We use a particular fractional generalization of the ordinary differential
equations that we apply to the Riccati equation of constant coefficients. By
this means the latter is transformed into a modified Riccati equation with the
free term expressed as a power of the independent variable which is of the same
order as the order of the applied fractional derivative. We provide the
solutions of the modified equation and employ the results for the case of the
cosmological Riccati equation of FRW barotropic cosmologies that has been
recently introduced by FaraoniComment: 7 pages, 2 figure

A L Madue o

H C Rosu

Hilfer R (ed)

J Socorro

Metzler R

Podlubny I

Ross B

Spanier J

English

arXiv

"We use a particular fractional generalization of the ordinary differential equations that we apply to the Riccati equation of constant coefficients. By this means the latter is transformed into a modified Riccati equation with the free term expressed as a power of the independent variable which is of the same order as the order of the applied fractional derivative. We provide the solutions of the modified equation and employ the results for the case of the cosmological Riccati equation of FRW barotropic cosmologies that has been recently introduced by Faraoni.

Rosu Barbus, Haret-Codratian

L. Madueno

García Díaz, José Socorro

Repositorio IPICYT

This is an author-created, un-copyedited version of an article accepted for publication in Journal of Physics A: Mathematical and General. The publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at https://doi.org/10.1088/0305-4470/36/4/316      Transform of Riccati equation of constant coefficients throughfractional procedureH.C. Rosu1∗, A.L. Madueño2, J. Socorro21 Dept. of Appl. Math., IPICyT, Apdo Postal 3-74 Tangamanga, San Luis Potośı, MEXICO2 Instituto de F́ısica, Universidad de Guanajuato, Apdo Postal E-143, León, MEXICOWe use a particular fractional generalization of the ordinary differentialequations that we apply to the Riccati equation of constant coefficients. Bythis means the latter is transformed into a modified Riccati equation with thefree term expressed as a power of the independent variable which is of thesame order as the order of the applied fractional derivative. We provide thesolutions of the modified equation and employ the results for the case of thecosmological Riccati equation of FRW barotropic cosmologies that has beenrecently introduced by Faraoni.J. Phys. A 36 (4), 1087-1094 (31 January 2003)PACS Numbers: 02.30.Hq; 04.20.Jb arXiv Number: math-ph/0112020I. INTRODUCTIONThe fractional calculus is a generalization of the ordinary differential and integral calculus[1]. The main point is how to think about the derivative of order r, where r is an arbitraryreal or complex number. In 1695, L′Hôspital was the first to ask in a letter to Leibnitz onthe possibility to perform calculations by means of a fractional derivative of order r = 12.Leibnitz answered that the question looked as a paradox to him but he predicted that in thefuture useful consequences might occur. In 1697, Leibnitz referring to the infinite productof Wallis for π/2 used the notation d1/2y and surmised that the fractional calculus could beused to get the same result.In 1819 the first mention of derivatives of arbitrary order occurred in a published text.The French mathematician S.F. Lacroix published 700 pages on the differential calculus,where one can find less than 2 pages dedicated to the fractional topic, which seems to bebased on a result of Euler dated 1730. He started withy = xn , (1)where n is an integer and wrote the mth derivative in the formdmydxm=n!