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Applied Mathematics and Computation 257 (2015) 12–33
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Applied Mathematics and Computation
journal homepage:www.elsevier.com/locate/amc
Lattice fractional calculus
Vasily E. Tarasov
Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia
article info abstract
Keywords: Integration and differentiation of non-integer orders for N-dimensional physical lattices
Fractional calculus with long-range particle interactions are suggested. The proposed lattice fractional deriv-
Lattice models atives and integrals are represented by kernels of lattice long-range interactions, such that
Fractional derivative their Fourier series transformations have a power-law form with respect to components of
Fractional integral wavevector.Continuouslimitsfortheselatticefractional derivatives and integrals give the
Nonlocal continuum continuumderivativesandintegralsofnon-integerorderswithrespecttocoordinates.Lat-
Fractional dynamics tice analogs of fractional differential equations that include suggested lattice differential
andintegral operators can serve as an important element of microscopic approach to non-
local continuum models in mechanics and physics.
2014Elsevier Inc. All rights reserved.
1. Introduction
The main approaches to describe nonlocal properties of media and materials are a macroscopic approach based on the
continuum mechanics [1–5], and a microscopic approach based on the lattice mechanics [6–9]. Continuum mechanics can
be considered as a continuous limit of lattice dynamics, where the sizes of continuum elements are much larger than the
distances between lattice particles.
Theory of derivatives and integrals of non-integer orders [10–19] has a long history and it goes back to the famous sci-
entist such as Leibniz, Riemann, Liouville, Letnikov, Weyl, Riesz and other. Fractional calculus and fractional differential
equations have a wide application in different areas of physics [20–31]. Fractional integro-differential equations are very
important to describe processes in nonlocal continua and media. Fractional integrals and differential operators with respect
to coordinates allow us to describe continuouslydistributed system with power-law type of nonlocality. Therefore fractional
calculus serve as a powerful tool in physics and mechanics of nonlocal continua. As it was shown in [41,42,25], the fractional
differential equations for nonlocal continua can be directly connected to models of lattice with long-range interactions of
power-lawtype. Interconnection between the equations for lattice with long-range interactions and the fractional differen-
tial equations for continuum is proved by special transform operator that includes a continuous limit, and the Fourier series
andintegraltransformations [41–44].In[55–59]thisapproachhasbeenappliedtolatticemodelsoffractionalnonlocalcon-
tinua in one-dimensional case only. In this paper we propose a lattice fractional calculus that allows us to extend these lat-
tice models to N-dimensional case.
Dynamics of physical lattices and discretely distributed systems with long-range interactions has been the subject of
investigations in different areas of science. Effect of synchronization for nonlinear systems with long-range interactions is
described in [32]. Non-equilibrium phase transitions for systems with long-range interactions are considered in [33]. Sta-
tionary states for fractional systems with long-range interactions are discussed in [46,34,35]. The evolution of soliton-like
E-mail address: tarasov@theory.sinp.msu.ru
http://dx.doi.org/10.1016/j.amc.2014.11.033
0096-3003/ 2014 Elsevier Inc. All rights reserved.
V.E. Tarasov/Applied Mathematics and Computation 257 (2015) 12–33 13
and breather-like structures in one-dimensional lattice of coupled oscillators with the long-range power are considered in
[36]. Kinks in the Frenkel–Kontorova model with long-range particle interactions is studied in [37]. In statistical mechanics
andnonlineardynamics,solvablemodelswithlong-rangeinteractionsaredescribedindetailinthereviews[38–40].Differ-
ent discrete systems and lattice with long-range interactions and its continuous limits are considered in [23,25]. It is impor-
tant that lattice models with long-range interactions of power-law type can lead to fractional nonlocal continuum models in
the continuous limit [41,42,25], Nonlocal continuum mechanics can be considered as a continuous limit of mechanics of lat-
tice with long-range interactions, whenthesizesofcontinuumelementaremuchlargerthanthedistancesbetweenparticles
of lattice.
It should be note that a calculus of operators of integer orders for physical lattice models has been considered in the
papers [48–50]. This lattice calculus of integer order is dened on a general triangulating graph by using discrete eld quan-
tities and differential operators roughly analogous to differential forms and exterior differential calculus. A scheme to derive
lattice differential operators of integer orders from the discrete velocities and associated Maxwell–Boltzmann distributions
that are used in lattice hydrodynamics has been suggested in the articles [51,52]. In this paper to formulate a lattice frac-
tional calculus, we use other approach that is based on models of physical lattices with long-range inter-particle interactions
and its continuum limit that are suggested in [41,42,25] (see also [43–47,55–59]).
