This paper first shows that this geometric distribution is only a particular distribution case and that many other distributions (an infinity) are in fact possible. From the networks obtained, a class of partial differential equations (heat equation with a spatially variable coefficient) is then deduced. This class of equations is thus another tool for power law type long memory behaviour modelling, that solves the drawback inherent in fractional heat equations that was proposed to model anomalous diffusion phenomena.
Trang 1Beyond the particular case of circuits with geometrically distributed
components for approximation of fractional order models: Application to
a new class of model for power law type long memory behaviour
modelling
Jocelyn Sabatier
IMS Laboratory, Bordeaux University, UMR CNRS 5218, 351 Cours de la liberation, 33400 Talence, France
g r a p h i c a l a b s t r a c t
Power law type behaviour ofuT 0;sðð0;sÞÞ
a r t i c l e i n f o
Article history:
Received 20 February 2020
Revised 1 April 2020
Accepted 4 April 2020
Available online 23 April 2020
Keywords:
Power law type long memory behaviours
Fractional models
Cauer networks
Foster networks
Heat equation
Poles and zeros geometric distributions
a b s t r a c t
In the literature, fractional models are commonly approximated by transfer functions with a geometric distribution of poles and zeros, or equivalently, using electrical Foster or Cauer type networks with com-ponents whose values also meet geometric distributions This paper first shows that this geometric dis-tribution is only a particular disdis-tribution case and that many other disdis-tributions (an infinity) are in fact possible From the networks obtained, a class of partial differential equations (heat equation with a spa-tially variable coefficient) is then deduced This class of equations is thus another tool for power law type long memory behaviour modelling, that solves the drawback inherent in fractional heat equations that was proposed to model anomalous diffusion phenomena
Ó 2020 The Authors Published by Elsevier B.V on behalf of Cairo University This is an open access article
under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Introduction
It is well known that the diffusion equation of the form
@/ x; tð Þ
@t ¼ Df@2
/ðx; tÞ
https://doi.org/10.1016/j.jare.2020.04.004
2090-1232/Ó 2020 The Authors Published by Elsevier B.V on behalf of Cairo University.
Peer review under responsibility of Cairo University.
E-mail address: Jocelyn.sabatier@u-bordeaux.fr
Contents lists available atScienceDirect
Journal of Advanced Research
j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / j a r e
Trang 2produces power law type long memory behaviours of order 0.5
(Df is a diffusion coefficient) That is why the Warburg impedance,
defined in the frequency domain (variablex) by Z jð Þ ¼ jx ð Þx1=2,
was introduced to model numerous diffusion-controlled processes
in many domains such as electrochemistry [44,23,42,36,3],
solid-state electronics and ionics [15,41,2] However, it is also
well-known that there are processes whose behaviour cannot be
modelled by the Warburg impedance as they exhibit a power law
type behaviour of the form
at least in a limited frequency range These behaviours are
con-nected to anomalous diffusion-based processes[10] To model this
kind of behaviour, ‘‘fractionalizations” of the diffusion equation
were introduced[4] These fractional (in time) diffusion equations
are defined by
@m
/ðx; tÞ
@tm ¼ Df@2
/ðx; tÞ
However, this class of equations has similar drawbacks to those
recently highlighted for fractional differential equations [34,35]
and for the resulting fractional models such as pseudo state space
descriptions[28] First, in relation(3), the fractional differentiation
operator@m
@tmis not defined uniquely More than 30 definitions were
listed in[9] Results presented in the literature are obtained by
choosing the most convenient definition to obtain them In most
cases, it is the Caputo definition that is chosen, as it can take into
account the initial conditions without taking into account all the
