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On a connection between the discrete fractional Laplacian and superdiffusion
AlgebraLetters and its49Applications 466 (2015) 102–116 Applied Linear Mathematics (2015) 119–125
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LinearMathematics Algebra andLetters its Applications Applied www.elsevier.com/locate/laa www.elsevier.com/locate/aml
Inverse eigenvalue problem of Jacobi matrix and On a connection between the discrete fractional Laplacian superdiffusion with mixed data Ying Lizama Wei 1 b,∗ , Luz Roncal a , Juan Luis Varona a ´ Oscar Ciaurri a , Carlos Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Departamento de Matem´ aticas y Computaci´ on, Universidad de La Rioja, 26004 Logro˜ no, Spain Nanjing 210016, PR China Universidad de Santiago de Chile, Facultad de Ciencias, Departamento de Matem´ atica y Ciencia de la Computaci´ on, Casilla 307, Correo 2, Santiago, Chile a
b
a r t i c l e article
info
i n f o
a b s t r a c t
abstract
Article history: In this paper, the inverse eigenvalue problem of reconstructing Received 16 January 2014 a Jacobi matrix from its eigenvalues, its leading principal Article history: Accepted 20 September We 2014 relate the fractional powers of discrete Laplacian with of a standard timesubmatrix andthepart of the eigenvalues its submatrix Received 30 March 2015Available online 22 October 2014 fractional derivative inisthe sense of Liouville by encoding the iterativeconditions nature of the considered. The necessary and sufficient for Received in revised form 12 May by Y. Weidiscrete operator through a time-fractional memory term. Submitted the existence and uniqueness of the solution are derived. 2015 Ltd. Allsome rightsnumerical reserved. © 2015 Elsevier Furthermore, a numerical algorithm and Accepted 12 May 2015 MSC: examples are given. Available online 21 May15A18 2015 © 2014 Published by Elsevier Inc. 15A57 Keywords: Discrete Laplacian Keywords: Superdiffusion Jacobi matrix Heat semigroup Eigenvalue Fractional Laplacian Inverse problem Modified Bessel functions Submatrix
1. Introduction Let 0 < α < 1 be given. Our concern in this paper is the study of the connection between two seemingly distinct classes of partial differential equations that have a mixed character, i.e. that can be modeled both in continuous time as well as in discrete space Dt v(n, t) = (−∆d )α v(n, t),
t > 0, n ∈ Z,
(1)
t > 0, n ∈ Z.
(2)
and 1
1/α Dt u(n, t) = (−∆d )u(n, t), E-mail address: [email protected]. Tel.: +86 13914485239.
1/α
In (1), Dt denotes the continuous derivative in the variable t, Dt denotes the fractional derivative of http://dx.doi.org/10.1016/j.laa.2014.09.031 α order 1/α in the sense of Liouville (left-sided) and (−∆ ) denotes the fractional powers of order α of the 0024-3795/© 2014 Published by Elsevier Inc.d ∗ Corresponding author. ´ Ciaurri), [email protected] (C. Lizama), [email protected] E-mail addresses: [email protected] (O. (L. Roncal), [email protected] (J.L. Varona).
http://dx.doi.org/10.1016/j.aml.2015.05.007 0893-9659/© 2015 Elsevier Ltd. All rights reserved.
120
´ Ciaurri et al. / Applied Mathematics Letters 49 (2015) 119–125 O.
unidimensional discrete Laplacian, introduced in [1] (where other operators in Harmonic Analysis, such as the discrete Riesz transform, square functions, conjugate harmonic functions and the Poisson semigroup were also studied). See Section 2 for definitions. For β := α1 > 1, Eq. (2) describes superdiffusive phenomena in time. It models anomalous superdiffusion in which a particle cloud spreads faster than the classical diffusion model predicts. The connection between order in time and space for partial differential equations is a surprising phenomena that seems not to be addressed for the discrete fractional Laplacian. It shows that the spatial-Laplacian of fractional order is entirely translated into temporal regularization. First studies on this unexpected property appear in [2] for the unidimensional continuous Laplacian. There, it was observed that ordinary PDEs of first order in space are transformed into PDEs of half-th order in time and second order in space, and that from an applied perspective (Stokes problem) this property shows numerical advantages. For the N -dimensional and continuous bi-Laplacian, the connection appeared in [3, Example 2.14], where also an abstract setting for higher powers is studied. For stochastic processes the relation seems to be more analyzed, but only very recently by H. Allouba [4] and B. Baeumer, M.M. Meerschaert and E. Nane [5, Theorems 3.1 and 3.9]. See also the references therein. In this paper we are able to prove that, for appropriate initial values in the Lebesgue space ℓ∞ (Z) and 0 < α < 1, the solution of problems (1) and (2) is the same, and that it admits an explicit representation in convolution form by means of a special kernel. This is shown in Theorem 3, that is the main result of this work. We define the discrete fractional Laplacian via the discrete Fourier transform, because this way is the most suitable to prove Theorem 3. Moreover, this definition coincides with the genuine definition of the fractional powers of a more general linear second order partial differential operator L by means of the semigroup generated by L, as shown by P.R. Stinga and J.L. Torrea in [6]. In particular, we provide the discrete counterpart of the formula for the fractional Laplacian via the semigroup due to Stinga and Torrea [6]; it is presented in Theorem 2 and has its own interest. 2. Preliminaries In order to establish and clarify the meaning of Eqs. (1) and (2), and the relationship between their solutions, we need to define several continuous and discrete operators. In particular, in what follows we are going to give precise definitions for several kinds of Fourier transforms and fractional operators on some spaces, and to provide some of their properties. For a given sequence f , we define the discrete Fourier transform FZ (f )(θ) = f (n)einθ , θ ∈ T, n∈Z
where T ≡ R/(2πZ) is the unidimensional torus, that we identify with the interval (−π, π]. The inverse discrete Fourier transform is obtained by the formula π 1 −1 FZ (ϕ)(n) = ϕ(θ)e−inθ dθ, n ∈ Z, 2π −π for a given function ϕ. Therefore f (n) =
1 2π
π
−π
FZ (f )(θ)e−inθ dθ,
n ∈ Z.