In this paper, we propose lattice analogs of differentiation and integration of non-integer orders based on N-dimensional
generalization of the lattice approach suggested in [41,42,25]. A general form of lattice fractional derivatives and integrals
that gives continuumderivatives and integrals of non-integer orders in continuous limit is suggested. These continuum frac-
tional operators of differentiations and integrations can be considered as fractional derivatives and integrals of the Riesz type
with respect to coordinates.
2. Lattice fractional differential operators
2.1. Lattice fractional partial derivatives
Let us consider an unbounded physical lattice characterized by N non-coplanar vectors ai;i ¼ 1;...;N, that are the short-
est vectors by which a lattice can be displaced and be brought back into itself. For simplication, we assume that
ai;i ¼ 1;...;N, are mutually perpendicular primitive lattice vectors. We choose directions of the axes of the Cartesian coor-
dinatesystemcoincideswiththevectora.Thena ¼a e,wherea ¼jajande;i¼1;...;N,isthebasisoftheCartesiancoor-
i i i i i i i
dinate system for RN. This simplication means that the lattice is a primitive N-dimensional orthorhombic Bravais lattice.
P
The position vector of an arbitrary lattice site is written rðnÞ¼ N n a, where n are integer. In a lattice the sites are num-
i¼1 i i i
beredbyn,sothatthevectorn¼ðn ;...;n Þcanbeconsideredasanumbervectorofthecorrespondinglatticeparticle.We
1 N
assumethattheequilibrium positions of particles coincide with the lattice sites rðnÞ. Coordinates rðnÞ of lattice sites differs
from the coordinates of the corresponding particles, when particles are displaced relative to their equilibrium positions. To
dene the coordinates of a particle, we dene displacement of n-particle from its equilibrium position by the scalar eld
PN
uðnÞ, or the vector eld uðnÞ¼ uðnÞe, where the vectors e ¼ a=jaj form the basis of the Cartesian coordinate system.
i¼1 i i i i i
The functions uiðnÞ¼uiðn1;...;nNÞ are components of the displacement vector for lattice particle that is dened by
n¼ðn ;...;n Þ. In many cases, we can assume that uðnÞ belongs to the Hilbert space l of square-summable sequences to
1 N 2
applytheFouriertransformations. For simplication, we will consider differential and integral operators for the lattice func-
tions u ¼ uðnÞ¼uðn ;...;n Þ. All transformations can be easily generalized to the case of the vector functions.
1 N
Let us give a denition of lattice partial derivative of arbitrary positive real order a in the direction e ¼ a =ja j in the
i i i
lattice.
Denition 1. A lattice fractional partial derivative is the operator D a such that
L i
þ1
D a u¼ 1 X Kðn mÞuðmÞ; ði¼1;...;NÞ; ð1Þ
L a a i i
i a
i m¼1
i
where a 2 R;a > 0, m 2 Z, and the interaction kernels KðnmÞ are dened by the equations
a
!
pa aþ1 1 aþ3 p2ðnmÞ2
KþðnmÞ¼ 1F2 ; ; ; ; a>0; ð2Þ
a aþ1 2 2 2 4
!
aþ1 2 2
p ðnmÞ aþ2 3 aþ4 p ðnmÞ
KaðnmÞ¼ aþ2 1F2 2 ;2; 2 ; 4 ; a>0; ð3Þ
where 1F2 is the Gauss hypergeometric function [63]. The parameter a > 0 will be called the order of the lattice
derivative (1).
Let us explain the reasons for denition the interaction kernels KðnmÞ in the forms (2), (3), and describe some
a
properties of these kernels.