past of the system However it was demonstrated in[25,30]that
all the system past (t! 1Þmust be taken into account to ensure
a proper initialisation, thus leading to difficulties in the definition
of this past This reflects the fact that the fractional differentiation
operator @m
@tm (but also the fractional integration operator) has an
infinite memory, and exhibits infinitely slow and infinitely fast
time constants Such a situation excludes the possibility of linking
the model internal variables to physical variables, even if fractional
models remain accurate fitting models Along the same lines, there
is no proof for the physical meaning of parameter units associated
to relation(3)(parameter Df) To conclude this list of drawbacks,
fractional differentiation and fractional integration operators are
defined using singular kernels[32,34], thus leading to difficulties
in the solution/simulation of the fractional diffusion equation(3)
For all these reasons, alternative models must be found to
model anomalous diffusion processes that exhibit power law type
long memory behaviours, as was done recently for fractional
differ-ential equations[35,37] To reach this goal, this paper first reminds
several solution that can be found in the literature for the
approx-imation of fractional integration operator The approxapprox-imation
based on a transfer function whose poles and zeros (frequency
modes) are geometrically distributed is used in the sequel This
approximation method is efficient but two limitations must be
mentioned:
– the resulting transfer function behaves exactly as a limited
fre-quency band fractional integration operator with an infinite
number of poles and zeros[20]but becomes sub-optimal for a
limited number of poles and zeros[31],
– the geometric distribution of poles and zeros is a particular
case, as an infinite number of distributions is possible
The latter limitation is demonstrated in this paper on the
inte-gral form of the impulse response of a fractional integrator This
integral form also permits a direct approximation with a Foster
network with resistors and capacitors whose values are linked
with defined functions
For a geometric distribution for the resistor and capacitor val-ues, it is also demonstrated in the paper that a Cauer network impedance also exhibits a power law type behaviour From this geometric distribution, it is then shown that an infinity of other distributions is permitted to produce a power law type behaviour
in the case of a Cauer network, reducing the geometric distribution
to a particular case
As a Cauer network can be viewed as the discrete form of a dif-fusion equation, the last part of the paper deduces difdif-fusion equa-tions with spatially variable coefficients that enable power law type behaviours to be produced It is argued that this class of equa-tions is a possible alternative to fractional diffusion equaequa-tions for anomalous diffusion process modelling
The main contributions of this paper are found in Sections
‘Beyond geometric distribution, Extension to Cauer type net-works, Heat equation with spatially variable coefficients for power law type long memory behaviour modelling and Discus-sions around some other distributions for further’ Section ‘Prior art on the approximation of fractional order integrators and the resulting electrical networks’ is dedicated to reminders on the approximation of a fractional order integrator and to the resulting electrical networks, results which will be used in the sequel of the paper As a first contribution, Section ‘Beyond geometric distribu-tion’ shows that a geometric distribution of poles and zeros for the approximation of a fractional order integrator is only a particular distribution and that an infinity of other distributions are permit-ted The analytic way to obtain these new distributions had never been presented before in the literature Due to the close link between the approximations obtained and Foster type networks, Section ‘Beyond geometric distribution’ also demonstrates as a new contribution that an infinity of Foster type networks can be used to produce power law type behaviours Section ‘Extension