(n − m)!xn−m . (2)Next, Lacroix changed the factorial using the Γ(x) function (introduced by Legendre). nwas changed from an integer to a real number, n → a, and m was chosen to be m = 12; thus∗hcr@ipicyt.edu.mx1d1/2ydx1/2=Γ(a + 1)Γ(a + 12)xa−12 . (3)In this way, he expressed the derivative of order one half by an arbitrary power of x. Asimple example given by Lacroix refers to the case y = xd1/2xdx1/2=2√x√π. (4)Along the years, great mathematicians, such as Euler, Fourier, Abel and others, did somework on the fractional calculus that nevertheless remained as a sort of curiosity.The modern epoch started in 1974, when a consistent formalism of the fractional calculushas been developed by K.B. Oldham and J. Spanier [2]. It has been found that Lacroix’sresult (4) coincides with that obtained by means of the present definition of the Riemann-Liouville fractional derivative.II. BASIC DEFINITIONS OF THE FRACTIONAL CALCULUS(i) One can define the fractional integral of order α > 0 as followsx0D−αx f(x) =1Γ(α)∫ xx0f(x′)dx′(x − x′)1−α. (5)In particular, for x0 = 0 one usually writesD−αx f(x) =1Γ(α)∫ x0f(x′)dx′(x − x′)1−α. (6)(ii) For β ≥ 0 one can define the fractional derivative of order β in the following waydβf(x)dxβ= Dβxf(x) =dndxnD−(n−β)x f(x) =dndxn1Γ(n − β)∫ x0f(x′)dx′(x − x′)1−n+β, (7)where n ≥ β. Thus the β fractional derivative is defined as an ordinary derivative of ordern of the fractional integral of order n − β.(iii) The chain rule has the formdβdxβf(g(x)) =∞∑k=0(βk)Γ(dβ−kdxβ−k1)dkdxkf(g(x)), (8)where k ∈ N and(βk)Γare the coefficients of the generalized binomial(βk)Γ=Γ(1 + β)Γ(1 + k)Γ(1 − k + β). (9)(iv) Leibnitz’s rule for the derivative of the product has the formdβdxβf(x)g(x) =∞∑k=0(βk)Γdkdxkf(x)dβ−kdxβ−kg(x) , (10)where k ∈ N .2III. ORDINARY DIFFERENTIAL EQUATIONS FROM THE FRACTIONALCALCULUSLet L̂ be a differential operator and let its action L̂[u(x)] = g(x) be used to define adifferential equation. We generalize L̂ to the fractional calculus by means of the fractionalderivative of order β = 1− δ, δ ∈ (0, 1), writing the following fractional differential equationD1−δx L̂[u(x)] = g(x). (11)When δ = 1, then D1−δx L̂[u(x)] = L̂[u(x)].One can obtain a solution of (11) by applying the fractional integral of the same orderto the leftD−(1−δ)x{D1−δx L̂[u(x)]}= D−(1−δ)x [g(x)], (12)L̂[u(x)] = D−(1−δ)x [g(x)], (13)L̂[u(x)] = Dδ−1x [g(x)]. (14)For example: L̂ = ddx+ p(x) impliesL̂[u(x)] =ddxu(x) + p(x)u(x), (15)ddxu(x) + p(x)u(x) = Dδ−1x [g(x)], (16)thenu(x) =1µ(x)[ ∫ xµ(s) Dδ−1s g(s)ds + c], (17)whereµ(x) = exp(∫ xp(s)ds). (18)Equations (17) and (18) provide the solution to the linear generalized equation of the firstorder. This is not the unique possible fractional generalization [3,4]. One could have takenD1−δx u(x) + p(x)u(x) = g(x) or some other procedure. However, the present approach leadsto analytical results in applications.3IV. APPLICATION TO THE RICCATI EQUATION OF CONSTANTCOEFFICIENTSThe ordinary Riccati equation of constant