The Higher-Order Heat-Type Equations via signed Lévy stable and generalized Airy functions
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t The Higher-Order Heat-Type Equations via signed L´evy stable and generalized Airyfunctions
K. G´orska ∗ and A. Horzela † H. Niewodnicza´nski Institute of Nuclear Physics, Polish Academy of Sciences,ul.Eljasza-Radzikowskiego 152, PL-31342 Krak´ow, Poland
K. A. Penson ‡ Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee,Universit´e Pierre et Marie Curie, CNRS UMR 7600Tour 13 - 5i`eme ´et., B.C. 121, 4 pl. Jussieu, F-75252 Paris Cedex 05, France
G. Dattoli § ENEA - Centro Ricerche Frascati, via E. Fermi, 45, IT-00044 Frascati (Roma), Italy
We study the higher-order heat-type equation with first time and M -th spatial partial derivatives, M = 2 , , . . . . We demonstrate that its exact solutions for M even can be constructed with the helpof signed L´evy stable functions. For M odd the same role is played by a special generalization ofAiry Ai function that we introduce and study. This permits one to generate the exact and explicitheat kernels pertaining to these equations. We examine analytically and graphically the spacial andtemporary evolution of particular solutions for simple initial conditions. PACS numbers: 05.10.Gg, 05.40.-a, 02.30.Uu
I. INTRODUCTION
The Brownian motion governed by the conventionalheat equation [1] has several generalizations. Some ofthem are related to the Markov processes described bythe second order partial differential equation [2–4]. Theother ones are connected with the so-called one- and two-sided L´evy stable distribution [5–7]. The last ones aregoverned by the so-called higher-order heat-type equa-tions (HOHTE) ∂∂t F M ( x, t ) = κ M ∂ M ∂x M F M ( x, t ) , (1)which for integer M >
2, are associated with a pseudo-Markov processes (signed processes). We choose to nor-malize F M ( x, t ) according to R ∞−∞ F M ( x, t ) dx = 1. Thepseudo Markov processes were introduced in the sixtiesof the last century and have been studied in many pa-pers starting from [8–10]. Eq. (1) for M = 4 is calledthe biharmonic heat equation [11, 12].HOHTE of order 3 or 4 (or higher) have been inten-sively investigated by several authors and display manyinteresting features [8, 10, 13–15] like e.g. oscillating na-ture, connection to the arc-sine law and its counterpart,the central limit theorem and so on. HOHTE arise inphysical phenomena, e.g. , in the fluctuation phenomenain chemical reactions [16, 17] or as a new method for ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] imaginary smoothing based on the biharmonic heat equa-tion [18, 19]. The biharmonic heat equation is used fordescribing the diffusion on the unit circle [20]. For fixedvalues of integer M > κ M are subject of constraints.Following [13, 15] we choose κ M = (cid:26) ( − M/ , M = 2 , , . . . , ± ( − [ M/ , M = 3 , , . . . . (2)The symbol [ n ] denotes the integer part of n . This choiceof Eq. (2) for κ M warrants, as we will see in Sec. II,the possibility of getting the solution of Eq. (1) via ap-propriate integral transform. Moreover, such a choice ofcoefficients κ M guarantees a holding of the classical arc-sine law for even M [8] and the counterpart to that lawin the case of odd M [13].From mathematical point of view Eq. (1), with theinitial condition F M ( x,
0) = f ( x ), is the Cauchy prob-lem. Its formal solution is obtained by using the exten-sion of the evolution operator formalism, introduced bySchr¨odinger, that gives F M ( x, t ) = ˆ U M ( t ) f ( x ) , ˆ U M ( t ) = exp (cid:18) κ M t ∂ M ∂x M (cid:19) (3)with f ( x ) being an infinitely differentiable function, oran appropriate limit of a sequence of such functions, seebelow.Below we shall employ Eqs. (3) to solve Eq. (1) forgiven initial conditions. For instance, for f ( x ) = x n ( n ∈ N ) a solution of HOHTE can be expressed with theHermite-Kamp´e de F´eri´et polynomials H ( M ) n ( x, y ) [23–25] as F ( n ) M ( x, t ) = H ( M ) n ( x, κ M t ) = n ! [ n/M ] X r =0 ( κ M t ) r x n − Mr r !