to Cauer type networks’ is dedicated to Cauer type networks It
is well known that a geometric distribution of the parameters in these networks produces power law type behaviours But the pre-sent work proposes an analytical proof never published and uses it
to produce other distributions (an infinity exist) of parameters that also lead to the same kind of behaviour As a Cauer type network can be obtained through the discretisation of a heat equation, Sec-tion ‘Heat equaSec-tion with spatially variable coefficients for power law type long memory behaviour modelling’ proposes a class of heat equation with spatially varying coefficients that pro-duce power law type behaviours This is another contribution of the paper that can be viewed as an alternative solution to frac-tional diffusion equation for anomalous diffusion modelling, with-out the limitations and drawbacks associated to fractional differentiation The paper ends with a discussion and propositions for finding other spatially varying coefficients for the heat equation leading to power law type long memory behaviours
Prior art on the approximation of fractional order integrators and the resulting electrical networks
Fractional models and consequently fractional order integrator are infinite dimensional, thus their simulations or their implemen-tations require their approximations Many methods were pro-posed in the literature to obtain approximate models and many have overlaps so that it is not easy to categorize them 28 methods are analysed in [39] Some of them are implemented in digital tools, a comparison of which is proposed in[14] Note also that dis-cussions about power law in electrical circuits and some power-law relations in Laplace transforms can be found in[38,40,9]
In most of these methods the fractional model is replaced by a classical integer model under various forms: continuous time model, discrete time model, electrical network (it is often simple
Trang 3to go from one form to another) Some apply to the full fractional
model such as frequency domain fitting based methods[1] But,
as a fractional integrator chain appears in a fractional model, a
large part of the proposed methods concentrates on the
approxi-mation of the fractional integrator of transfer function s m Among
all these methods, the following are the most common
– power series expansion (PSE) techniques based on Taylor series,
Maclaurin series, etc.[40,21,6],
– continued fractional expansion based methods[43],
– impulse response based method[29],
– time moments based approaches[12],
– Carlson method based approaches[7],
– optimisation based methods[5],
– frequency distribution mode approach[16,11,19,24,8,45]
In this last class of method, a widely used is the one known in
the literature as the Oustaloup method[20] although a similar
approach was proposed by Manabe[16] This method, based on a
geometric distribution of mode is widespread because it comes
in the form of a simple algorithm, given below
Algorithm 1 In the frequency band½xl;xh, the limited frequency
band fractional integrator of transfer function ImLbð Þ defined ins (4)
can be approximated by the transfer function ImNð Þs
ImLbð Þ ¼ Cs 0
1þ s
x h
1þ s
x l
!m
Im
Nð Þ ¼ Cs 00
QN k¼1 1þ s
x0k
QN k¼1 1þ s
x k
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xh
2
r
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x l
2
r
0
B
B
1 C C
m
ð4Þ
As shown in[20], the corner frequenciesxk and x0
k (respec-tively the poles and zeros of the transfer function ImNð Þ) are geo-s
metrically distributed to obtain the required frequency behaviour:
xl
x1¼g1 =2xl x0
1¼ax1 x0
kþ1¼ rx0
k xkþ1¼ rxk k
Remark 1 As shown in[20] this algorithm is exact as N tends
towards infinity:
ImLbð Þ ¼ Cs 0lim
N!1
QN
k¼1 1þ s
x0k
QN
k¼1 1þ s
xk
but becomes sub-optimal[31]with a finite number of corner
fre-quencies xk and x0
k In this case, the sub optimality relates to the absolute and relative error between ImLbð Þ and Is m
Nð Þ for a givens N
Using fraction expansion, approximation (4)can be rewritten
as:
ImNð Þ ¼s XN
k¼1
Rk
1þ s
xk
ð8Þ
If it is assumed that the transfer function ImNð Þ links an inputs