It is easily verified that FZ (f ∗ g)(θ) = FZ (f )(θ)FZ (g)(θ), where ∗ denotes the usual convolution in Z.
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121
We are going to motivate our definition of the discrete fractional Laplacian; the details can be seen in [1]. Observe that from the discrete Laplacian ∆d f (n) := f (n + 1) − 2f (n) + f (n − 1), (such as defined in [7]), and using the identity 2 sin2
θ 2
n∈Z
= 1 − cos θ, we obtain
FZ (−∆d f )(θ) = 4 sin2 (θ/2)FZ (f )(θ). Let f ∈ ℓ∞ (Z) be given. By taking the inverse discrete Fourier transform, the discrete fractional Laplacian of order α > 0 is then defined by (−∆d )α f (n) := K α (n − k)f (k), n ∈ Z, (3) k∈Z
where K α (n) :=
1 2π
π
(4 sin2 (θ/2))α e−inθ dθ,
n ∈ Z.
−π
Then, it is clear that FZ (K α )(θ) = (4 sin2 (θ/2))α ,
θ ∈ (−π, π].
(4)
Remark 1. Using [8, formula 2.5.12 (22), p. 402] we obtain the following explicit expression for the kernel K α (n) in terms of the Gamma function: K α (n) =
(−1)n Γ (2α + 1) , Γ (1 + α + n)Γ (1 + α − n)
n ∈ Z.
(5)
In particular, it shows that definition (3) coincides with the generalized fractional difference corresponding to the type 1 central derivative considered by M.D. Ortigueira in [9,10]. See also [11, formulas (2) and (36)]. Because of this connection, some properties for the discrete Laplacian can be directly deduced. For example, associativity (−∆d )α (−∆d )β = (−∆d )α+β provided α + β > −1. Moreover, |K α (n)| ∼
Γ (2α + 1) −2α−1 |n| , π
n → ±∞,
see [9, formula (4.24)]. In particular, it shows that the series on the right hand side of (3) converges for f ∈ ℓ∞ (Z). For other properties, we refer to [11, Section 2.2]. In Fig. 1 we show the aspect of K α (n) (for −15 ≤ n ≤ 15) for several values of the parameter α; actually, to compensate the big decay of K α (n) when n → ±∞ and to get a more significative picture, we represent the kernel multiplied by 1 + |n|2α+1 . In particular, we clearly observe that K α (n) > 0 only when n = 0 (of course, this can be also proved from (5)). From this and the identity (−∆d )α f (n) := K α (n − k)f (k) = K α (n − k)f (k) + K α (0)f (n) k̸=n
k∈Z
we deduce that (−∆d )α f (n0 ) ≤ 0 whenever f (k) ≥ 0 for all k ̸= n0 and f (n0 ) ≤ 0. It extends [1, Theorem 1]. In what follows, we present the definition and some properties of the Liouville fractional derivative on the whole axis R. More detailed information may be found in the book [12, Chapter II, Section 2.3]. We denote gβ (t) :=
tβ−1 , Γ (β)
t > 0, β > 0,
(6)
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Fig. 1. Graphical representation of (1 + |n|2α+1 )K α (n) for α = 0.3, 0.7, 0.9 and 1.
and in case β = 0 we set g0 (t) := δ0 , the Dirac measure concentrated at the origin. Recall also the well-known formula for the Gamma function ∞ 1 (7) e−λt gβ (t) dt = β , λ > 0, β > 0; λ 0 see, for instance, [8, formula 2.3.3 (1), p. 322]. The Liouville (left-sided) fractional derivative of order α > 0 is defined by t dn Dtα h(t) = n gn−α (t − s)h(s) ds dt −∞ where n = ⌊α⌋ + 1, t ∈ R. In particular, when α = m ∈ N0 , then Dt0 h(t) = h(t) and Dtm h(t) = h(m) (t) where h(m) (t) is the usual derivative of h(t) of order m. Let h be in the N -dimensional Schwartz’s class S. The Fourier transform of h is given by 1 FRN (h)(ξ) = h(x)e−ix·ξ dx, ξ ∈ RN . (2π)N/2 RN We recall that the continuous fractional Laplacian (−∆)α with 0 < α < 1 can be defined in several equivalent ways. It is defined via the Fourier transform as FRN ((−∆)α h)(ξ) = |ξ|2α FRN (h)(ξ). An equivalent definition, obtained by computing the inverse Fourier transform (see [13]), is given by the singular integral h(x) − h(y) dy, (−∆)α h(x) = CN,α P.V. N +2α RN ∥x − y∥ where CN,α is an explicit positive constant. A comparable formula, that avoids the computation of the inverse Fourier transform, and provides the pointwise formula above in a simple way, can be found for example in [14, Lemma 2.1, p. 35]: ∞ 1 dt (−∆)α h(x) = (et∆ h(x) − h(x)) 1+α Γ (−α) 0 t where et∆ is the heat-diffusion semigroup.
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3. The discrete Laplacian and superdiffusion We recall from [1, Section 2] that the discrete heat semigroup is defined by e−2t In−m (2t)f (m), et∆d f (n) := m∈Z
where Ik is the modified Bessel function of the first kind and order k ∈ Z, defined as ∞ t 2m+k 1 Ik (t) = . m! Γ (m + k + 1) 2 m=0 Several properties of Ik are listed in [1, Section 8]. In [1, Proposition 1] it was proved that {et∆d }t≥0 is a positive Markovian diffusion semigroup. Moreover, for each ϕ ∈ ℓ∞ , the function u(n, t) = et∆d ϕ(n) is a solution of the discrete heat equation, that is (1) with α = 1. The following result corresponds to the discrete counterpart of the formula with the semigroup for the fractional Laplacian [6, Lemma 5.1]. Theorem 2. For all 0 < α < 1 and f ∈ ℓ∞ (Z) the following holds: ∞ 1 dt (−∆d )α f (n) = (et∆d f (n) − f (n)) 1+α , Γ (−α) 0 t
n ∈ Z.
Proof. By definition (3) and the identity (4) we have FZ ((−∆d )α f )(θ) = FZ (K α )(θ)FZ (f )(θ) = (4 sin2 (θ/2))α FZ (f )(θ). On the other hand, we have 2
FZ (et∆d f )(θ) = e−4t sin
(θ/2)
FZ (f )(θ),
see the proof of (iii) in [1, Proposition 1]. The claim then follows from the identity ∞ 2 1 dt (e−4t sin (θ/2) − 1) 1+α = (4 sin2 (θ/2))α , Γ (−α) 0 t that can be proven using integration by parts and the formula (7).
Let α > 0 be given. We define 1 := 2π
Ktα (n)
π
α 2 θ et 4 sin 2 e−inθ dθ,
t ∈ R, n ∈ Z.
−π
The following is the main result of this paper. Theorem 3. For every α that satisfies 0 < α < 1 and ϕ ∈ ℓ∞ (Z), the function u(n, t) = Ktα (n − k)ϕ(k), t ≥ 0, n ∈ Z,
(8)
k∈Z
solves the problems
Dt u(n, t) = (−∆d )α u(n, t), u(n, 0) = ϕ(n), n ∈ Z,
t > 0, n ∈ Z,
(9)
t > 0, n ∈ Z,
(10)
and 1/α Dt u(n, t) = (−∆d )u(n, t), u(n, 0) = ϕ(n), n ∈ Z.
´ Ciaurri et al. / Applied Mathematics Letters 49 (2015) 119–125 O.
124
Proof. We first prove that (8) solves (9). Indeed, by taking discrete Fourier transform in the variable n, Eq. (9) becomes
θ α FZ (u(·, t))(θ), Dt FZ (u(·, t))(θ) = 4 sin2 2 F (u(·, 0))(θ) = F (ϕ)(θ), Z Z
t > 0,
(11)
and a solution to (11) is FZ (u(·, t))(θ) = e
t 4 sin2
θ 2
α
FZ (ϕ)(θ),
t > 0.
Now, by applying inverse discrete Fourier transform, we have 1 u(n, t) = 2π =
π
e
t 4 sin2
θ 2
α
e
−π
ϕ(k)
k∈Z
1 2π
π
et
π α 1 t 4 sin2 θ2 FZ (ϕ)(θ) dθ = e−inθ ϕ(k)eikθ dθ e 2π −π k∈Z α θ −i(n−k)θ α 2 e dθ = ϕ(k)Kt (n − k).
−inθ
4 sin2
−π
k∈Z
Secondly, we prove that (8) solves (10). In fact, we have 1/α
Dt
u(n, t) =
1/α
Dt
Ktα (n − k)ϕ(k)
k∈Z
where, by interchanging the order of integration, 1/α
Dt
Ktα (n) =
1 2π
π
1/α (·) 4 sin2
Dt
e
(·) 4 sin2
e
θ 2
α
dm+1 (t) = m+1 dt
α
(t)e−inθ dθ.
−π
1 Since 0 < α < 1, there exists m ∈ N such that m+1 <α≤ Remembering the notation for gβ in (6), we obtain 1/α Dt
θ 2
t
1 m
for m ∈ N \ {1}, or
1 m+1
<α<
1 m
for m = 1.
α 2 θ gm+1−1/α (t − s)es 4 sin 2 ds
−∞
α ∞ α 2 θ 2 θ dm+1 = m+1 et 4 sin 2 gm+1−1/α (τ )e−τ 4 sin 2 dτ dt 0 α (m+1)α 2 θ θ 1 = 4 sin2 et 4 sin 2 2 θ α m+1−1/α 2 4 sin 2 α θ t 4 sin2 θ2 = 4 sin2 e , 2 where we used (7) in the third equality. (It is well known that the β-order Liouville (left-sided) fractional derivative Dtβ of eλt is λβ eλt , for λ > 0; see, for example, [12, formula (2.3.11), p. 88] or [15, Example 2.6]. We give here a direct proof for the sake of completeness.) Therefore 1/α
Dt
u(n, t) =
α 1 π 2 θ (2 − 2 cos θ)et 4 sin 2 e−i(n−k)θ dθ ϕ(k). 2π −π
k∈Z
(12)
´ Ciaurri et al. / Applied Mathematics Letters 49 (2015) 119–125 O.
125
On the other hand, using (8) we have ∆d u(n, t) = u(n + 1, t) − 2u(n, t) + u(n − 1, t) = Ktα (n + 1 − k) − 2Ktα (n − k) + Ktα (n − 1 − k) ϕ(k) k∈Z
1 π t4 sin2 θ α 2 e e−i(n−k+1)θ − 2e−i(n−k)θ + e−i(n−k−1)θ dθ ϕ(k) 2π −π k∈Z 1 π t4 sin2 θ α 2 =− e (2 cos θ − 2)e−i(n−k)θ dθ ϕ(k). 2π −π =
(13)
k∈Z
Combining (12) with (13) we obtain (10).
Remark 4. We note that if α ≥ 12 , then 1 < α1 ≤ 2 in (10), and then the equation should have two initial conditions. By using (3), a calculation shows that in such case the second initial condition reads ut (n, 0) = (−∆d )α ϕ(n). More generally, if
1 m+1
≤ α <
1 m,
m ∈ N, then we have m extra initial conditions in (10) and they are (m)
ut (n, 0) = (−∆d ) ϕ(n), utt (n, 0) = (−∆d )2α ϕ(n), . . . , ut α
(n, 0) = (−∆d )mα ϕ(n).
Acknowledgments The first, third and fourth authors are partially supported by DGI grant number MTM2012-36732C03-02. The third author is also partially supported by a mobility grant “Jos´e Castillejo” (CAS14/00037) from Ministerio de Educaci´ on, Cultura y Deporte of Spain. The second author is partially supported by FONDECYT grant number 1140258 and Project Anillo PBCT-ACT 1112. References ´ Ciaurri, T.A. Gillespie, L. Roncal, J.L. Torrea, J.L. Varona, Harmonic analysis associated with a discrete Laplacian, [1] O. J. Anal. Math. (2015) in press. arXiv:1401.2091. [2] V.V. Kulish, J.L. Lage, Application of fractional calculus to fluid mechanics, J. Fluids Eng. 124 (2002) 803–806. [3] V. Keyantuo, C. Lizama, On a connection between powers of operators and fractional Cauchy problems, J. Evol. Equ. 12 (2012) 245–265. k [4] H. Allouba, Time-fractional and memoryful ∆2 SIEs on R+ × Rd : how far can we push white noise? Illinois J. Math. 57 (2013) 919–963. [5] B. Baeumer, M.M. Meerschaert, E. Nane, Brownian subordinators and fractional Cauchy problems, Trans. Amer. Math. Soc. 361 (2009) 3915–3930. [6] P.R. Stinga, J.L. Torrea, Extension problem and Harnack’s inequality for some fractional operators, Comm. Partial Differential Equations 35 (2010) 2092–2122. [7] F.A. Gr¨ unbaum, P. Iliev, Heat kernel expansions on the integers, Math. Phys. Anal. Geom. 5 (2002) 183–200. [8] A.P. Prudnikov, A.Y. Brychkov, O.I. Marichev, Integrals and Series. Vol. 1. Elementary Functions, Gordon and Breach Science Publishers, New York, 1986. [9] M.D. Ortigueira, Riesz potentials operators and inverses via centred derivatives, Int. J. Math. Math. Sci. (2006) Article ID 48391. [10] M.D. Ortigueira, Fractional central differences and derivatives, J. Vib. Control 14 (2008) 1255–1266. [11] M.D. Ortigueira, J.J. Trujillo, A unified approach to fractional derivatives, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 5151–5157. [12] A.A. Kilbas, H.R. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-Holland Math. Studies, vol. 44, Elsevier, 2006. [13] N.S. Landkof, Foundations of modern potential theory, in: Die Grundlehren der Mathematischen Wissenschaften, Band 180, Springer-Verlag, New York, 1972. [14] P.R. Stinga, Fractional powers of second order partial differential operators: extension problem and regularity theory (Ph.D. thesis), Madrid, 2010. [15] M.M. Meerschaert, A. Sikorskii, Stochastic Models for Fractional Calculus, in: De Gruyter Studies in Mathematics, vol. 43, De Gruyter, Berlin, 2012.