coefficients is du(x)dx+ au2(x) = b, where a andb are constants. Thus the operator of the Riccati type is L̂R =ddx+ au(x) acting always inthe space of functions u(x), i.e., L̂Ru(x) =du(x)dx+ au2(x) = g(x). Fractional considerationsrelated to this operator can be found in the work of Metzler et al [4].The fractional Riccati equation according to the scheme proposed in the previous sectionisD1−δx[ddx+ au(x)]u(x) = b, (19)ordu(x)dx+ au2(x) = D−(1−δ)x b. (20)The right hand side can be writtenb[D−(1−δ)x 1] =bΓ(1 − δ)∫ x0dt(x − t)δ=b x1−δΓ(1 − δ)(1 − δ)=bx1−δΓ(2 − δ). (21)Thus, solving the fractional Riccati equation is equivalent to solving the following type ofparticular, ordinary Riccati equation that we call the δ-modified Riccati equationdu(x)dx+ a[u(x)]2 =bx1−δΓ(2 − δ). (22)A. Solution of the δ-modified Riccati equationIn order to solve (22) we use the transformation u = y′/ay; u′ = [y′′y − (y′)2]/[ay2]. Onegets the associated linear second order differential equationd2y(x)dx2−abΓ(2 − δ)x1−δy = 0. (23)Multiplying by x2 leads tox2d2y(x)dx2−abΓ(2 − δ)x3−δy = 0. (24)The latter has solutions expressed in terms of Bessel functions. To see this we use thefollowing known result. The equationx2y′′ + (1 − 2p)xy′ +[q2r2x2r +(p2 − n2r2) ]y = 0 (25)4has for real q the linear independent solutionsy1 = xpJn(qxr), (26)y2 = xpYn(qxr), (27)where Jn(x) and Yn(x) are the Bessel functions of the first and second type, respectively.We shall also use the following properties of the Bessel functionsJ ′n(x) = Jn−1(x) −nxJn(x) (28)J ′n(x) =12[Jn−1(x) − Jn+1(x)](29)J ′n(x) =nxJn(x) − Jn+1(x). (30)The same holds for Yn(x). Then, using (28) one getsy′1 = y1[p − nrx+ qrxr−1Jn−1(qxr)Jn(qxr)], (31)y′2 = y2[p − nrx+ qrxr−1Yn−1(qxr)Yn(qxr)]. (32)On the other hand, using (30) one getsy′1 = y1[p + nrx− qrxr−1Jn+1(qxr)Jn(qxr)], (33)y′2 = y2[p + nrx− qrxr−1Yn+1(qxr)Yn(qxr)]. (34)In order to find the Riccati solutions we can identify the parameters from comparison of(24) and (25)p =12, q =23 − δ√−abΓ(2 − δ), r =3 − δ2, n =13 − δ. (35)It is more convenient to work with equations (31) and (32) because p − nr = 0. Thereforeu1(x) =y′1ay1=1a[qrxr−1Jn−1(qxr)Jn(qxr)], (36)u2(x) =y′2ay2=1a[qrxr−1Yn−1(qxr)Yn(qxr)], (37)5and thusu1(x) =1a[√abΓ(2 − δ)x1−δ2J δ−23−δ( 23−δ√abΓ(2−δ)x3−δ2 )J 13−δ( 23−δ√ab|Γ(2−δ)|x3−δ2 )], (38)u2(x) =1a[√abΓ(2 − δ)x1−δ2Y δ−23−δ( 23−δ√abΓ(2−δ)x3−δ2 )Y 13−δ( 23−δ√abΓ(2−δ)x3−δ2 )]. (39)For positive b one gets an imaginary q parameter that turns the J and Y functions in the Iand K Bessel functions, respectively. If we consider normal (nondivergent) initial conditionsas a criterium for physical solutions than this selects the expressions containing the J andI functions.V. APPLICATION TO FRW BAROTROPIC COSMOLOGYRecently Faraoni [5] showed that the equations describing the FRW barotropic cosmolo-gies can be combined in a simple Riccati equation of constant coefficients. In addition, Rosu[6] discussed in some detail the cosmological Riccati solutions and used nonrelativistic su-persymmetry (generalized Darboux transformations) to get cosmological Riccati equationsof nonconstant coefficients. Faraoni’s Riccati equation isdHdη+ cH2 = −kc, (40)where H(η) = dR/dηRis the Hubble parameter (R is the scale factor of the universe) andη is the conformal time, c is related to the adiabatic index of the cosmological fluid underconsideration, c = 32γ − 1. k = 0,−1, 1 are the curvature indices of the FRW universes,plane, open, and closed, respectively.Applying the results of the previous section, for a = c, b = −kc, we get the followingsolutions for the δ- modified Hubble parameter:For k = 1 (the closed case), the q parameter is real and we getH(+)1 (η; δ) =√1Γ(2 − δ)η1−δ2J δ−23−δ( 2c3−δ√1Γ(2−δ)η3−δ2 )J 13−δ( 2c3−δ√1Γ(2−δ)η3−δ2 ), (41)H(+)2 (η; δ) =√1Γ(2 − δ)η1−δ2Y δ−23−δ( 2c3−δ√1Γ(2−δ)η3−δ2 )Y 13−δ( 2c3−δ√1Γ(2−δ)η3−δ2 ). (42)For k = −1 (the open case), the q parameter is imaginary and we getH(−)1 (η; δ) =√1Γ(2 − δ)η1−δ2I δ−23−δ( 2c3−δ√1Γ(2−δ)η3−δ2 )I 13−δ( 2c3−δ√1Γ(2−δ)η3−δ2 ), (43)6H(−)2 (η; δ) =√1Γ(2 − δ)η1−δ2K δ−23−δ( 2c3−δ√1Γ(2−δ)η3−δ2 )K 13−δ( 2c3−δ√1Γ(2−δ)η3−δ2 ). (44)The case k = 0 corresponds to b = 0, therefore it does not enter the present generalizationin the sense that there is no change in the Riccati equation.The formulas (41-44) can be considered a generalization of the results obtained byFaraoni. Nondivergent initial data correspond to (41) and (43). Three-dimensional plots ofthese formulas are given in Figures (1) and (2). For δ = 1 we get H(+)1 (η; δ) = J−1/2/J+1/2 =cotan(cη) and H(−)1 (η; δ) = I−1/2/I+1/2 = cotanh(cη) that correspond to the ordinary cal-culus. We mention that there are various works in the literature on the issue of geometricand physical interpretation of the fractional derivative and fractional integral, see, e.g., Pod-lubny [7]. In the cosmological case, the new parameter δ is introduced in the cosmologicalevolution of the Hubble parameter as a consequence of applying a special fractional calculusto cosmological realms. In principle, as in statistical mechanics [8], the fractional calculuscan be considered as the macroscopic manifestation of randomness. This has been arguedto be so [8] when there is no definite time-scale separation between the macroscopic and themicroscopic level of description and this could be the case of cosmology itself.FIG. 1. Fractional Hubble parameter calculated according to the formula (41).7FIG. 2. Fractional Hubble parameter calculated according to the formula (43).[1] Ross B 1997 Lecture Notes in Mathematics 457 (Springer)[2] Spanier J and Oldham K B 1974 The Fractional Calculus (Academic Press)For recent reviews, see Metzler R and Klafter J 2000 Phys. Rep. 339 1 Hilfer R ed. 2000Applications of Fractional Calculus in Physics (World Scientific)[3] Barkai E 2001 Phys. Rev. E 63 046118 Metzler R, Barkai E and Klafter J 1999 Phys. Rev. Lett.82 3563[4] Metzler R, Glöckle W G and Nonnenmacher T F 1997 Fractals 4 597[5] Faraoni V 1999 Am. J. Phys. 67 732 [physics/9901006][6] Rosu H C 2001 Mod. Phys. Lett. A 16 2029 and 2000 Mod. Phys. Lett. A 15 979 [both ingr-qc/0003108][7] Podlubny I 1999 Fractional Differential Equations (Academic Press) and 2002 Fractional Cal-culus and Appl. Analysis 5 367 [math.CA/0110241][8] Grigolini P, Rocco A and West B J 1999 Phys. Rev. E 59 2603 [cond-mat/9809075]8

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Transform of Riccati equation of constant coefficients through
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Transform of Riccati equation of constant coefficients through fractional procedure

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