current I sð Þ to an output voltage U sð Þ, then, from relation(8)the
following relation holds:
U sð Þ ¼XN k¼1
Ukð Þ with Us kð Þ ¼s Rk
1þ s x k
The transfer function ImNð Þ can thus be represented by the elec-s trical network of Fig 1 by introducing parameters Ck such that
RkCk¼xk Corner frequenciesxkare linked by the ratio r so that
xkþ1¼ rxk For large values of N,Fig 2shows that the following relations also hold:
and the transfer function ImNð Þ exhibits a power law behaviour.s This geometric distribution of corner frequencies or of compo-nents in the electrical network ofFig 1(electrical networks with resistors and inductors are also possible), now admitted by all, is however a particular case among an infinity of other possible dis-tributions Other distributions are presented in the next section: some can improve the optimality problem mentioned in remark
1[31], and can be applied to more complex transfer functions such
as the one given by relation(4)(seeAppendix A.1)
– Beyond geometric distribution
Using the Cauchy method, the impulse response h tð Þ of a frac-tional model of transfer function H sð Þ can be written under the form
h tð Þ ¼ L1fHðsÞg ¼
0
As an example, consider the transfer function
H sð Þ ¼1
It can be shown that[22]
h tð Þ ¼
0
lð Þex txdx with lð Þ ¼x sinðmpÞ
and thus
H sð Þ ¼
0
lð Þx
Remark 2 If h tð Þ is the impulse response of a model whose input
is u tð Þ, the convolution product of relation(11)with an input u tð Þ means that the model output can be written as:
y tð Þ ¼
0
lð Þw t; xx ð Þdx; _w t; xð Þ ¼ xw t; xð Þ þ u tð Þ ð15Þ
which is in fact the diffusive representation introduced by[18] and [17]
From the discretization of integral(14), it is easy to deduce an electrical network whose transfer function is an approximation
of H sð Þ on a given frequency range x½ min; xmax Using the Euler approximation method (but many other methods of higher order can be used), integral(14)can be approximated as follows:
U k (s)
R 0
C 0
R k
C k
R N
C N
Fig 1 Electrical network (Foster type) whose impedance is I m ð Þ s
Trang 4H sð Þ ¼
Z1
0
lð Þx
sþ xdx Hað Þ ¼s XN
k¼0
lð Þxk Dx
sþ xk ; with x0¼ xmin
ð16Þ
with
x0¼ xmin Dx¼xmax xmin
Applied to transfer function(12), the following approximation
can be obtained
H sð Þ Hað Þ ¼s sinðmpÞ
p
XN k¼0
xm
k
For m¼ 0:4, on the frequency range ½xl;xh with
xl¼ 0:001 rd=s and xh¼ 1000 rd=s the approximation Hað Þs
Bode diagrams are shown in Fig 3 with several values of N,
(N¼ 105
,N¼ 5 105
,N¼ 106
), showing that a large number N is required for the power law type behaviour appears
As relation(15)can be rewritten as
Hað Þ ¼s XN k¼0
sin ðmpÞ
p xm1 k s
it permits a realization using an RC network like the one inFig 1 with
Rk¼sinðmpÞ
m1
k Dx; Ck¼sinðmpÞ 1
p xm1
k Dx and
xk¼ 1
RkCk
as the circuit impedance inFig 1is
G sð Þ ¼XN k¼0
Rk
Note that an RL (resistor-self) circuit can also be designed Also note that this approximation method can be applied to many other transfer functions of which a non-exhaustive list is given in Appen-dix A.1,Table A1 For instance, the impulse response of transfer function(4)is defined by (seeAppendix A.1)
Fig 2 Logarithm of ratiosRkþ1
R k and Ckþ1
C k in relation (10) forxl ¼ 1,xh ¼ 10 6
;m¼ 0:3 witha¼ 1:0208,g¼ 1:0493and N = 200 (a) ora¼ 1:0021,g¼ 1:0048 and N = 2000 (b).
ð Þ (given by relation
Trang 5L1 Imað Þ ¼ Cs 0
s
xhþ 1
s
x lþ 1
8
>
>
9
>
>
¼ C0
xl
xh
dðtÞ þsinðmpÞ
p
Z xh
x b
ðxh xÞm
ðx xlÞmðs þ xÞdx
0
B
1
where dðtÞ is the Dirac impulse function According to the previous
comments,
ImLbð Þ as 0þXN
k¼0
ak s
with
a0¼ C0ðx l
x hÞm; ak¼ C0ðx l
x hÞmsinðmpÞ
p
ð x h x k Þ m
ð xk xlÞ m xkDx;
xk¼xhþ kDx; Dx¼x h x b
Fraction expansion of ImNð Þ in relations (4)can also be written as
ImLbð Þ ¼ as 0
0þXN
k¼0
a0 k s
A comparison of coefficients ak and a0kis given by Fig 4 It
reveals that, for a large value of N, ak a0
kand that the two approx-imations are very close
If the transfer function H sð Þ of relation(12)is considered again
(for simplicity but a similar analysis can be done with the transfer
functions ofTable A1), relation(20)highlights that the capacitors
and resistors are linked by the recurrence relations:
Rkþ1
Rk
¼xkþ1m1
xm1
k
Ckþ1
Ck
¼xkm1
xm1 kþ1
ð26Þ
It can be noticed, that unlike relation(10), the ratios linking two
resistors or two capacitors are not constant and depend on k This
discretization can be viewed as an alternative solution to algorithm
1, but as parameter N must be very large to have an accurate
approximation on a large frequency band, it requires a very large
number of components in the network ofFig 1 Such a defect is
due to the fixed step discretization of integral(11) To overcome
this defect, it is possible to search for a change of variable that
con-tracts the frequency domain, thus making the fixed step
discretiza-tion more efficient
As a first try, the following change of variable is used in relation (14):
x¼ az¼ ezln a ð Þ with a2 R
þ thus dx¼ ln að Þezln a ð Þdz: ð27Þ
H sð Þ can be rewritten as:
H sð Þ ¼
1
lezln a ð Þ
sþ ezln a ð Þln að Þezln a ð Þdz¼
1
ln að Þlezln að Þ
s
e zln a ð Þþ 1dz: ð28Þ
This transfer function can be approximated by:
Hað Þ ¼s XN k¼0
ln að Þlezkln að Þ
s
with
z0¼ln xð minÞ
ln að Þ Dx¼
ln x ð max Þ
ln a ð Þ ln x ðminÞ
ln a ð Þ
N zk¼ln xð minÞ
Such a discretisation permits the realization ofFig 1with:
Rk¼sinðmpÞ
p ln að Þemz k ln a ð ÞDz Ck¼sinðmpÞ 1
p ln að Þeð mþ1 Þz k ln a ð ÞDz ð31Þ
and
xk¼ 1
RkCk
If¼ 0:4; a ¼ 10, N ¼ 10, xl¼ 0:001 rd=s, xh¼ 1000 rd=s, the Bode diagrams of the approximation Hað Þ in relations (29) with change of variable(27)are shown inFig 5 They are very similar
to those of Fig 3obtained with relation (18)and N¼ 106, thus showing the interest of the change of variable (27) in reducing the size of the approximation
Remark 3 Whatever the value of a, and as:
zkþ1 zk¼ln xð minÞ
ln að Þ þ k þ 1ð ÞDzln xð minÞ
It can be noticed that
Rkþ1
Rk
¼emzkþ1ln að Þ
emzkln a ð Þ ¼ emDzln a ð Þ Ckþ1
Ck
¼1eðmþ1Þzkþ1ln að Þ
eð mþ1 Þz k ln a ð Þ ¼ eð mþ1 Þln a ð Þ
ð34Þ
And
xkþ1
xk
RkCk
The previous relation highlights a geometric distribution of the values of resistors, capacitors and corner frequencies, defined by the following ratios:
a¼ emDzln a ð Þ g¼ eð mþ1 Þln a ð Þ: ð36Þ
This geometric distribution generalises the one introduced by Oustaloup[19,20] The latter is indeed a particular case obtained with a¼ 10, among the infinite number of distributions obtained for all the other values of a, and for other changes of variable that can be proposed instead of relation(27) Among this infinity, the following one is interesting as it also makes it possible to contract the frequency domain
Using the following change of variable
relation(14)can be rewritten as:
H sð Þ ¼sinðmpÞ
p
0
zmn
sþ znnzn1dz¼sinðmpÞ
p
0
nz
mn1 s
z nþ 1dz ð38Þ
Fig 4 Comparison of coefficients a k and a 0
k , with N ¼2000 and m¼ 0:3,
x¼ 1 rd=s,x ¼ 10 6
rd=s (zoom inside the figure).
Trang 6and permits the network ofFig 1with:
Rk¼sinðmpÞ
p nz
mn1
k Dz; Ck¼sinðmpÞ 1
p nzmn1þn
and
xk¼ 1
RkCk
¼ zn
If m¼ 0:4; n ¼ 60, N ¼ 10, xl¼ 0:001 rd=s, xh¼ 1000 rd=s is
the Bode diagrams of the approximation Hað Þ obtained by discreti-s
sation of integral(38)and change of variable(37) are shown in
Fig 6 They are compared with the Bode diagrams obtained with
change of variable (27) The comparison reveals that the two
changes of variable are of equivalent quality with the same
com-plexity (N¼ 10)
As an infinity of changes of variable can be proposed, an infinity
of Foster type networks can be used to generate a power law
beha-viour The following section shows that Cauer networks can also
generate this type of behaviour with an infinity of different
distributions
Extension to Cauer type networks The Cauer network ofFig 7is considered
For the geometric distribution, such as the one defined by rela-tions(5) and (6), an analytical result can be obtained to show that a Cauer type network generates a power law behaviour Considering the network inFig 7, the following relations hold
1
sCk
Ik1ð Þ Is kð Þs
and
From relations(41) and (42)respectively, it can be written that
Ukð Þs
Uk1ð Þs ¼
1
sC k
1þ 1
sC k
Ikð Þ s
U k ð Þ s
ð43Þ
and
Fig 5 Bode diagram of H a ð Þ with change of variable s (27) and comparison with the Bode diagrams of approximation (18).
ð Þ with the change of variable
Trang 7Ikð Þs
Ukð Þs ¼
1
k
1þ1
k
Uiþ1ð Þ s
I i ð Þ s
Combining relations (43) and (44), it can be shown that the
input admittance of the network inFig 7is defined by the
contin-ued fraction:
H sð Þ ¼ I0ð Þs
U0ð Þs ¼
1 1
1þ sC1R11
1þ 1 sC1R2 1þ 1 sC2R2 1þ
Suppose that inFig 7, the resistors and capacitors are
geomet-rically distributed and linked by the following ratios (as in[19]):
Rkþ1
Rk
Ck
and if Z sð Þ ¼ 1
sC 1 R 1relation(45)becomes
H sð Þ ¼ I0ð Þs
U0ð Þs ¼
1 1
1þ Z s ð Þ= r
1þ Z sð Þ=rq 1þ Z sð Þ=r2q 1þ Z s ð Þ= r 2 q 2
1þ
Z s ð Þ= r N q N 1þZ s ð Þ= r Nþ1 q N
Introducing the function
g Z sð ð Þ;r;qÞ ¼ Z sð Þ
1þ Z s ð Þ=r
1þ Z s ð Þ= rq
1þ Z sð Þ=r2q 1þ Z s ð Þ= r 2 q 2
1þ
Z s ð Þ= r N q N 1þZ s ð Þ= r Nþ1 q N
ð48Þ
relation(48)becomes
H sð Þ ¼ I0ð Þs
U0ð Þs ¼
1 1
If N tends towards infinity, function g Z sð ð Þ;r;qÞ meets the
fol-lowing property:
Property 1 Function g Z sð ð Þ;r;qÞ meets the following relation
g Z sð ð Þ;r;qÞ ¼ Z s ð Þ
1þg Z s ð ð Þ;q;rÞ
Thanks to property 1, the function g Z sð ð Þ;r;qÞ can be written
under the form of a rational function with descending powers
Theorem 1 With N! 1
g Z sð ð Þ;r;qÞ ¼ Kðr;qÞZ sð Þm
1þP1
k¼1C2k1ðr;qÞ Z s ð Þ
r
mkþ1
þ C2kðr;qÞ Z s ð Þ
r
k
1þP1
k¼1C2k1ðq;rÞ Z sð ð ÞÞmkþ1þ C2kðq;rÞ Z sð ð ÞÞk
2
64
3
Using theorem 1, demonstrated inAppendix A.2, for Z sj ð Þj 1,
(asr> 1), admittance H sð Þ meets the relation
H sð Þ
1
1
If the network results from an infinitesimal slicing of a contin-uous medium of abscissa z, the ratio of two consecutive compo-nents (capacitor or resistor) denoted Fis given by:
Fkþ1
Fk
¼F kdzð þ dzÞdz
where dz denotes the thickness of the considered slices, with
dz! 0
Given that
F kdzð þ dzÞ F kdzð Þ
where F0ð Þ denotes the derivative of F zz ð Þ and thus
the ratio of relation(52)becomes
Fkþ1
Fk
¼ 1 þF0ðkdzÞ
If this ratio, only a function of dz, is assumed constant8k as in relation(46)and equal toK, using relation(55),
K¼ 1 þF
0
kdz
with
F0ðkdzÞ
After resolution of the differential equation(57), function F kdzð Þ
is given by
This shows that the lineic characteristics of the discretized medium that produces the network ofFig 7are defined by:
R zð Þ ¼ R0ek R z and C zð Þ ¼ C0ek C z z2 0; 1½ ½: ð59Þ
The ratio of two consecutive resistors and capacitors is thus defined by:
Rkþ1
Rk
¼R kðð þ 1ÞdxÞ
R kdxð Þ ¼ ekRDz and
Ckþ1
Ck
¼C kðð þ 1ÞdxÞ
C kdxð Þ ¼ ekCDz:
ð60Þ
Now consider the change of variable z¼ log xð Þ, x 2 1; 1½ ½, then relation(59)becomes
R xð Þ ¼ R0xk R and C xð Þ ¼ C0xk C x2 1; 1½ ½ ð61Þ
With an infinitesimal slicing of the continuous medium, the system can be characterised by the network ofFig 7with:
Rk¼ R kdxð Þ ¼ R0ðkdxÞk Rdx and Ck¼ C kdxð Þ ¼ C0ðkdxÞk Cdx
ð62Þ
The ratio of two consecutive resistors and capacitors is thus defined by:
Rkþ1
Rk
¼R0ððkþ 1ÞdxÞk Rdx
R0ðkdxÞk Rdx ¼ðkþ 1Þ
k R
k
Ckþ1
Ck
¼C0ððkþ 1ÞdxÞk Cdx
C0ðkdxÞk Cdx ¼ðkþ 1Þ
k C
k
These ratios are similar to those given by relation(26)for the Foster circuit ofFig 1
The following change of variable z¼ log xð Þ, x 2 1; 1n ½ ½, n 2 N
þ
is now considered Relation(62)thus becomes
Fig 7 Cauer type RC network.
Trang 8R xð Þ ¼ R0xnk R and C xð Þ ¼ C0xnk C x2 1; 1½ ½: ð64Þ
With an infinitesimal slicing of the continuous medium, the
system can be characterised by the network ofFig 7with:
Rk¼ R kdxð Þ ¼ R0ðkdxÞnkR
dx and Ck¼ C kdxð Þ ¼ C0ðkdxÞnkC
dx:
ð65Þ
The ratio of two consecutive resistors and capacitors is thus
defined by:
Rkþ1
Rk
¼R0ððkþ 1ÞdxÞnk Rdx
R0ðkdxÞnkR
dx ¼ðkþ 1Þ
nk R
k
Ckþ1
Ck
¼C0ððkþ 1ÞdxÞnk Cdx
C0ðkdxÞnk Cdx ¼ðkþ 1Þ
nk C
k
These ratios are similar to those given by relation(39)for the
Foster circuit ofFig 1
These networks and the associated distributions are used in the
next section to introduce a class of heat equation that exhibits a
power law type long memory behaviour
Heat equation with spatially variable coefficients for power law
type long memory behaviour modelling
The following heat equation with spatially dependent
parame-ters is now considered
@T z; tð Þ
@t ¼cð Þ @z @z bð Þ @z
T z; tð Þ
@z
ð67Þ
with z2 Rþ
This equation is a simplified form of the equation studied in
[13] Let
uðz; tÞ ¼ b zð Þ @T z; tð Þ
Discretisation of equation(68)with a discretisation stepDzleads
to:
uðz; tÞ ¼ b zð ÞT zð þ dz; tÞ T z; tð Þ
and thus:
T z; tð Þ T z þ dz; tð Þ ¼ Dz
Using relation(69), relation(67)can be rewritten as:
@T z; tð Þ
@t ¼cð Þ @z
uðz; tÞ
Spatial discretisation of Eq.(71)with a discretisation step Dz
leads to:
@T z; tð Þ
@t ¼cð Þz
uðzþ dz; tÞ uðz; tÞ
Dz
¼cð Þz
For z¼ kDz and if the following notations are introduced
Ck¼ Dz
cðkDzÞ¼ C kð DzÞDz and Rk¼ Dz
bðkDzÞ¼ R kð DzÞDz ð73Þ
discretisation of Eq.(67)thus leads to the Cauer network ofFig 8
As Ck¼ C kð DzÞDz and Rk¼ R kð DzÞDz, according to relations
(46),(62) and (72), the transfer functionuð0; sÞ=T 0; sð Þ of the Cauer
network ofFig 8exhibits a power law type long memory
beha-viour if
Rkþ1
Rk
¼ ek R Dz and Ckþ1
Ck
Rkþ1
Rk
¼ðkþ 1Þ
k R
k
Ck
¼ðkþ 1Þ
k C
k
Rkþ1
Rk
¼ðkþ 1Þ
nk R
k
Ck
¼ðkþ 1Þ
nk C
k
and according to the relations(59),(60) and (63), the heat equation (67) exhibits a power law type long memory behaviour if (as
cð Þ ¼ 1=C zz ð Þ and b zð Þ ¼ 1=R zð Þ according to relation(73))
cð Þ ¼ z 1
C0ek C z and b zð Þ ¼ 1
R0ek R z z
cð Þ ¼ z 1
C0zk C and bð Þ ¼ z 1
R0zk R z
cð Þ ¼ z 1
C0znkC and b zð Þ ¼ 1
R0znkR z
Of course, as previously explained, many other spatially varying coefficients can be obtained using other changes of variable than those proposed at the end of Section ‘Extension to Cauer type networks’
Discussions around some other distributions for further Now, among the infinity of distributions that can be obtained using changes of variable as shown in Section ‘Beyond geometric distribution’, the following is studied:
z¼ xm; or x ¼ z1=m thus dx¼ 1
mz
1
Using this change of variable, relation(14)becomes:
H sð Þ ¼sinðmpÞ
p
Z 0 þ1
z
sþ z 1
mz
1
m 1
dz
¼sinðmpÞ p
0
1 m
z 1
m
sþ z 1
or after simplification
H sð Þ ¼sinðmpÞ
mp
0
1
s
and permits the realization ofFig 1with:
Hað Þ ¼s PN
k¼0 R k CRkksþ1 Rk¼sin ðmpÞ
mp Dz¼ Cte
Ck¼ mpz
1
m
k sin ð mp Þ
k C k¼ z1m
If¼ 0:4; N ¼ 10; 000,xl¼ 0:001 rd=s,xh¼ 1000 rd=s the Bode diagrams of the approximation Hað Þ are shown ins Fig 9 They are compared with the Bode diagrams of approximation(18)and the one obtained with change of variable(37) As for approximation (18), parameterNmust be very large to have an accurate approxi-mation of s mon a large frequency band, but the interest of this change of variable is not there
The distribution of resistors and capacitors of relation(83) is now used to build the Cauer network of Fig 7, with m¼ 0:4,
N = 1000,Dz¼ 2 and
Trang 9The resulting Bode diagram of the transfer function I0ð Þ=Us 0ð Þs
is represented byFig 10 This diagram shows yet again that a
power law behaviour can be obtained without a geometric
distri-bution of resistors and capacitors In this circuit, all the resistors
have the same values and the capacitors are linked by the
follow-ing relation
Ckþ1
Ck
¼ððN k 1ÞDzÞ1
m
N k
¼ðN k 1Þ
1
m
N k
This class of components distribution, that cannot be deduced using a change of variable in relation(59), and the resulting class
of spatially varying coefficients in relation(67)will be studied by the author in future work
(0,t)
T(0,t)
k
Ck
(z,t)
T(k z,t)
((k+1) z,t)
R
(k z,t)
Fig 8 Cauer type RC network resulting from the discretization of relation (67).
Fig 9 Bode diagram of H a ð Þ of relation s (83) with change of variable (80) and comparisons with the Bode diagrams of approximation (18) and the one obtained with change
of variable (37).
Fig 10 Bode diagram of transfer function I ð Þ=U s ð Þ of the Cauer type RC network with distribution of relation s (83).
Trang 10This paper shows that an infinity of
– pole and zero distributions (frequency modes) in classical
inte-ger transfer functions,
– passive component value distributions (such as capacitors or
resistors) in Foster type networks,
can generate power law type long memory behaviours Hence, the
geometric distributions[19,20]often encountered in the literature
are a particular case among an infinity of distributions
For the Foster type network the proof is easy to establish using
several changes of variables, as this network results directly from
the discretisation of a filter transfer function that exhibits a power
law behaviour The proof for the Cauer type network is more
tedious and is developed in the paper
Due to the close link between Cauer type networks and heat
equations (through discretisation), this paper also shows the
abil-ity of heat equations with a spatially variable coefficient to have a
power law type long memory behaviour This class of equation is
thus another tool for power law type long memory behaviour
mod-elling that solves the drawback inherent in fractional heat
equa-tions This class of equation will be more deeply studied by the
author
Finally, this paper shows, without proof, that other distributions
and thus heat equations with spatially variable coefficients also
exhibit power law type long memory behaviours Moreover, by
increasing the number of components in each branch of the Cauer
network, it is possible to keep a power law behaviour, which
sug-gests that there are a very large number of partial differential
equa-tions, other than the heat equaequa-tions, which can produce a power
law type long memory behaviour, some were already proposed
in[27]
With reference to other papers recently published by the author
[33,35], this work is a new contribution to the dissemination of
models not based on fractional differentiation but which exhibit
power law type long memory behaviours
Compliance with ethics requirements
This article does not contain any studies with human or animal
subjects
Declaration of Competing Interest
The author has declared no conflict of interest
Appendix A.1 Impulse response of some transfer functions that
exhibit power law type long memory behaviours
The approximations given in Section ‘Beyond geometric
distribu-tion’ are made on the integral form of the impulse response of the
transfer function HðsÞ ¼1
sm The methodology used to derive the approximations and the change of variable used in Sections
‘Beyond geometric distribution’ and ