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Phys. Scr. T136 (2009) 014027 (7pp) doi:10.1088/0031-8949/2009/T136/014027
Self-organization phenomena
in reaction–diffusion systemswith non-integer order time derivatives
B Y Datsko1 and V V Gafiychuk2
1 Institute for Applied Problems in Mechanics and Mathematics of the National Academy of Sciences,
Naukova 3b, Lviv 79063 Ukraine2 Physics Department, New York City College of Technology, CUNY, 300 Jay, NY, USA
E-mail: b [email protected] and [email protected]
Received 27 January 2009
Accepted for publication 28 January 2009
Published 12 October 2009
Online at stacks.iop.org/PhysScr/T136/014027
Abstract
This paper considers a one-dimensional fractional reaction–diffusion model with a different
order of derivative indices. The corresponding linear stability analysis is presented for any
number of fractional derivative indices that are greater than zero but less than two. It was
shown that instability can arise for the Hopf and the Turing modes. The self-organization
phenomena are more diverse when the activator derivative index is greater than the inhibitor
one than when the inhibitor variable index is greater. This analysis was supported by the
results of computer simulations of the fractional nonlinear reaction–diffusion
problem.
PACS number: 02.10.Ud
1. Introduction
Recently, the study of fractional diffusion has attracted much
interest [1–9]. Part of this interest comes from the attempt to
understand the phenomena of fractal and irregular systems [4],
and the relaxation processes [10] in nature. In fact, fractional
derivatives are used for a description of heterogeneous porous
systems [11], tumor growth [12], plasma [13], polymers [14],turbulence [15], disordered semiconductors [16], magnetic
resonance imaging [17], etc.
However, the study of the fractional reaction–diffusion
system (RDS) in general is of interest in other fields, like the
complex system and self-organization [19–25]. The present
paper is devoted to the investigation of self-organization
phenomena, widely studied in the standard systems [26–28],
in the media described by fractional systems. We will show
that in the fractional RDS with different orders of derivatives
we have phenomena that are not possible to find in RDSs with
integer derivatives. We confirm the linear stability analysis
by numerical simulation of a Bonhoeffer–van der Pol type
fractional RDS.
2. Mathematical model
The starting point of our consideration is the coupled
reaction–diffusion equations with indices of different orders
τ αu∂ αu( x , t )
∂t α= l2 ∂ 2u( x , t )
∂ x 2+ W (u, v, A ), (1)
τ βv∂β v( x , t )
∂t β= L2 ∂ 2v( x , t )
∂ x 2+ Q(u, v, A ), (2)
subject to the Neumann:
∂u/∂ x | x =0,l x = ∂v/∂ x | x =0,l x
= 0 (3)
boundary conditions and with certain initial conditions. Here,
u( x , t ), v( x , t ) are activator and inhibitor variables, 0 x l x , τ u , τ v, l, L are the characteristic times and lengths of
the system, correspondingly, A is an external parameter,
and W (u, v, A ), Q(u, v, A ) are the nonlinear sources of the
system modeling their production rates.
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Phys. Scr. T136 (2009) 014027 B Y Datsko and V V Gafiychuk
Time derivatives ∂α u( x ,t )
∂ t α, ∂β v( x ,t )
∂t βon the left-hand side
of equations (1) and (2), instead of the standard ones,
are the Caputo fractional derivatives in time of the order
0 < α, β < 2 and, for a certain rational value of α, are
represented as [29, 30]
∂α
f (t )∂t α
: s = 1(m − α)
t
0
f (m)
(τ )(t − τ )α+1−m
dτ,
where m − 1 < α < m, m = 1, 2.
The system (1) and (2) with arbitrary rational α and β,
by a certain substitution, can be transformed into the system
of differential equations with the same order of fractional
derivative index. In fact, if the new fractional derivative index
γ is the greatest common factor of α (α = pγ ) and β(β =r γ ), m, p ∈N, we obtain the system of p + r differential
equations
τ γ u
∂γ u p( x , t )
∂t γ
=l2 ∂ 2u( x , t )
∂ x 2
+ W (u, v, A ), (4)
τ γ v
∂ γ vr ( x , t )
∂t γ = L2 ∂2v( x , t )
∂ x 2+ Q(u, v, A ), (5)
where the derivatives on the right-hand side generate
recurrent equations for ui , i = p, p − 1, . . . , 1 and v j , j =r , r − 1, . . . , 1
τ γ u
∂γ ui−1( x , t )
∂t γ = ui ( x , t ), p i > 1, u1 ≡ u, (6)
τ γ v
∂γ v j−1( x , t )
∂t γ = v j ( x , t ), r j > 1, v1 ≡ v. (7)
This system of equations (4)–(7) is equivalent to the system
(1) and (2) and our analysis is devoted to the investigation of
its possible solutions for given forms W (u, v, A ), Q(u, v, A ).
We consider the RDS with two specific variables: one of
them is a variable with positive feedback and the other one
is a variable with negative feedback. Such systems possess
a variety of nonlinear phenomena investigated in the last
decades [26–28].
3. Linear stability analysis
3.1. Standard RDS
Let us start with α = β = 1, and we obtain a system (1)
and (2) with standard derivatives, where we can analyze its
nullclines
W (u, v, A ) = 0, Q(u, v, A ) = 0. (8)
The simultaneous solution of the system (8) leads to
homogeneous distribution of u and v. The stability of the
steady-state solutions of the system (1) and (2) corresponding
to homogeneous equilibrium state (u, v) is determined by the
eigenvalue problem for the matrix
F (k ) = a11
(k )/τ u
a12
/τ u
a21/τ v a22(k )/τ v
,
k = πl x
j, j = 1, 2, . . . , a11(k ) = a11 − k 2l2, a11 = W u , a12 =W v, a21 = Q
u , a22(k ) = a22 − k 2 L2, a22 = Qv (all derivatives
are taken at homogeneous equilibrium states W = Q = 0).
For this square matrix, eigenvalues are given by the quadratic
equation and can be determined as
λ1,2 = 12
(tr F ±√
tr2 F − 4det F ). (9)
For α
=β
=1 and k
=0 under the conditions
trF (0) > 0, det F (0) > 0, (10)
we can have a Hopf bifurcation. For k = 0 it is possible to
obtain that at a certain value of k 0 eigenvalues λ1,2 are real
and one of them is greater than zero (a Turing bifurcation).
The conditions of this instability are
trF < 0, det F (0) > 0, det F (k 0) < 0. (11)
We can rewrite the inequality (10) as a11 > −a22τ u /τ vaccording to the time frequency oscillation ω =√
det F (0)/(τ u τ v ) and the inequality (11) as [26–28]
a11 > −a22(l2/ L2) + 2
det F (0)(l/ L) (12)
according to wave numbers
k 0 = 4
det F (0)/√
l L. (13)
Instability conditions for these two types of bifurcations are
realized due to positive feedback (a11 > 0) and at τ u /τ v → 0
and l/ L → 0 they coincide and approach the extremum point
(or points) of W (u, v, A ) = 0.
3.2. Fractional RDS
Now we will analyze the fractional RDS with arbitrary
rational α, β. By introducing a new parameter γ , we have
a system of m + p differential equations (4)–(7). In this
case, the linearization of the system in the equilibrium
conditions described by vectors u = (u, 0, . . . , 0) p×1, v =(v, 0, . . . , 0)r ×1 leads to the characteristic equation det( J −λ I ) = 0 of the (r + p) degree polynomial
(−λ)r + p + (−1)r −1 a22(k )
τ βv
(−λ) p + (−1) p+1 a11(k )
τ αu(−λ)r
+(−1)r + p det F = 0. (14)
In the common case, the solution of such a type of equationcan be obtained numerically. This solution will determine a
stability of the system (4)–(7) [34]. At a small value of =detF (this situation takes place when nullclines W (u, v, A ) =0, Q(u, v, A ) = 0 are practically tangent to each other), it is
always possible to find the roots with the value close to zero
λi ≈ (τ βv /a22)1/ p, α < β, i ∈ ¯1, p, (15)
λi ≈ (τ αu /a11)1/r , α > β, i ∈ ¯1, r , (16)
and to determine where they are greater than zero. At
det F
≈0, the condition det F (0) > 0 can be rewritten
as dv/du|Q=0 > dv/du|W =0, which means that the secondnullcline (Q = 0) has a greater slope than the first one ( W =0). Here, we would like to consider the roots of several
specific cases, described by third degree polynomials, namely,
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Phys. Scr. T136 (2009) 014027 B Y Datsko and V V Gafiychuk
(a) (b)
Figure 1. Imaginary (gray lines) and real parts (black lines) of eigenvalues as function u at α = β for k = 0 (a), and k = 1 (b).The other parameters are B = 2, τ αu = 12, τ βv = 1, l2 = 0.1, L2 = 1.