( n − M r )! . (4)For M = 2, H (2) n ( x, y ) are known as the heat polyno-mials [1, 26]. Any initial function in the form of powerseries, i.e. f ( x ) = P ∞ n =0 a n x n , allows F M ( x, t ) to berepresented as the following expansion F M ( x, t ) = ∞ X n =0 a n H ( M ) n ( x, κ M t ) . (5)In the general case the above representation of F M ( x, t )can lead to the divergent series even for the well-definedinitial condition f ( x ). Nevertheless, Eq. (5) givesthe correct asymptotic expansion of F M ( x, t ). Theformal solution (5) is not effective because it con-verges for short times only, e.g. , for M = 2 and f ( x ) = exp[ − x / (2 σ )] / ( √ πσ ) the convergence is lim-ited to t < σ / (4 k ) [27, 28]. (However, observe that suchinitial conditions as those before Eq. (5) are not inte-grable.) The correct long time behavior of a solution ofthe heat equation is provided by the Gauss-Weierstrasstransform for M = 2 [1]. Therefore we look for an anal-ogous transform for M = 3 , , . . . .The main purpose of the paper is to find the newtype of integral transform which will be furnishing thelong time-behavior of the formal solution (3) for inte-ger M >
2. The paper is organized as follows. In Sec.II we will develop the operational methods initiated in[29, 31, 32] for generalizing the Gauss-Weierstrass trans-form. We will show that such a new type of integraltransform is well-defined as its kernel is converging. Next,we will find the integral representation of the evolutionoperator. Secs. III and IV are devoted to finding the ex-act and explicit forms of the integral kernels of obtainedtransform. The asymptotic expansion for small and largevalues of argument as well as the Mellin transform of theintegral kernel will be considered. In Sec. III we studythe L´evy signed functions associated with the even valuesof M . The case of odd M and the generalized Airy func-tion is investigated in Sec. IV. The specific examples ofsolutions of HOHTE are presented in Sec. V. In this Sec-tion we also derive some new formulas of Glaisher-type.We conclude the paper in Sec. VI. II. INTEGRAL TRANSFORM
Following the example for M = 3 developed in [32], weconsider the integral p M ( x, t ) = 12 πi Z c + i ∞ c − i ∞ e xs + κ M ts M ds (6)with s = c + iτ and c , τ ∈ R . The quantity p M ( x, t )of Eq. (6) will play the role of higher-order heat kernel needed to represent the solutions of Eq. (1) via appro-priate integral transform. If s = | s | e iϕ , | s | = √ c + τ and ϕ = arctan( τc ), then for t >
0, the following relationshold | p M ( x, t ) | ≤ Z ∞−∞ e ℜ ( xs + κ M ts M ) dτ = Z ∞−∞ e xc + κ M t | s | M cos( Mϕ ) dτ. (7)For large τ we have | s | ≈ τ , ϕ ≈ π − cτ andcos( M ϕ ) ≈ cos( M π )+ M cτ sin( M π ), where for M = 2 m ,cos(2 mϕ ) ≈ ( − m and for M = 2 m +1, cos[(2 m +1) ϕ ] ≈ (2 m + 1)( − m cτ . For large τ , that gives Eq. (7) in theform | p M ( x, t ) | ≤ Z ∞−∞ e xc + κ m ( − m tτ m dτ, (8)where M = 2 m , and | p M ( x, t ) | ≤ Z ∞−∞ e xc +(2 m +1) κ m +1 ( − m ctτ m dτ (9)with M = 2 m + 1. The integral of Eq. (8) converges onlyfor κ m = ( − m +1 , whereas Eq. (9) converges for twovalues of κ M , i.e. for c > κ m +1 = ( − m +1 and for c < κ m +1 = ( − m . That substantiates theconditions specified in Eq. (2).Moreover, applying to Eq. (6) the Cauchy’s theoremwith integration over the rectangle s = ± iR , c ± iR , itcan be shown that in Eq. (6) we can take c = 0. As | R | tends to infinity, the integrals over horizontal sidesapproach zero. It boils down to two inequalities: (cid:12)(cid:12)(cid:12) Z c ± iR ± iR e x ˜ s + κ m t ˜ s m ds (cid:12)(cid:12)(cid:12) ≤ e − tR m Z c e xσ dσ, (10)for M = 2 m and ˜ s = σ ± iR , and (cid:12)(cid:12)(cid:12) Z c ± iR ± iR e x ˜ s + κ m +1 t ˜ s m +1 ds (cid:12)(cid:12)(cid:12) ≤ Z c e xσ ∓ (2 m +1) tσR m dσ, (11)for M = 2 m + 1, that vanish in the limit of | R | → ∞ .Consequently the integral Eq. (6) converges (for M = 2 m absolutely) when c = 0. Thus for t > p M ( x, t ) = 12 π Z ∞−∞ e ixτ + κ M ( iτ ) M t dτ, (12)which, after substitution of κ M from Eq. (2) and chang-ing the variable y = τ t /M , can be expressed in the form p M ( x, t ) = 1 t /M g M (cid:16) xt /M (cid:17) , (13) g M ( u ) = ℜ (cid:20) π Z ∞ e iuy + κ M ( iy ) M dy (cid:21) . (14)It is clear that p M ( x,