14 V.E. Tarasov/Applied Mathematics and Computation 257 (2015) 12–33
ThekernelsKðnÞarereal-valuedfunctionsofintegervariablen 2 Z.ThekernelKþðnÞiseven(orsymmetricwithrespect
a a
to zero) function and KðnÞ is odd (or antisymmetric with respect to zero) function such that
a
KþðnÞ¼þKþðnÞ; KðnÞ¼KðnÞ ð4Þ
a a a a
hold for all n 2 Z. ^þ þ
The Fourier series transforms KaðkÞ of the kernels KaðnÞ in the form
þ1 1
^þ X ikn þ Xþ þ
KaðkÞ¼ e KaðnÞ¼2 KaðnÞcosðknÞþKað0Þð5Þ
n¼1 n¼1
satisfy the condition
^þ a
KaðkÞ¼jkj ; ða > 0Þ: ð6Þ
^
The Fourier series transforms KaðkÞ of the kernels KaðnÞ in the form
þ1 1
^ X ikn X
KaðkÞ¼ e KaðnÞ¼2i KaðnÞ sinðknÞð7Þ
n¼1 n¼1
satisfy the condition
^ a
KaðkÞ¼isgnðkÞjkj ; ða > 0Þ: ð8Þ
Notethatweusetheminussignintheexponentsof(5)and(7)insteadofplusinordertohavetheplussignforplanewaves
and for the Fourier series.
Theform(2)oftheinteractiontermKþðnmÞiscompletelydeterminedbytherequirement(6).Ifweuseaninverserela-
a a
^þ
tion to (5) with KaðkÞ¼jkj that has the form
KþðnÞ¼1 Z pka cosðnkÞdk; ða 2 R; a>0Þ; ð9Þ
a p 0
then we get Eq. (2) for the interaction kernel KþðnmÞ. Note that
a
pa
Kþð0Þ¼ : ð10Þ
a aþ1
The form (3) of the interaction term KðnmÞ is completely determined by (6). If we use an inverse relation to (7) with
a a
^
KaðkÞ¼isgnðkÞjkj that has the form
1 Z p a
KaðnÞ¼ k sinðnkÞdk ða2R; a>0Þ; ð11Þ
p 0
then we get Eq. (3) for the interaction kernel KðnmÞ. Note that Kð0Þ¼0.
a a
Theinteractionswith(2)and(3)forintegerandnon-integerordersacanbeinterpretedasalong-rangeinteractionsofn-
particle with all other particles.
Properties of the interaction kernels (2) and (3), can be visualized by plots of the functions
y 2 2
fþðx;yÞ¼ p 1F2 yþ1;1;yþ3;p x ; ð12Þ
yþ1 2 2 2 4
yþ1 2 2
p x yþ2 3 yþ4 p x
fðx;yÞ¼yþ21F2 2 ;2; 2 ; 4 ð13Þ
for positive values of variables x and y. The function (12) is given on Figs. 1, 3 and 5, and the function (13) is presented by
Figs. 2, 4 and 6.
^
Let us give exact forms of the kernels KaðkÞ for integer positive a 2 N. Eqs. (2) and (3) for the case a 2 N can be simplied.
^
Wecanuseinverserelations (9) and (11) for integer positive a 2 N. to dene exact form of the kernels KaðkÞ. To obtain the
^
simplied expressions for kernels KaðkÞ with positive integer a ¼ m, we use the integrals (see Section 2.5.3.5 in [62]) of the
form
Z p nþ2 ½ðm1Þ=2 k ½ðmþ1Þ=2
xm cosðnxÞdx ¼ ð1Þ X ð1Þ m! ðpnÞm2k1þð1Þ m! ðÞ2½ðm þ1Þ=2m ; ðm2NÞ; ð14Þ
mþ1 mþ1
0 n k¼0 ðm2n1Þ! n
Z p nþ1 ½m=2 k ½m=2
xm sinðnxÞdx ¼ ð1Þ Xð1Þ m! ðpnÞm2kþð1Þ m! ðÞ2½m=2mþ1 ; ðm2NÞ; ð15Þ
mþ1 mþ1
0 n k¼0 ðm2nÞ! n
V.E. Tarasov/Applied Mathematics and Computation 257 (2015) 12–33 15
Fig. 1. Plot of the function f ðx;yÞ (12) for the range x 2½0;6 and y ¼ a 2½0;6 that represents the kernels of the lattice fractional derivatives Dþ a with
a¼y. þ L i
Fig. 2. Plot of the function f ðx;yÞ (13) for the range x 2½0;6 and y ¼ a 2½0;6 that represents the kernels of the lattice fractional derivatives D a with
a¼y. L i
Fig. 3. Plot of the function f ðx;yÞ (12) for the range x 2½0;6 and y ¼ a 2½0;8;1:2 that represents the kernels of the lattice fractional derivatives Dþ a
with a ¼ y. þ L i
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