(a) (b)
Figure 2. Imaginary (gray lines) and real parts (black lines) of eigenvalues as a function of u1 obtained from the solution of equation (19) at α = 2β for k = 0 (a) and k = 1 (b). The otherparameters are B = 1.1, τ αu = 0.1, τ βv = 1, l2 = 0.1, L2 = 1.
α = 2β = 2γ and β = 2α = 2γ . In this case, the characteristic
equation (14) has the form of the cubical equation
λ3 + λ2b + λc + d = 0, (17)
where
b = −a22(k )/τ βv , c = −a11(k )/τ αu , d = det F (k ) (18)
for α = 2β = 2γ , and
b = −a11(k )/τ u , c = −a22(k )/τ v, d = det F (k ) (19)
for β = 2α = 2γ (0 < γ < 1).
4. Eigenvalue problem analysis
for a Bonhoeffer–van der Pol type RDS
As an example, we consider here a Bonhoeffer–van der
Pol type RDS with cubical nonlinearity (see [24, 25, 27,
28]). In this case, the source term for the activator variable
is nonlinear, W = u − u3/3 − v, and it is linear for the
inhibitor one, Q = −v + Bu +A . The homogeneous solution
of variables u and v can be obtained from the system W =Q = 0, and for the determination of u, we have a cubic
algebraic equation
( B − 1)u + (u3/3) + A = 0. (20)
Calculation of the coefficients ai j : a11 = (1 − u2), a12 =−1, a21 = B, a22 = −1 at the homogeneous state (20) makes
it possible to investigate the eigenvalues of the system
explicitly. As a result, we can see that at τ αu /τ βv → 0
(a) (b)
Figure 3. Imaginary (gray lines) and real parts (black lines) of eigenvalues as a function of u1 at 2α = β for k = 0 (a) and k = 1(b). The other parameters are B = 1.1, τ αu = 0.1, τ βv = 1, l2 = 0.1, L2 = 1.
(a)
(b)
Figure 4. The oscillatory structures obtained from numericalsimulations of the system (1) and (2). (a) Dynamics of the variableu1 on the time interval (0,10) for α = 1.9, β = 1.75, τ αu = 0.1,τ βv = 1, l2 = 0.1, L2 = 1, A = −10, B = 3. (b) Dynamics of variable u1 on the time interval (0,60) for α
=1.9, β
=1.85,
τ αu = 12.5, τ βv = 1, l2 = 5.0, L2 = 1, A = −0.2, B = 2.
and l/ L → 0 in a standard RDS the instability domain
is determined at |u| < 1. In this case, the simultaneous
conditions of the Hopf (10) and the Turing (11) bifurcations
are realized.
4.1. The case α = β
The linear stability analysis of the system when fractional
derivative indices are equal is considered in [32, 33]. Here,
we would like to single out the special case when the index
is greater than one and a new type of bifurcation arises. Thereal and imaginary parts of eigenvalues for k = 0 obtained
numerically for each particular point u as a solution of
the equation (17) for certain values B, τ αu , τ βv , l2, L2 are
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Phys. Scr. T136 (2009) 014027 B Y Datsko and V V Gafiychuk
(a) (b)
(c) (d)
Figure 5. Pattern formation scenario for A = −0.1, α1 = 1.4, α2 = 0.7, l2 = 0.1, L2 = 1.0, B = 1.1, τ βv = 1, l x = 4π and different valuesof τ αu ((a) τ αu = 0.2, (b) τ αu = 0.25, (c) τ αu = 0.3 and (d) τ αu = 0.35).
presented in figure 1(a). For these parameters, we see thatthe real part of the roots is always less than zero and the
imaginary one on some interval of u becomes nonzero. In
this case, when the fractional derivative index becomes greater
than some critical value γ 0 = π2
tan−1(Im λ/Re λ): γ > γ 0, we
have homogeneous oscillations. In this case, at the values
of γ 0 > 1.5 instability conditions take place on the interval
1.8 |u| 4.5.
A similar plot can be presented for the eigenvalues when
wave number k = 0 (for example, k = 1). On figure 1(b),
imaginary parts of the eigenvalues are presented for the same
parameters of the system. As we can see from this figure,
when nullclines have an intersection point on the interval|u| between 4.5 and 6, the system is stable according to
homogeneous oscillations. At a certain value of α, instability
conditions are possible to realize for k = 1. This means that
at least perturbations with this wave number are unstable.