1) = g M ( x ). (In [33] the integralkernels of the form of Eq. (13) have been already treated,however under the restriction M = α < f ( x ) = δ ( x ) [13, 15], because in the limit of t = 0 + , p M ( x, t ) = δ ( x ), where δ ( x ) is Dirac delta function. For M = 2 m Eq. (12) can be rewritten in the form g m ( u ) = 1 π Z ∞ cos( uy ) e − y m dy, (15)where g m ( u ) = g m ( − u ) is the so-called symmetric L´evystable signed function [6, 34, 35]. Note that g m ( u ) of Eq.(15) here, are denoted by g (2 m, , u ) in Ref. [6]. It shouldbe stressed that the functions g ± m +1 ( u ) do not belong tothe family of L´evy stable functions considered in Ref. [6].For M = 2 m + 1 g ± m +1 ( u ) = 1 π Z ∞ cos( uy ∓ y m +1 ) dy (16)defines the generalization of the Airy function in the senseof [29, 30, 36]. The functions g ± m +1 ( u ) are not any-more even functions and Eq. (16) implies g +2 m +1 ( − u ) = g − m +1 ( u ). Without loss of the generality, in this pa-per we will be studying the case of g +2 m +1 ( u ). We shalladopt the notation of [29] and henceforth set g +2 m +1 ( x ) ≡ Ai (2 m +1) ( x ). The notation with ± will be used whennecessary. The explicit and exact form of g M ( u ) will bederived in the next two Sections.It seems that obtaining p M ( x, t ) for arbitrary integer M ≥ M , i.e. M = 2 [37, seeformula 2.3.15.11 on p. 344], M = 3 [29, 32], M = 4[38], and M = 6 [38] the explicit forms of p M ( x, t ) wereknown. The ref. [6] provides an explicit solution for allrational values of admissible parameters, which include all integer M . We remark that p ( x, t ) and p ( x, t ) playan important role [38] in a theory of energy correlationsin the ensembles of Hermitian random matrices [39, 40]and are related to the problem of phase transitions inchiral QCD models [41].Let us now consider the two-sided (bilateral) Laplacetransform [42, 43] of p M ( x, t ), compare Eq. (6). For t > c = 0 the integral (see Eqs. (8) and (9)) Z ∞−∞ e − sy p M ( y, t ) dy = e κ M ts M (17)is converging absolutely. The validity of Eq. (17) is easyto demonstrate for two cases: M = 2, where we useformula 2.3.15.11 on p. 344 of [37], and M = 3, thatis proved in [32]. For the arbitrary integer M ≥ g M ( x ) at infinity, namely g a M ( x ) ≈ exp( − Ax β ), β ≥
1, see Eqs. (10) and (11) for t = 1.Eq. (17) is a crucial formula of our paper because bymaking the substitution s = ∂ x we obtainˆ U M ( t ) = Z ∞−∞ exp( − y ∂∂x ) p M ( y, t ) dy, (18) where exp( − y ∂∂x ) is the shift operator. That gives the in-tegral representation of the evolution operator of Eq. (3)and, as a consequence, the general form of F M ( x, t ): F M ( x, t ) = Z ∞−∞ exp( − y ∂∂x ) p M ( y, t ) f ( x ) dy = Z ∞−∞ p M ( y, t ) f ( x − y ) dy. (19)Eq. (19) for M = 2 is the Gauss-Weierstrass transform,see [1], and for M = 3 it is the Airy Ai transform, see[24, 29, 32]. Several examples of f ( x ) such that Eq. (19)can be evaluated analytically are given in Sec. V. III. THE SIGNED L´EVY STABLE LAWS
This Section is devoted to the exact and explicit formsof g m ( u ) for m = 1 , , , . . . , see Eq. (15), and thereafterwe look closer at their asymptotic behavior at infinityand the associated Hamburger moment problem .In the spirit of Refs. [5, 6] and [38] we shall provide theexact expression of g m ( u ) using the Mellin transform.We start with supposing that for certain values of com-plex s the Mellin transform of g m ( u ) exists: g ⋆ m ( s ) = M [ g m ( u ) , s ] = Z ∞ u s − g m ( u ) du, (20)and g m ( u ) = M − [ g ⋆ m ( s ) , u ]. Then, using Eq. (15) and[37, see formulas 2.5.3.10 on p. 387 and 2.3.3.1 on p. 322]we have g ⋆ m ( s ) = 12 πm Γ( s )Γ (cid:18) − s m (cid:19) cos (cid:16) sπ (cid:17) . (21)With the help of the second Euler’s reflection formulawe express the cosine via gamma function. Inverting theMellin transform of g ⋆ m ( s ) we obtain g m ( u ) = M − [ g ⋆ m ( s ) , u ]= 12 πi Z L u − s Γ( s − (cid:0) − s − m (cid:1) Γ (cid:0) s − (cid:1) Γ (cid:0) − s − (cid:1) ds, (22)with the contour L lying between the poles of Γ( s − (cid:16) − s − m (cid:17) . After applying the Gauss-Legendre multiplication formula to the gamma functionsin Eq. (22) we can express g m ( u ) in terms of the MeijerG functions G m,np,q (cid:16) x (cid:12)(cid:12)(cid:12) ( α p )( β q ) (cid:17) [44] g m ( u ) = r mπ u G , m m,m +1 (cid:18) (2 m ) m u m (cid:12)(cid:12)(cid:12) ∆(2 m, , ∆( m, , ∆( m, (cid:19) , (23) For the purpose of this paper we use this terminology for signedfunctions. where ∆( k, a ) = ak , a +1 k , . . . , a + k − k is a special list of k elements. Furthermore, it turns out that g m ( u ) is afinite sum of m generalized hypergeometric functions oftype p F q (cid:16) ( α p )( β q ) (cid:12)(cid:12)(cid:12) z (cid:17) : g m ( u ) = m X j =1 c j ( m ) u − j F m , j − m ∆(2 m, j ) (cid:12)(cid:12)(cid:12) z ! (24)with z = ( − m [ u/ (2 m )] m and coefficients c j ( m ) read c j ( m ) = p m/π (2 m ) j − π m Γ (cid:18) j − m (cid:19) × hQ j − i =1 Γ (cid:0) i − j m (cid:1)i hQ mi =2 j +1 Γ (cid:0) i − j m (cid:1)ihQ m − i =0 sin (cid:0) π im − π j − m (cid:1)i − . (25)The formulas Eq. (24) and (25) follow from the Eq.8.2.2.3 of [44] and are in agreement with Eq. (2.4) of[45]. Here, we have used the compact notation of p F q ’sfunctions where the upper (lower) list of parameters cor-responds to the first (second) list of parameters in stan-dard notation. We remark that in the lists of parametersof F m two cancellations of the same terms appear dueto the obvious identity p + r F q + r (cid:16) ( α p ) , ( γ r )( β p ) , ( γ r ) (cid:17) = p F q (cid:16) ( α p )( β p ) (cid:17) ,where ( γ r ) is an arbitrary sequence of r parameters. ThusEq. (24) finally reads as the sum of m generalized hyper-geometric functions of type F m − (cid:16) − ( β m − ) (cid:12)(cid:12)(cid:12) z (cid:17) [45].Formulas (24) and (25) reconstruct the explicitlyknown cases presented in [6] and give an unlimited num-ber of new exact solutions g m ( u ), e.g. for m = 4 g ( u ) = c (4) u F (cid:18) − , , , , , (cid:12)(cid:12)(cid:12) u (cid:19) + c (4) u F (cid:18) − , , , , , (cid:12)(cid:12)(cid:12) u (cid:19) + c (4) u F (cid:18) − , , , , , (cid:12)(cid:12)(cid:12) u (cid:19) + c (4) F (cid:18) − , , , , , (cid:12)(cid:12)(cid:12) u (cid:19) . (26)The coefficients c j (4) for j = 1 , . . . , √ (cid:0) π (cid:1) / [4Γ (cid:0) (cid:1) ], −√ (cid:0) π (cid:1) / [8Γ (cid:0) (cid:1) ], √ (cid:0) π (cid:1) sin (cid:0) π (cid:1) Γ (cid:0) (cid:1) / (96 π ),and −√ (cid:0) π (cid:1) sin (cid:0) π (cid:1) Γ (cid:0) (cid:1) / (2880 π ), respectively.In Fig. 1 g ( u ) (I; red line) and g ( u ) (II; black line)are presented. The tails of symmetric function g ( u ) os-cillate. The amplitude of these oscillations is decreasingwith increasing values of | u | . The analogous behavior isobserved in the other examples of g m ( u ), for instancesee Fig. 1 in [38] where functions g ( u ) and g ( u ) areexhibited. Figs. 1 and 2 in [38] well illustrate the consid-erations presented in [14]. FIG. 1. (Color online) Plot of the function g m ( u ) for m = 1and m = 4; i.e. g ( u ) = exp( − u / / (2 √ π ) (I; black line)and g ( u ) (II; red line) given in Eq. (26). A. Asymptotics of g m ( u ) . The asymptotics of g ( u ) and g ( u ) for large u are pre-sented in [34, 38], whereas the general formula of g a2 m ( u )( m = 1 , , . . . ) for large u is given in [15, 35]. Here, fol-lowing [15, 35], we just present their compact form: g a2 m ( u ) ∼ √ m ) − m − p (2 m − π u − m − m − e − sin h π m − i Z × cos (cid:26) cos h π m − i Z − m − m − π (cid:27) ,Z = (2 m − u/ (2 m )] m m − , which guarantees the abso-lute convergence of (17). For small u , the L´evy signedfunction can be represented by the series˜ g a2 m ( u ) ∼ mπ ∞ X r =0 ( − r (2 r )! Γ (cid:18) rm + 12 m (cid:19) u r , (27)which has the infinite radius of convergence. B. Hamburger moment problem for signed g m ( u ) . Let us consider the following integral defining Ham-burger moments of g M ( u ) h M ( µ ) = Z ∞−∞ u µ g M ( u ) du, (28)for M = 2 , , . . . and arbitrary real µ . In this Sectionwe will look closer at the case of even M = 2 m , whereasEq. (28) for odd M = 2 m + 1 will be studied in Sec. IVbelow.Since g M ( u ) is an even function for M = 2 m , the inte-gral (28) vanishes by symmetry for all odd µ = (2 n + 1),regardless of the value of m ( m = 1 , , . . . ). The calcu-lation of h m (2 n ) is a somewhat subtler problem whichhas been carefully analyzed in [34]. Decomposing h m ( µ )into two symmetric parts, employing Eq. (21) and mak-ing some simple transformation, we express Eq. (28) for M = 2 m , ( m = 1 , , . . . ), in the form, compare Eq. (20): h m ( µ ) = [1 + ( − µ ] g ⋆ m ( µ + 1)= 12 m [1 + ( − µ ] Γ (cid:0) µ (cid:1) Γ (cid:0) µ m (cid:1) sin (cid:0) πµ (cid:1) sin (cid:0) πµ m (cid:1) . (29)The moments h m ( µ ) vanish for µ = 2( mp + r ), p = 1 , , . . . , r = 1 , , . . . , m −
