛿𝑘^4−𝛿. (28)(i) Let 𝛿=2. For this 𝛿 (27) becomes the differential equation 𝑧𝑑²𝜙𝑑𝑧²+(𝜏−𝑧)𝑑𝜙+𝜏𝑑𝑧𝜎𝑧−2𝜙(𝑧)=0. (29) For (29), we use the substitution 𝜙(𝑧)=𝑒^𝑧/2𝑧^−𝜏/2𝑤(𝑧). (30) Thus, we have 𝑑𝜙𝑑𝑧=𝑒^𝑧/2𝑧^{−𝜏/2−1}𝑧𝑑𝑤+1𝑑𝑧2()𝑤(𝑧,𝑑𝑧−𝜏)₂𝜙𝑑𝑧²(𝑧)=𝑒^𝑧/2𝑧^{−𝜏/2−2}×𝑧²𝑑²𝑤𝑑𝑧²+𝑧(𝑧−𝜏)𝑑𝑤+1𝑑𝑧4(𝑧−𝜏)²+2𝜏𝑤(𝑧). (31)(32) After substituting (30), (31), and (32) into (29) and doing some simplifications, we obtain at the differential equation 𝑧²𝑑²𝑤𝑑𝑧²+2𝜏−𝜏²4+1𝜎−4𝑧²𝑤(𝑧)=0. (33) The transformation 𝑤(𝑧)=𝑧^1/2𝜑(𝑧) (34) has first and second derivative 𝑑𝑤𝑑𝑧=𝑧^1/2𝑑𝜑+1𝑑𝑧𝜑(𝑧,𝑑2𝑧₂)𝑤𝑑𝑧²=𝑧^1/2𝑑²𝜑𝑑𝑧²+1𝑧𝑑𝜑−1𝑑𝑧4𝑧²𝜑(𝑧). (35)(36) Finally, substituting (34) and (36) into (33) and doing simplifications we arrived at the equation 𝑧²𝑑²𝜑𝑑𝑧²+𝑧𝑑𝜑+√𝑑𝑧4𝜎−12𝑧²−𝜏−12²𝜑(𝑧)=0. (37)(ii) Let 𝛿=4. For this 𝛿 (27) becomes the following differential equation: 𝑧𝑑²𝜙𝑑𝑧²+(𝜏−𝑧)𝑑𝜙+𝜎𝑑𝑧𝑧−𝜏2𝜙(𝑧)=0. (38) For (38), we use the substitution 𝜙(𝑧)=𝑒^𝑧/2𝑧^−𝜏/2𝑢(𝑧). (39) Therefore, we obtain 𝑑𝜙𝑑𝑧=𝑒^𝑧/2𝑧^{−𝜏/2−1}𝑧𝑑𝑢+1𝑑𝑧2()𝑢(𝑧,𝑑𝑧−𝜏)₂𝜙𝑑𝑧²=𝑒^𝑧/2𝑧^{−𝜏/2−2}×𝑧²𝑑²𝑢𝑑𝑧²+𝑧(𝑧−𝜏)𝑑𝑢+1𝑑𝑧4(𝑧−𝜏)²+2𝜏𝑢(𝑧). (40) After substituting (39) and (40) into (38) and doing simplifications, we arrived at the equation 4𝑧²𝑑²𝑢𝑑𝑧²+4𝜎−𝜏²+2𝜏−𝑧²𝑢(𝑧)=0. (41) Similarly, for (41), we use the transformation 𝑢(𝑧)=𝑧^1/2𝜑(𝑧). (42) Thus, we have 𝑑𝑢𝑑𝑧=𝑧^1/2𝑑𝜑+1𝑑𝑧𝜑(𝑧,𝑑2𝑧₂)𝑢𝑑𝑧²=𝑧^1/2𝑑²𝜑𝑑𝑧²+1𝑧𝑑𝜑−1𝑑𝑧4𝑧²𝜑(𝑧). (43)(44) Finally, substitute (42) and (44) into (41) and do simplifications to obtain the equation 𝑧²𝑑²𝜑𝑑𝑧²+𝑧𝑑𝜑−𝑧𝑑𝑧2²+(𝜏−1)²−4𝜎2²𝜑(𝑧)=0. (45)