Moreover, they are unstable for oscillatory fluctuations. This
situation is qualitatively different from the integer RDS,
whether either the Turing (k = 0) or the Hopf bifurcations
(k = 0) take place. This depends on which conditions are
more easy to realize. In the system under consideration, we
can choose the parameter 4.5 < u < 6 when we do not have
a standard Hopf or Turing bifurcation at all. Nevertheless,
we obtained that the conditions for the Hopf bifurcation can
be realized for non-homogeneous perturbations with wave
numbers k = 0 [24, 25]. This phenomenon is inherent to the
system with different orders of fractional derivatives if the
difference between them is not so substantial.
4.2. The case α > β(0 < α, β < 2)
The real and imaginary parts of eigenvalues for k = 0,
depending on parameter u, for α = 2β and parameters typical
for pattern formation in regular systems, are presented in
figure 2(a). We see that one real root is always less than zero.
The roots of two others at |u| > u0 are complex conjugate
roots. At |u| < u0 these roots become real and positive. As
a result of these conditions, the instability in the system takes
place practically for any value of α.
Except homogeneous oscillations, the condition of the
Turing instability becomes true if the ratio l/ L 1. As an
example, the plot of eigenvalues for k
=1 is presented in
figure 2(b) where at |u| < uk all the roots are real and one of them corresponds to stationary non-homogeneous structures.
At |u| < uk , we could have non-homogeneous
oscillations of the structures if the conditions of this
instability are softer than conditions of homogeneous
oscillations.
4.3. The case α < β(0 < α, β < 2)
The dependence of eigenvalues as a function of u for 2α = β
and some other parameters are presented in figure 3. Similar to
the case considered above at |u| > u0, two roots are complex
and one is real. At certain values of u, all three roots become
real, two of them are positive and the system loses its stabilityat any value α.
At the same time, at l/ L 1 and |u| < 1, the system
is unstable according to the Turing instability. The plot of
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Phys. Scr. T136 (2009) 014027 B Y Datsko and V V Gafiychuk
(a)
(b)
Figure 6. Pattern formation scenario for A = −0.1, α = 1.0,β = 0.5, l2 = 0.1, L2 = 1.0, τ αu = 0.15, τ βv = 1, l x = 7.5 anddifferent values of B ((a) B = 2.0 and (b) B = 1.05).
the roots of the characteristic equation (17) for k
=1 is
presented in figure 3(b). At |u| < uk the system has onepositive real root of great value and two complex conjugate
roots. Namely, the first root is responsible for stationary
pattern formation. By increasing α, we can always obtain
homogeneous oscillations that, on the one hand, will lead
to oscillations of the structures, and on the other hand, can
destroy them and lead to homogeneous oscillations.
5. Nonlinear dynamics of Bonhoeffer–van der Poltype RDS
For a numerical simulation of the system (1) and (2) with
corresponding initial and boundary conditions, we used the
finite difference schemes based on the Grünwald–Letnikov
and Riemann–Liouville definition [29–31] the application of
which is considered in detail in the paper [32].
5.1. The case α β
The results of computer simulation of nonlinear
non-homogeneous oscillations for α β are presented
in figure 4. Realized non-homogeneous oscillations emerge
spontaneously at the given parameters. This agrees with the
theoretical investigation presented in section 4. It should be
noted that the presented phenomena are typical for differentrelationships between the system parameters and are realized
outside the increasing path of nullcline W = 0 (|u| > 1)
(figure 4(b)) as well as for l > L (figure 4(a)).
(a)
(b)
Figure 7. Pattern formation scenario for (a) α = 1.1, β = 0.55,A = −0.01, B = 1.1 and (b) α = 1.0, α = 0.5, A = −0.01, B = 1.1.The other parameters are τ αu = 0.1, τ βv = 1, l2 = 0.1, L2 = 1.0.
5.2. The case α > β(0 < α, β < 2)
Computer simulation of the system for this case is presented
in figures 5–7. In figure 5(a), we can mostly see the simplepattern formation scenario corresponding to homogeneousoscillations when the homogeneous solution u is close to zero.
By moving out of this point by increasing A , homogeneousoscillations are modulated by non-homogeneous mode.
Computer simulation shows that the interplay between
the Hopf and the Turing bifurcations, which leads tocomplicated dynamics, is typical for a wide spectrum of parameter α from a small one such as α = 0.1 to α = 1.