1. The only non-zeroterms of h m ( µ ) occur for µ = 2 mp for whichlim µ → mp h m ( µ ) = ( − (1+ m ) p (2 mp )! p ! . (30)We see that h m (0) = 1, that is g m ( u ) is normalizedto unity. The first few non-zero terms of h m ( µ ) for m = 1 , , µ = 0 , , . . . ,
12 are presented in Tab. I. µ h ( µ ) 1 2 12 120 1680 30240 665280 h ( µ ) 1 -24 20160 -79833600 h ( µ ) 1 720 239500800TABLE I. The values of the integrals (28) for M = 2 , , µ = 0 , , . . . , IV. GENERALIZED AIRY FUNCTIONS
To derive the exact and explicit form of Ai (2 m +1) ( u ) weapply the method od Sec. III with the Mellin transform[ Ai (2 m +1) ( s )] ⋆ = M [ Ai (2 m +1) ( u ); s ] for complex s . Then,using Eq. (16) and formulas 2.5.3.10 on page 387 and2.3.3.1 on page 322 of [37], we get h Ai (2 m +1) ( s ) i ⋆ = Γ( s )Γ( − s m +1 ) π (2 m + 1) cos (cid:20) π (2 ms + 1)2(2 m + 1) (cid:21) (31)and performing the Mellin inversion as in Eq. (22), weobtain the exact and explicit expression for Ai (2 m +1) ( u ) in terms of Meijer G function: Ai (2 m +1) ( u ) = r m + 12 π u × G , m +13 m +1 ,m +1 (cid:18) ( − m ˜ z (cid:12)(cid:12)(cid:12) ∆(2 m + 1 , , ∆( m, , ∆( m, (cid:19) (32)with ˜ z = ( − m [ u/ (2 m +1)] m +1 . Furthermore, the Mei-jer G function is converted to the finite sum of the gen-eralized hypergeometric functions: Ai (2 m +1) ( u ) = m X j =1 b j ( m ) u − j F m +1 , j m +1 ∆(2 m + 1 , j ) (cid:12)(cid:12)(cid:12) ˜ z ! (33)with b j ( m ) = p (2 m + 1)(2 m + 1) j π m +1 Γ (cid:18) j m + 1 (cid:19) × hQ ji =1 Γ (cid:0) i − j − m +1 (cid:1)i hQ m +1 i = j +2 Γ (cid:0) i − j − m +1 (cid:1)ihQ mi =0 sin (cid:0) π im +1 − π j m +1 (cid:1)i − . (34)In obtaining Eqs. (33) and (34) we have again usedthe formula 8.2.2.3 of [44]. We point out that simi-larly as in the case of the L´evy signed functions of theprevious section, in the list of parameter F m +1 thecancellation of the same two terms occurs. That givesthe finite sum of 2 m hypergeometric functions of type F m − (cid:16) − ( β m − ) (cid:12)(cid:12)(cid:12) ˜ z (cid:17) . Eqs. (33) and (34) are in agree-ment with the case of t = 1 of the formula for u n +1 ( x, t )on page 2 of [46].The formulas (33) and (34) reproduce the well-knowncase of conventional Airy Ai function (see Eqs. (33) and(34) for m = 1) and give the exact and explicit formof the generalized Airy Ai functions Ai (2 m +1) ( u ) whichwere only numerically obtained in [36]. Without loss ofthe generality below we are writing out Ai (2 m +1) ( u ) for m = 2: Ai (5) ( u ) = b (2) F (cid:18) − , , (cid:12)(cid:12)(cid:12) u (cid:19) + b (2) u F (cid:18) − , , (cid:12)(cid:12)(cid:12) u (cid:19) + b (2) u F (cid:18) − , , (cid:12)(cid:12)(cid:12) u (cid:19) + b (2) u F (cid:18) − , , (cid:12)(cid:12)(cid:12) u (cid:19) . (35)The coefficients b j (2), j = 1 , . . . ,
4, are equalto √ A/ [10Γ( ) sin( π )], −√ B/ [10Γ( ) sin( π )], −√ B Γ( ) / (20 π ), and √ A Γ( ) / (60 π ), where A = sin( π ) / sin( π ) and B = sin( π ) / sin( π ), re-spectively.In Fig. 2 the functions Ai (2 m +1) ( u ) are presented fromEqs. (33) and (34) for m = 1 (I; black line), m = 2 (II;blue line), and m = 3 (III; red line). For these generalizedAiry Ai functions we observe the oscillation for positive and negative u , which constitutes a good illustration ofconsiderations presented in [15]. FIG. 2. (Color online) Plot of the functions Ai (2 m +1) ( u ) for m = 1 , , i.e. Ai (3) ( u ) = 3 − / Ai ( − u/ − / ) (I; black line)and Ai (5) ( u ) (II; blue line) given in Eq. (35), and Ai (7) ( u )(III; red line) calculated from Eqs. (33) and (34) for m = 3,respectively. Note that Ai (3) ( x ) is not oscillating for x < The comparison between functions g m ( u ) and Ai (2 m +1) ( u ) for fixed values of m is shown in Figs. 3;on Fig. 3a it is done for m = 8, whereas the Fig. 3b isfor m = 50. It turns out that for large m the differencebetween g m ( u ) and Ai (2 m +1) are negligible. A. Asymptotics of Ai (2 m +1) ( u ) . The asymptotic expansions of the generalized Airy Aifunctions for large negative and positive values of argu-ment are given as Ai (2 m +1)a − ( u ) ∼ √ mπ (cid:2) (2 m + 1) | u | m − (cid:3) − m e − sin (cid:0) π m (cid:1) | ˜ Z | × cos (cid:20) cos (cid:0) π m (cid:1) | ˜ Z | − m − m π (cid:21) , (36)for u → −∞ , and Ai (2 m +1)a + ( u ) ∼ √ mπ (cid:2) (2 m + 1) u m − (cid:3) − m cos (cid:16) ˜ Z − π (cid:17) , (37) FIG. 3. (Color online) Comparison of the functions g m ( u )(blue line) and Ai (2 m +1) ( u ) (red line) for given m . In Fig. 3awe present the case m = 8, whereas in Fig. 3b we presentthe case m = 50. It is seen that for large m the differencebetween even and odd case becomes negligible. for u → ∞ ; ˜ Z = 2 m [ u/ (2 m + 1)] m +12 m , see Proposition2 in [15]. The symbol Ai (2 m +1)a − ( u ) [ Ai (2 m +1)a + ( u )] denotesthe asymptotic estimate for large negative (positive) u .For m = 1 Eqs. (36) and (37) lead to the known formulason the asymptotic behavior of Ai (3) ( u ), see e.g. [32]. For m = 2 we have Ai (5)a − ( u ) ∼ − / √ π | u | − / e −| ˜ Z | sin √ | ˜ Z | + 3 π ! ,Ai (5)a + ( u ) ∼ − / √ π u − / sin (cid:16) ˜ Z + π (cid:17) , (38)with ˜ Z = 4 (cid:0) u/ (cid:1) / . We point out that for large valuesof m , Ai (2 m +1)a − ( u ) approaches Ai (2 m +1)a + ( u ).The asymptotic behavior for small values of u is ob-tained by using the Taylor expansion of Ai (2 m +1) ( u ) orformula 2.5.21.6 on p. 430 of [37]. That gives˜ Ai (2 m +1)a ( u ) ∼ π − (2 m + 1) ∞ X r =0 u r r ! Γ (cid:18) r m + 1 (cid:19) × cos (cid:18) − mr m + 2 π (cid:19) . (39)The series in Eq. (39) has the infinite radius of conver-gence. For a given m the summation can be carried outand it agrees with the general formula of Eq. (33). B. Hamburger moment problem for Ai (2 m +1) ( u ) . Now, we look closer at the Hamburger moment prob-lem of Ai (2 m +1) ( u ). For M = 2 m + 1, Eq. (28) decom-poses into two integrals according to the sign of u . Thatgives the sum h m +1 ( µ ) = g + ,⋆ m +1 ( µ + 1) + ( − µ g − ,⋆ m +1 ( µ + 1) , (40)where g ± ,⋆ m +1 ( s ) denotes the Mellin transform of g ± m +1 ( u )given in Eq. (16). Considering separately the case ofeven and odd moments, after using Eqs. (20), (16) andformula 2.3.3.1 on page 322 of [37] we have h m +1 (2 n ) = (2 n − (cid:16) n m +1 (cid:17) sin( πn )sin (cid:16) πn m +1 (cid:17) , (41) h m +1 (2 n + 1) = − (2 n )!Γ (cid:16) n +12 m +1 (cid:17) sin( πn )cos (cid:16) π n +12 m +1 (cid:17) . (42)Let us analyze properties of Eqs. (41) and (42). Atfirst we see that h m +1 (2 n ) vanish for integer n exceptfor n being the multiple of (2 m + 1). For n = (2 m + 1) k , k = 0 , , . . . the ratio of sines in h m +1 (2 n ) is finite and itgoes to (2 m +1). The only non-zero terms can be writtenin the form lim n → (2 m +1) k h m +1 (2 n ) = [(2 m + 1)2 k ]!(2 k )! . (43)It is obvious that h m +1 (0) = 1 for k = 0. Analogicalsituation as in the above case appears for h m +1 (2 n + 1)for which for n = (2 m + 1) k + m the ratio of sine andcosine goes to ( − m +1 (2 m + 1). That giveslim n → (2 m +1) k + m h m +1 (2 n +1) = ( − m [(2 m + 1)(2 k + 1)]!(2 k + 1)! . (44)The first few values of non-vanishing terms of h m +1 ( µ )for m = 1 , µ h ( µ ) 1 -6 360 -60480 h ( µ ) 1 120 1814400 h ( µ ) 1 -5040TABLE II. The values of the integrals (28) for M = 3 , , µ = 0 , , . . . , V. SPECIFIC EXAMPLES
The content of this Section concerns the formaldeliberations on the relations between F M ( x, t ), for fixedinitial condition, expressed by Eqs. (5) and (19), andthe integral transform approach. (A) First we observe that Eq. (19) with f ( x ) = x n andthe kernel given in Eq. (13) lead to F ( n ) M ( x, t ) = n X k =0 (cid:16) nk (cid:17) ( − k x n − k t k/M h M ( k ) , (45)where h M ( k ) are defined in (28) and for the initial condi-tion we have used the Newton’s binomial theorem. With-out loss of generality, let us look at the first few termsof (45) for M = 3. Using to that purpose h ( k ) ex-hibited in Tab. II we get: F (0)3 ( x, t ) = 1, F (1)3 ( x, t ) = x , F (2)3 ( x, t ) = x , and F (3)3 ( x, t ) = x +6 t . That reproducesthe Hermite-Kamp´e de F´eri´et polynomials H (3) n ( x, t ) for n = 0 , . . . ,
3. In the general case, from Tabs. I and II itemerges that only [ n/M ]th terms of h M ( µ ) are differentfrom zero. After changing the summation index in Eq.(45) as k = M p ( p = 0 , , . . . , [ n/M ]) and employing for-mulas (30), (43) and (44) we convert Eq. (45) into Eq.(4). Making similar consideration for the initial condi-tion formally written in the form of the power series wecan express the representation of F M ( x, t ) in the form ofEq. (5).Our approach suggests a new way of looking at theHermite-Kamp´e de F´eri´et polynomials, which accordingto Eq. (45) can be defined by the operational form H ( M ) n ( x, κ M t ) = ( x − t /M ˆ h M ) n ϕ , (46)with ˆ h kM ϕ = h M ( k ) , (47)where ϕ is the ’vacuum’ and h M ( k ) are given by Eqs.(30), (43) and (44). See also [47] for related considera-tions in the context of lacunary Laguerre polynomials. (B) The next example, which shows the validity of themethod based on the integral transform (19), we considerthe generalization of the Glaisher formula whose originalform reads: F ( x, t ) = exp (cid:18) κ t ∂ ∂x (cid:19) e − αx = exp( − αx α t ) √ α t . (48)for t > − / (4 α ) and κ = 1. For that purpose we takeas the initial condition f ( x ) = g M ( αx ), α >
0. For sucha choice, we get F M ( x, t ) = exp (cid:18) κ M t ∂ M ∂x M (cid:19) g M ( αx )= (cid:0) α M t (cid:1) − /M g M (cid:20) αx (1 + α M t ) /M (cid:21) , (49) −∞ < x < ∞ and t > − α − M . The constant κ M ishidden in the definition of g M ( u ) given in Eq. (14). Eq.(49) neatly illustrates the scaling character of the timeevolution from this initial condition. In certain sense itconstitutes a rather far-reaching extension of Glaisher-type relations [48], which for M = 3 can be compactlywritten as F ( x, t ) = exp (cid:18) κ t ∂ ∂x (cid:19) Ai ( − αx ) = Ai h − αx (1+3 α t ) / i (1 + 3 α t ) / (50)for t > − (3 α ) − and κ = 1. The analogous expressionfor κ = − α to − α with the restriction on t : t < (3 α ) − .These relations are evocative of the reproducing propertyof L´evy laws under the L´evy transform [33]. We remindthe reader that Ai ( z ) is not a L´evy signed function. Theconditions on validity of Eqs. (48), (49), and (50) are al-ways satisfied for t >
0, compare with Eq. (12). Below,we will sketch the proof of Eq. (49) in two independentways: first by using the integral transform of Eq. (19)and then by the study of the action of the evolution op-erator on the initial condition g M ( αx ). At first, we useEq. (19): F M ( x, t ) = 1 t /M Z ∞−∞ g M (cid:16) yt /M (cid:17) g M [ α ( x − y )] dy. (51)Substituting Eq. (14) into Eq. (51) we have three in-tegrals to calculate, which after changing the order ofintegration, can be written as F M ( x, t ) = ℜ (cid:26)Z ∞ e κ M ( iy ) M dy π Z ∞ e κ M ( iy ) M + iαxy dy π × Z ∞−∞ e i (cid:0) y t /M − αy (cid:1) y dyt /M (cid:27) . (52)The integral over y in Eq. (52) is equal to2 πδ ( y − αt /M y (cid:1) . That simplifies the integration over y in Eq. (52) and, in consequence, gives formula (49).Otherwise, F M ( x, t ) can be obtained by employing Eqs. (3) and (14), illustrated below: F M ( x, t ) = ℜ (cid:26) π Z ∞ exp (cid:18) κ M t ∂ M ∂x M (cid:19) e iαxy e κ M ( iy ) M dy (cid:27) = ℜ (cid:20) π Z ∞ e iαxy + κ M ( iy ) M (1+ α M t ) dy (cid:21) . After introducing y = u (1 + α M t ) − /M we will recoverEq. (49). (C) In the last two examples, we choose, rather arbitrar-ily, the initial condition given by the Cauchy distribution f ( x ) = π ( α + x ) , α >
0. For that choice of f ( x ) and foreven M we are able to find the exact and explicit formof F M ( x, t ). Using Eq. (19) with the integral kernel p m ( y, t ) defined in Eqs. (13) and (14), we obtain F m ( x, t ) = ℜ (cid:26)Z ∞−∞ duπ ( α + u ) (cid:20)Z ∞ e i ( x − u ) zt / (2 m ) − z m dzt m (cid:21)(cid:27) , where u = ( x − y ). Calculating at first the integral over u and thereafter using formula 2.3.2.13 of [37], we get F m ( x, t ) = ℜ (cid:26)Z ∞ exp (cid:20) − α − ixt / (2 m ) z − z m (cid:21) dzπαt / (2 m ) (cid:27) = 12 mαπ ℜ m X j =1 ( − j − ( j − (cid:18) j m (cid:19) ( α − ix ) j − t j/ (2 m ) × F m , j m ∆(2 m, j ) (cid:12)(cid:12)(cid:12) ( α − ix ) m t (2 m ) m !) . (53)For odd M the function F m +1 ( x, t ) given by F m +1 ( x, t ) = 1 παt / (2 m +1) Z ∞ exp h − zαt / (2 m +1) i × cos (cid:16) xzt / (2 m +1) − z m +1 (cid:17) dz (54)can be calculated only numerically.For m = 3 the functions F ( x, t ) given in Eq. (53) for α = 1 and t = 0 . , . , α = 1 there ex-ists the border time t ∼ = 0 .