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

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: 𝐴𝑧²𝑑²𝜑𝑑𝑧²+(𝐵𝑧+𝐶)𝑑𝜑+𝑑𝑧𝐷𝑧²+𝐸𝑧+𝐹𝜑(𝑧)=0(𝑧∈ℂ⧵{0}), (46) If the first two lines and last lines of (47) substitute into (46) (𝐴,𝐷≠0), (37) and (45) is obtained, respectively. 𝐴=𝐵=1,𝐷=4𝜎−14,𝐶=𝐸=0,𝐹=−𝜏−12²𝐴=𝐵=1,𝐷=−14,𝐶=𝐸=0,𝐹=𝜎−𝜏−12². (47) 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 𝑓 satisfies the constraint (4) and 𝑓_−𝜈≠0, then the nonhomogeneous linear ordinary differential equation 𝐴𝑧²𝑑²𝜑𝑑𝑧²+𝐵𝑧𝑑𝜑+𝑑𝑧𝐷𝑧²𝜑(𝑧)(𝑧)+𝐸𝑧+𝐹=𝑓)(𝐴≠0,𝐷≠0 (48) has a particular solution in the form 𝜑(𝑧)=𝑧^𝜌𝑒^𝜆𝑧𝐴⁻¹𝑧^{−𝜈−1+(2𝐴𝜌+𝐵)/𝐴}𝑒^2𝜆𝑧×𝑧^−𝜌−1𝑒^−𝜆𝑧𝑓(𝑧)_−𝜈−1×𝑧^{𝜈−(2𝐴𝜌+𝐵)/𝐴}𝑒^−2𝜆𝑧_𝜈−1(𝐴≠0,𝐷≠0,𝑧∈ℂ⧵{0}), (49) where 𝜌 and 𝜆 are given by 𝜌=𝐴−𝐵±(𝐴−𝐵)²−4𝐴𝐹2𝐴,𝜆=±𝑖𝐷𝐴,𝜈=2𝐴𝜌+𝐵𝜆+𝐸,2𝐴𝜆 (50)(51) provided that the second member of (49) exists.

Furthermore, the homogeneous linear ordinary differential equation 𝐴𝑧²𝑑²𝜑𝑑𝑧²+𝐵𝑧𝑑𝜑+𝑑𝑧𝐷𝑧²+𝐸𝑧+𝐹𝜑(𝑧)=0 (52) has solutions of the form 𝜑(𝑧)=𝐾𝑧^𝜌𝑒^𝜆𝑧𝑧^{𝜈−(2𝐴𝜌+𝐵)/𝐴}𝑒^−2𝜆𝑧_𝜈−1(𝐴≠0,𝐷≠0,𝑧∈ℂ⧵{0}), (53) 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 Theorem 6, the homogeneous linear ordinary differential equation 𝑧²𝑑²𝜑𝑑𝑧²+𝑧𝑑𝜑+√𝑑𝑧4𝜎−12𝑧²−𝜏−12²𝜑(𝑧)=0 (54) has a particular solituon in the form 𝜑(𝑧)=𝑁𝑧^𝜈−1/2𝑒^𝜆𝑧𝑧^−𝜈𝑒^−2𝜆𝑧_𝜈−1(𝜈∈ℝ,𝑧∈ℂ⧵{0}), (55) where 𝑁 is an arbitrary constant and 𝜌 and 𝜆 are given by ±()𝜌=𝜏−12√,𝜆=±𝑖4𝜎−12,𝜈=2𝜌+12, (56) provided that the second member of (55) exists.

Theorem 8. Under the hypotheses of Theorem 6, the homogeneous linear ordinary differential equation 𝑧²𝑑²𝜑𝑑𝑧²+𝑧𝑑𝜑−𝑧𝑑𝑧2²+(𝜏−1)²−4𝜎2²𝜑(𝑧)=0 (57) has a particular solution of the form 𝜑(𝑧)=𝐻𝑧^𝜈−1/2𝑒^𝜆𝑧𝑧^−𝜈𝑒^−2𝜆𝑧_𝜈−1(𝜈∈ℝ,𝑧∈ℂ⧵{0}), (58) where 𝐻 is an arbitrary constant and 𝜌 and 𝜆 are given by 𝜌=±𝜏−12²1−𝜎,𝜆=∓2,𝜈=2𝜌+12, (59) provided that the second member of (58) exists.