Contents lists at ScienceDirect Contents lists available at available ScienceDirect
LinearMathematics Algebra andLetters its Applications Applied www.elsevier.com/locate/laa www.elsevier.com/locate/aml
Inverse eigenvalue problem of Jacobi matrix and On a connection between the discrete fractional Laplacian superdiffusion with mixed data Ying Lizama Wei 1 b,∗ , Luz Roncal a , Juan Luis Varona a ´ Oscar Ciaurri a , Carlos Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Departamento de Matem´ aticas y Computaci´ on, Universidad de La Rioja, 26004 Logro˜ no, Spain Nanjing 210016, PR China Universidad de Santiago de Chile, Facultad de Ciencias, Departamento de Matem´ atica y Ciencia de la Computaci´ on, Casilla 307, Correo 2, Santiago, Chile a
b
a r t i c l e article
info
i n f o
a b s t r a c t
abstract
Article history: In this paper, the inverse eigenvalue problem of reconstructing Received 16 January 2014 a Jacobi matrix from its eigenvalues, its leading principal Article history: Accepted 20 September We 2014 relate the fractional powers of discrete Laplacian with of a standard timesubmatrix andthepart of the eigenvalues its submatrix Received 30 March 2015Available online 22 October 2014 fractional derivative inisthe sense of Liouville by encoding the iterativeconditions nature of the considered. The necessary and sufficient for Received in revised form 12 May by Y. Weidiscrete operator through a time-fractional memory term. Submitted the existence and uniqueness of the solution are derived. 2015 Ltd. Allsome rightsnumerical reserved. © 2015 Elsevier Furthermore, a numerical algorithm and Accepted 12 May 2015 MSC: examples are given. Available online 21 May15A18 2015 © 2014 Published by Elsevier Inc. 15A57 Keywords: Discrete Laplacian Keywords: Superdiffusion Jacobi matrix Heat semigroup Eigenvalue Fractional Laplacian Inverse problem Modified Bessel functions Submatrix
1. Introduction Let 0 < α < 1 be given. Our concern in this paper is the study of the connection between two seemingly distinct classes of partial differential equations that have a mixed character, i.e. that can be modeled both in continuous time as well as in discrete space Dt v(n, t) = (−∆d )α v(n, t),
t > 0, n ∈ Z,
(1)
t > 0, n ∈ Z.
(2)
and 1
1/α Dt u(n, t) = (−∆d )u(n, t), E-mail address: [email protected]. Tel.: +86 13914485239.
1/α
In (1), Dt denotes the continuous derivative in the variable t, Dt denotes the fractional derivative of http://dx.doi.org/10.1016/j.laa.2014.09.031 α order 1/α in the sense of Liouville (left-sided) and (−∆ ) denotes the fractional powers of order α of the 0024-3795/© 2014 Published by Elsevier Inc.d ∗ Corresponding author. ´ Ciaurri), [email protected] (C. Lizama), [email protected] E-mail addresses: [email protected] (O. (L. Roncal), [email protected] (J.L. Varona).
http://dx.doi.org/10.1016/j.aml.2015.05.007 0893-9659/© 2015 Elsevier Ltd. All rights reserved.
120
´ Ciaurri et al. / Applied Mathematics Letters 49 (2015) 119–125 O.
unidimensional discrete Laplacian, introduced in [1] (where other operators in Harmonic Analysis, such as the discrete Riesz transform, square functions, conjugate harmonic functions and the Poisson semigroup were also studied). See Section 2 for definitions. For β := α1 > 1, Eq. (2) describes superdiffusive phenomena in time. It models anomalous superdiffusion in which a particle cloud spreads faster than the classical diffusion model predicts. The connection between order in time and space for partial differential equations is a surprising phenomena that seems not to be addressed for the discrete fractional Laplacian. It shows that the spatial-Laplacian of fractional order is entirely translated into temporal regularization. First studies on this unexpected property appear in [2] for the unidimensional continuous Laplacian. There, it was observed that ordinary PDEs of first order in space are transformed into PDEs of half-th order in time and second order in space, and that from an applied perspective (Stokes problem) this property shows numerical advantages. For the N -dimensional and continuous bi-Laplacian, the connection appeared in [3, Example 2.14], where also an abstract setting for higher powers is studied. For stochastic processes the relation seems to be more analyzed, but only very recently by H. Allouba [4] and B. Baeumer, M.M. Meerschaert and E. Nane [5, Theorems 3.1 and 3.9]. See also the references therein. In this paper we are able to prove that, for appropriate initial values in the Lebesgue space ℓ∞ (Z) and 0 < α < 1, the solution of problems (1) and (2) is the same, and that it admits an explicit representation in convolution form by means of a special kernel. This is shown in Theorem 3, that is the main result of this work. We define the discrete fractional Laplacian via the discrete Fourier transform, because this way is the most suitable to prove Theorem 3. Moreover, this definition coincides with the genuine definition of the fractional powers of a more general linear second order partial differential operator L by means of the semigroup generated by L, as shown by P.R. Stinga and J.L. Torrea in [6]. In particular, we provide the discrete counterpart of the formula for the fractional Laplacian via the semigroup due to Stinga and Torrea [6]; it is presented in Theorem 2 and has its own interest. 2. Preliminaries In order to establish and clarify the meaning of Eqs. (1) and (2), and the relationship between their solutions, we need to define several continuous and discrete operators. In particular, in what follows we are going to give precise definitions for several kinds of Fourier transforms and fractional operators on some spaces, and to provide some of their properties. For a given sequence f , we define the discrete Fourier transform FZ (f )(θ) = f (n)einθ , θ ∈ T, n∈Z
where T ≡ R/(2πZ) is the unidimensional torus, that we identify with the interval (−π, π]. The inverse discrete Fourier transform is obtained by the formula π 1 −1 FZ (ϕ)(n) = ϕ(θ)e−inθ dθ, n ∈ Z, 2π −π for a given function ϕ. Therefore f (n) =
1 2π
π
−π
FZ (f )(θ)e−inθ dθ,
n ∈ Z.
It is easily verified that FZ (f ∗ g)(θ) = FZ (f )(θ)FZ (g)(θ), where ∗ denotes the usual convolution in Z.
´ Ciaurri et al. / Applied Mathematics Letters 49 (2015) 119–125 O.
121
We are going to motivate our definition of the discrete fractional Laplacian; the details can be seen in [1]. Observe that from the discrete Laplacian ∆d f (n) := f (n + 1) − 2f (n) + f (n − 1), (such as defined in [7]), and using the identity 2 sin2
θ 2
n∈Z
= 1 − cos θ, we obtain
FZ (−∆d f )(θ) = 4 sin2 (θ/2)FZ (f )(θ). Let f ∈ ℓ∞ (Z) be given. By taking the inverse discrete Fourier transform, the discrete fractional Laplacian of order α > 0 is then defined by (−∆d )α f (n) := K α (n − k)f (k), n ∈ Z, (3) k∈Z
where K α (n) :=
1 2π
π
(4 sin2 (θ/2))α e−inθ dθ,
n ∈ Z.
−π
Then, it is clear that FZ (K α )(θ) = (4 sin2 (θ/2))α ,
θ ∈ (−π, π].
(4)
Remark 1. Using [8, formula 2.5.12 (22), p. 402] we obtain the following explicit expression for the kernel K α (n) in terms of the Gamma function: K α (n) =
(−1)n Γ (2α + 1) , Γ (1 + α + n)Γ (1 + α − n)
n ∈ Z.
(5)
In particular, it shows that definition (3) coincides with the generalized fractional difference corresponding to the type 1 central derivative considered by M.D. Ortigueira in [9,10]. See also [11, formulas (2) and (36)]. Because of this connection, some properties for the discrete Laplacian can be directly deduced. For example, associativity (−∆d )α (−∆d )β = (−∆d )α+β provided α + β > −1. Moreover, |K α (n)| ∼
Γ (2α + 1) −2α−1 |n| , π
n → ±∞,
see [9, formula (4.24)]. In particular, it shows that the series on the right hand side of (3) converges for f ∈ ℓ∞ (Z). For other properties, we refer to [11, Section 2.2]. In Fig. 1 we show the aspect of K α (n) (for −15 ≤ n ≤ 15) for several values of the parameter α; actually, to compensate the big decay of K α (n) when n → ±∞ and to get a more significative picture, we represent the kernel multiplied by 1 + |n|2α+1 . In particular, we clearly observe that K α (n) > 0 only when n = 0 (of course, this can be also proved from (5)). From this and the identity (−∆d )α f (n) := K α (n − k)f (k) = K α (n − k)f (k) + K α (0)f (n) k̸=n
k∈Z
we deduce that (−∆d )α f (n0 ) ≤ 0 whenever f (k) ≥ 0 for all k ̸= n0 and f (n0 ) ≤ 0. It extends [1, Theorem 1]. In what follows, we present the definition and some properties of the Liouville fractional derivative on the whole axis R. More detailed information may be found in the book [12, Chapter II, Section 2.3]. We denote gβ (t) :=
tβ−1 , Γ (β)
t > 0, β > 0,
(6)
122
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Fig. 1. Graphical representation of (1 + |n|2α+1 )K α (n) for α = 0.3, 0.7, 0.9 and 1.
and in case β = 0 we set g0 (t) := δ0 , the Dirac measure concentrated at the origin. Recall also the well-known formula for the Gamma function ∞ 1 (7) e−λt gβ (t) dt = β , λ > 0, β > 0; λ 0 see, for instance, [8, formula 2.3.3 (1), p. 322]. The Liouville (left-sided) fractional derivative of order α > 0 is defined by t dn Dtα h(t) = n gn−α (t − s)h(s) ds dt −∞ where n = ⌊α⌋ + 1, t ∈ R. In particular, when α = m ∈ N0 , then Dt0 h(t) = h(t) and Dtm h(t) = h(m) (t) where h(m) (t) is the usual derivative of h(t) of order m. Let h be in the N -dimensional Schwartz’s class S. The Fourier transform of h is given by 1 FRN (h)(ξ) = h(x)e−ix·ξ dx, ξ ∈ RN . (2π)N/2 RN We recall that the continuous fractional Laplacian (−∆)α with 0 < α < 1 can be defined in several equivalent ways. It is defined via the Fourier transform as FRN ((−∆)α h)(ξ) = |ξ|2α FRN (h)(ξ). An equivalent definition, obtained by computing the inverse Fourier transform (see [13]), is given by the singular integral h(x) − h(y) dy, (−∆)α h(x) = CN,α P.V. N +2α RN ∥x − y∥ where CN,α is an explicit positive constant. A comparable formula, that avoids the computation of the inverse Fourier transform, and provides the pointwise formula above in a simple way, can be found for example in [14, Lemma 2.1, p. 35]: ∞ 1 dt (−∆)α h(x) = (et∆ h(x) − h(x)) 1+α Γ (−α) 0 t where et∆ is the heat-diffusion semigroup.
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3. The discrete Laplacian and superdiffusion We recall from [1, Section 2] that the discrete heat semigroup is defined by e−2t In−m (2t)f (m), et∆d f (n) := m∈Z
where Ik is the modified Bessel function of the first kind and order k ∈ Z, defined as ∞ t 2m+k 1 Ik (t) = . m! Γ (m + k + 1) 2 m=0 Several properties of Ik are listed in [1, Section 8]. In [1, Proposition 1] it was proved that {et∆d }t≥0 is a positive Markovian diffusion semigroup. Moreover, for each ϕ ∈ ℓ∞ , the function u(n, t) = et∆d ϕ(n) is a solution of the discrete heat equation, that is (1) with α = 1. The following result corresponds to the discrete counterpart of the formula with the semigroup for the fractional Laplacian [6, Lemma 5.1]. Theorem 2. For all 0 < α < 1 and f ∈ ℓ∞ (Z) the following holds: ∞ 1 dt (−∆d )α f (n) = (et∆d f (n) − f (n)) 1+α , Γ (−α) 0 t
n ∈ Z.
Proof. By definition (3) and the identity (4) we have FZ ((−∆d )α f )(θ) = FZ (K α )(θ)FZ (f )(θ) = (4 sin2 (θ/2))α FZ (f )(θ). On the other hand, we have 2
FZ (et∆d f )(θ) = e−4t sin
(θ/2)
FZ (f )(θ),
see the proof of (iii) in [1, Proposition 1]. The claim then follows from the identity ∞ 2 1 dt (e−4t sin (θ/2) − 1) 1+α = (4 sin2 (θ/2))α , Γ (−α) 0 t that can be proven using integration by parts and the formula (7).
Let α > 0 be given. We define 1 := 2π
Ktα (n)
π
α 2 θ et 4 sin 2 e−inθ dθ,
t ∈ R, n ∈ Z.
−π
The following is the main result of this paper. Theorem 3. For every α that satisfies 0 < α < 1 and ϕ ∈ ℓ∞ (Z), the function u(n, t) = Ktα (n − k)ϕ(k), t ≥ 0, n ∈ Z,
(8)
k∈Z
solves the problems
Dt u(n, t) = (−∆d )α u(n, t), u(n, 0) = ϕ(n), n ∈ Z,
t > 0, n ∈ Z,
(9)
t > 0, n ∈ Z,
(10)
and 1/α Dt u(n, t) = (−∆d )u(n, t), u(n, 0) = ϕ(n), n ∈ Z.
´ Ciaurri et al. / Applied Mathematics Letters 49 (2015) 119–125 O.
124
Proof. We first prove that (8) solves (9). Indeed, by taking discrete Fourier transform in the variable n, Eq. (9) becomes
θ α FZ (u(·, t))(θ), Dt FZ (u(·, t))(θ) = 4 sin2 2 F (u(·, 0))(θ) = F (ϕ)(θ), Z Z
t > 0,
(11)
and a solution to (11) is FZ (u(·, t))(θ) = e
t 4 sin2
θ 2
α
FZ (ϕ)(θ),
t > 0.
Now, by applying inverse discrete Fourier transform, we have 1 u(n, t) = 2π =
π
e
t 4 sin2
θ 2
α
e
−π
ϕ(k)
k∈Z
1 2π
π
et
π α 1 t 4 sin2 θ2 FZ (ϕ)(θ) dθ = e−inθ ϕ(k)eikθ dθ e 2π −π k∈Z α θ −i(n−k)θ α 2 e dθ = ϕ(k)Kt (n − k).
−inθ
4 sin2
−π
k∈Z
Secondly, we prove that (8) solves (10). In fact, we have 1/α
Dt
u(n, t) =
1/α
Dt
Ktα (n − k)ϕ(k)
k∈Z
where, by interchanging the order of integration, 1/α
Dt
Ktα (n) =
1 2π
π
1/α (·) 4 sin2
Dt
e
(·) 4 sin2
e
θ 2
α
dm+1 (t) = m+1 dt
α
(t)e−inθ dθ.
−π
1 Since 0 < α < 1, there exists m ∈ N such that m+1 <α≤ Remembering the notation for gβ in (6), we obtain 1/α Dt
θ 2
t
1 m
for m ∈ N \ {1}, or
1 m+1
<α<
1 m
for m = 1.