415 for which F ( x, t ), t < t ,is positive, see line I in Fig. 4. For t > t the function F ( x, t ) is negative, see lines III and IV in Fig. 4. Theline II in Fig. 4 presents the border case between thetwo previous situations. The function F ( x, t ) is pos-itive with the roots at the points x ∼ = ± . F ( x, t ≥ t ) and the depth of negative partsof F ( x, t > t ) depend on values of parameter α andthey can be minimized for the appropriate choice of α for given t .In Fig. 5 for m = 1 the function F ( x, t ) is presented,as numerically calculated from Eq. (54) for α = 2 and t = 0 . , . , α , we canalso find the border time ˜ t for which F ( x, ˜ t ) ceasesto be strictly positive and starts to have the roots. For FIG. 4. (Color online) Plot of F ( x, t ) given by Eq. (53) for α = 1 and fixed value of t ; line I is for t = 0 .
125 (red line),line II is for t = 0 .
415 (green line), line III is for t = 3 (yellowline), and line IV (blue line) is for t = 10. α = 2 the border time ˜ t is equal to 0 . F ( x, t < ˜ t ) is shown as the line I inFig. 5, F ( x, ˜ t ) is presented in line II, whereas two signedfunctions F ( x, t > ˜ t ) are illustrated in lines III and IV.The existence of the negative parts in Fig. 5 are relictsof the oscillations of the integral kernel p ( x, t ) and theycan be minimized by the suitable choice of α for fixed t . x -5 0 5 10 1500.020.040.06 F ( x, t ) I II III IV
FIG. 5. (Color online) Plot of F ( x, t ) given in Eq. (54) for α = 2 and fixed value of t ; line I is for t = 0 . t = 0 .
926 (green line), line III is for t = 2 (brownline), and line IV (blue line) is for t = 4. (D) The formalism developed so far shows a wide flexi- bility. It can be applied for solving a large class of partialdifferential equations in the form ∂∂t ˜ F M ( x, t ) = ˆ O M ˜ F M ( x, t ) (55)with ˆ O M the differential operator of order M being thefunction of x and ∂∂x and with the initial condition˜ F ( x,
0) = ˜ f ( x ). According to the technique proposedhere the formal solution of Eq. (55) can be expressed by˜ F M ( x, t ) = Z ∞−∞ ˜ p M ( y, t ) e − y ˆ O M ˜ f ( x ) dy, (56)where the kernel ˜ p M ( y, t ) has to be adapted to a preciseform of the operator ˆ O M . The Eq. (55) encompasses alarge class of Fokker-Planck type operators, see [28] forrelated considerations. Furthermore the method can beshown to be applicable to the solution of problems wherefractional evolution differential equations occur as in thecase of anomalous diffusion [28] and relativistic quantummechanics [56, 57]. VI. CONCLUSIONS
In this paper we have shown that the formalism of evo-lution equation and of the associated integral transformsis a very efficient tool to deal with evolution problems in-volving generalizations of the heat equations through theintroduction of higher-order derivatives. We have seenhow the formalism is capable of including popular trans-forms like Gauss-Weierstrass and Airy via the so-calledsigned L´evy stable and generalized Airy Ai functions.The key result of the paper is the construction of anew technique of solving the HOHTE which furnishesthe long-time behavior of Eq. (3). We have also shownthat our technique reconstructs the Hermite-Kamp´e deF´eri´et polynomials being the formal solution of Eq. (1)with the initial condition f ( x ) = x n . For the initialcondition given by the L´evy signed function and the gen-eralized Airy Ai function we observe the scaling characterof the time evolutions which are the natural extension ofGlaisher-type relations. The next interesting result is re-lated to the existence the border time in which the timeevolution calculated for the Cauchy distribution beginsto possess the negative parts.Most of the formalism developed in the paper canbe applied to non-standard forms of evolution equationswhich are encountered in physical problems concerninganomalous diffusion and quantum mechanical relativis-tic effects. Regarding the first point, we note that manyproblems concerning the anomalous transport (in par-ticular sub-diffusive) can be treated using HOHTE withnot necessarily integer derivatives. Fractional transportis within the capabilities of the present formalism, whichpotentially offers possibility of treating in a unified waydifferent phenomena occurring in economics [49, 50], pop-ulation mobility [51, 52], infectious disease propagation[53], metastatic cancer spread [54, 55] etc. e.g. in the analysis of the relativistic Schr¨odinger equa-tion (see Refs. [56, 57], where some aspects of the un-derlying problems have started to be explored.) Beforeclosing the paper we want to mention the possibility oflooking at old problems with fresh eyes. The presentformalism may yield unique tools to merge two aspectsof anomalous diffusion and non-local quantum mechanicsthrough the emergent L´evy generators [58], non-local innature, which are naturally suited to provide a bridge be- tween anomalous transport and pseudo-differential evo-lution in semi-relativistic quantum mechanics. It will bediscussed in a forthcoming investigation. VII. ACKNOWLEDGEMENTS
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