For example, typical spatio-temporal structures for differentparameters of τ u are presented in figures 5(c)–(d). Thesespatio-temporal patterns emerge as a result of the interplay
between the Hopf and the Turing bifurcations. In fact,
according to figure 2, eigenvalues for k = 0 and k = 1, thesystem parameters are practically the same in the vicinity of
the point |u| = 0.By changing the system parameters, for example β,
spatio-temporal patterns (see figure 6) are similar to those
displayed in figures 5(c) and (d) and we have a profile of complex ‘zigzag’ oscillations. It should be noted that the formof these structures is strongly dependent on the fractional
derivative indices as well as other parameters of the systemunder consideration.
At a certain value of A , where the Hopf and the
Turing instabilities have the same realization conditions, wehave stable space-time oscillations of the form presented in
figures 7(a) and (b). It should be noted that for a successiveincrease of A to u 1, we obtain a stable homogeneousdistribution, which for integer derivative indices, is unstableand we have homogeneous oscillations.
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Phys. Scr. T136 (2009) 014027 B Y Datsko and V V Gafiychuk
(a) (b)
(c) (d)
Figure 8. Pattern formation scenario for A = −0.1, l2 = 0.1, L2 = 1.0, τ αu = 0.15, τ βv = 1, B = 1.1 and different values of derivativeorders: (a) α = 0.4, β = 0.8 and (b) α = 0.75, β = 1.5. Oscillating dissipative structures for (c) α = 0.75, β = 1.5, A = −0.12, l2 = 0.05, L2 = 1.0, τ αu = 0.55, τ βv = 1, B = 1.1 and (d) α = 0.7, β = 1.4, A = −0.1, l2 = 0.05, L2 = 1.0, τ αu = 0.55, τ βv = 1, B = 1.1.
The obtained scenario of pattern formation is typical
for the general case α > β . We can conclude that at
certain parameters the solutions may have a simple form
of homogeneous oscillations or stationary non-homogeneous
structures, and can correspond to spatio-temporal structures
similar to those presented in figures 5–7. In addition to
homogeneous oscillations or stationary structure formation
inherent to the standard system with integer derivatives, the
system considered here with indices α > β possesses more
complicated nonlinear dynamics.
5.3. The case α < β(0 < α, β < 2)
The typical scenario of the structure formation in the case
2α = β is not so diverse as for the case considered above,
and at a wide limit of system parameters we have either
homogeneous oscillation or stationary dissipative structures.
Nevertheless, even homogeneous oscillation can have a
variety of forms depending on the relationship between α
and β (figure 8(a)–(d)). On the other hand, at the parameters
close to realization of the Turing bifurcation we can obtain
the oscillatory non-homogeneous structures period the shape
of which depends on derivative indices.
For α β the diversity of the structure formation
increases. This is due to the fact that the Turing and theHopf bifurcations have independent parameters for their
realization. The Turing bifurcation depends on the ratio of
the characteristic lengths and is connected with the instability
domain |u| < 1. The Hopf bifurcation is not connected with
the domain |u| < 1 and is realized at a wide spectrum of
parameters α < β . This makes it possible to find the scenario
of a complicated pattern formation due to the interplay
between these two types of instabilities.
References
[1] Zaslavsky G M 2002 Chaos, fractional kinetics, andanomalous transport Phys. Rep. 371 461–580
[2] Zaslavsky G M 2005 Hamiltonian Chaos and Fractional Dynamics (Oxford: Oxford University Press)
[3] Korabel N, Zaslavsky G M and Tarasov V E 2007 Coupledoscillators with power-law interaction and their fractionaldynamics analogues Commun. Nonlinear. Sci. Numer. Simul.12 1405–17
[4] Metzler R and Klafter J 2000 The random walk’s guide toanomalous diffusion: a fractional dynamics approach Phys. Rep. 339 1–77
[5] Uchaikin V V 2003 Self-similar anomalous diffusion andLevy-stable laws Phys.—Usp. 46 821–49
[6] Hornung G, Berkowitz B and Barkai N 2005 Morphogengradient formation in a complex environment: an anomalousdiffusion model Phys. Rev. E 72 041916
[7] Ray S S 2007 Exact solutions for time-fractionaldiffusion-wave equations by decomposition method Phys.Scr. 46 53–61
[8] Chen W 2006 Time-space fabric underlying anomalousdiffusion Chaos Solitons Fractals 28 923–9
[9] Povstenko Y Z 2008 Fundamental solutions tothree-dimensional diffusion-wave equation and associateddiffusive stresses Chaos Solitons Fractals 36 961–72
6
7/28/2019 physscr9_T136_014027
http://slidepdf.com/reader/full/physscr9t136014027 7/7
Phys. Scr. T136 (2009) 014027 B Y Datsko and V V Gafiychuk
[10] Mainardi F and Gorenflo R 2007 Time-fractional derivativesin relaxation processes: a tutorial survey Fract. Calc. Appl. Anal. 10 269–308
[11] Valdes-Parada J F, Ochoa-Tapia J A and Alvarez-Ramirez J2006 Effective medium equation for fractional Cattaneo’sdiffusion and heterogeneous reaction in disorder porousmedia Physica A 369 318–28
[12] Iomin A 2006 Toy model of fractional transport of cancer celldue to self-entrapping Phys. Rev. E 73 061918
[13] del-Castillo-Negrete D, Carreras B A and Lynch V E 2003Front dynamics in reaction–diffusion systems with Levyflights: a fractional diffusion approach Phys. Rev. Lett.91 018302
[14] Amblard F, Maggs A C, Yurke B, Pargellis A N and Leibler S1996 Subdiffusion and anomalous local viscoelasticity inacting networks Phys. Rev. Lett. 77 4470–3
[15] Sokolov I M, Klafter J and Blumen A 2000 Ballistic versusdiffusive pair dispersion in the Richardson regime Phys. Rev. E 61 2717–22
[16] Uchaikin V V and Sibatov R T 2008 Fractional theory fortransport in disorder semiconductors Commun. Nonlinear Sci. Numer. Simul. 13 715–27
[17] Magin L M, Abdullah O, Baleanu D and Zhou X J 2008Anomalous diffusion expressed through fractional orderdifferential operators in the Bloch–Torrey equation J. Magn. Reson. 190 255–70
[18] Hornung G, Berkowitz B and Barkai N 2005 Morphogengradient formation in a complex environment: an anomalousdiffusion model Phys. Rev. E 72 041916
[19] Golovin A A, Matkowsky B J and Volpert V A 2008 Turingpattern formation in the Brusselator model withsuperdiffusion SIAM J. Appl. Math. 69 251–72
[20] Langlands T A M, Henry B I and Wearne S L 2007 Turingpattern formation with fractional diffusion and fractionalreactions J. Phys.: Condens. Matter 19 065115–35
[21] Langlands T A M, Henry B I and Wearne S L 2008Anomalous subdiffusion with multispecies linear reaction
dynamics Phys. Rev. E 77 021111
[22] Henry B I, Langlands T A M and Wearne S L 2005 Turingpattern formation in fractional activator–inhibitor systemsPhys. Rev. E 72 026101
[23] Gafiychuk V and Datsko B 2006 Pattern formation in afractional reaction–diffusion system PhysicaA 365 300–6
[24] Gafiychuk V and Datsko B 2007 Stability analysis andoscillatory structures in time-fractional reaction–diffusionsystems Phys. Rev. E 75 055201
[25] Gafiychuk V and Datsko B 2008 Inhomogeneous oscillatorystructures in fractional reaction–diffusion Phys. Lett. A 372619–22
[26] Nicolis G and Prigogine I 1977 Self-Organization in Non-Equilibrium Systems (New York: Wiley)
[27] Cross M C and Hohenberg P C 1993 Pattern formation out of equilibrium Rev. Mod. Phys. 65 851–1112
[28] Kerner B S and Osipov V V 1994 Autosolitons (Dordrecht:Kluwer)
[29] Podlubny I 1999 Fractional Differential Equations (New York:Academic)
[30] Samko S G, Kilbas A A and Marichev O I 1993 Fractional Integrals and Derivatives: Theory and Applications(Newark, NJ: Gordon and Breach)
[31] Diethelm K, Ford N J, Freed A D and Luchko Y 2005Algorithms for the fractional calculus: a selection of numerical methods Comput. Methods Appl. Mech. Eng.194 742–73
[32] Gafiychuk V and Datsko B 2008 Inhomogeneous oscillatorysolutions in fractional reaction–diffusion systems andtheir computer modeling Appl. Math. Comput. 198251–60
[33] Gafiychuk V, Datsko B, Meleshko V and Blackmore D 2008Analysis of the solutions of coupled nonlinear fractionalreaction–diffusion equatios Chaos Solitons Fractalsdoi:10.1016/j.chaos.2008.04.039
[34] Matignon D 1996 Computational Engineering in Systems and Application Multiconference vol 2 (Lille, France: IMACS,
IEEE-SMC) p 63
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