α 2 θ gm+1−1/α (t − s)es 4 sin 2 ds
−∞
α ∞ α 2 θ 2 θ dm+1 = m+1 et 4 sin 2 gm+1−1/α (τ )e−τ 4 sin 2 dτ dt 0 α (m+1)α 2 θ θ 1 = 4 sin2 et 4 sin 2 2 θ α m+1−1/α 2 4 sin 2 α θ t 4 sin2 θ2 = 4 sin2 e , 2 where we used (7) in the third equality. (It is well known that the β-order Liouville (left-sided) fractional derivative Dtβ of eλt is λβ eλt , for λ > 0; see, for example, [12, formula (2.3.11), p. 88] or [15, Example 2.6]. We give here a direct proof for the sake of completeness.) Therefore 1/α
Dt
u(n, t) =
α 1 π 2 θ (2 − 2 cos θ)et 4 sin 2 e−i(n−k)θ dθ ϕ(k). 2π −π
k∈Z
(12)
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125
On the other hand, using (8) we have ∆d u(n, t) = u(n + 1, t) − 2u(n, t) + u(n − 1, t) = Ktα (n + 1 − k) − 2Ktα (n − k) + Ktα (n − 1 − k) ϕ(k) k∈Z
1 π t4 sin2 θ α 2 e e−i(n−k+1)θ − 2e−i(n−k)θ + e−i(n−k−1)θ dθ ϕ(k) 2π −π k∈Z 1 π t4 sin2 θ α 2 =− e (2 cos θ − 2)e−i(n−k)θ dθ ϕ(k). 2π −π =
(13)
k∈Z
Combining (12) with (13) we obtain (10).
Remark 4. We note that if α ≥ 12 , then 1 < α1 ≤ 2 in (10), and then the equation should have two initial conditions. By using (3), a calculation shows that in such case the second initial condition reads ut (n, 0) = (−∆d )α ϕ(n). More generally, if
1 m+1
≤ α <
1 m,
m ∈ N, then we have m extra initial conditions in (10) and they are (m)
ut (n, 0) = (−∆d ) ϕ(n), utt (n, 0) = (−∆d )2α ϕ(n), . . . , ut α
(n, 0) = (−∆d )mα ϕ(n).
Acknowledgments The first, third and fourth authors are partially supported by DGI grant number MTM2012-36732C03-02. The third author is also partially supported by a mobility grant “Jos´e Castillejo” (CAS14/00037) from Ministerio de Educaci´ on, Cultura y Deporte of Spain. The second author is partially supported by FONDECYT grant number 1140258 and Project Anillo PBCT-ACT 1112. References ´ Ciaurri, T.A. Gillespie, L. Roncal, J.L. Torrea, J.L. Varona, Harmonic analysis associated with a discrete Laplacian, [1] O. J. Anal. Math. (2015) in press. arXiv:1401.2091. [2] V.V. Kulish, J.L. Lage, Application of fractional calculus to fluid mechanics, J. Fluids Eng. 124 (2002) 803–806. [3] V. Keyantuo, C. Lizama, On a connection between powers of operators and fractional Cauchy problems, J. Evol. Equ. 12 (2012) 245–265. k [4] H. Allouba, Time-fractional and memoryful ∆2 SIEs on R+ × Rd : how far can we push white noise? Illinois J. Math. 57 (2013) 919–963. [5] B. Baeumer, M.M. Meerschaert, E. Nane, Brownian subordinators and fractional Cauchy problems, Trans. Amer. Math. Soc. 361 (2009) 3915–3930. [6] P.R. Stinga, J.L. Torrea, Extension problem and Harnack’s inequality for some fractional operators, Comm. Partial Differential Equations 35 (2010) 2092–2122. [7] F.A. Gr¨ unbaum, P. Iliev, Heat kernel expansions on the integers, Math. Phys. Anal. Geom. 5 (2002) 183–200. [8] A.P. Prudnikov, A.Y. Brychkov, O.I. Marichev, Integrals and Series. Vol. 1. Elementary Functions, Gordon and Breach Science Publishers, New York, 1986. [9] M.D. Ortigueira, Riesz potentials operators and inverses via centred derivatives, Int. J. Math. Math. Sci. (2006) Article ID 48391. [10] M.D. Ortigueira, Fractional central differences and derivatives, J. Vib. Control 14 (2008) 1255–1266. [11] M.D. Ortigueira, J.J. Trujillo, A unified approach to fractional derivatives, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 5151–5157. [12] A.A. Kilbas, H.R. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-Holland Math. Studies, vol. 44, Elsevier, 2006. [13] N.S. Landkof, Foundations of modern potential theory, in: Die Grundlehren der Mathematischen Wissenschaften, Band 180, Springer-Verlag, New York, 1972. [14] P.R. Stinga, Fractional powers of second order partial differential operators: extension problem and regularity theory (Ph.D. thesis), Madrid, 2010. [15] M.M. Meerschaert, A. Sikorskii, Stochastic Models for Fractional Calculus, in: De Gruyter Studies in Mathematics, vol. 43, De Gruyter, Berlin, 2012.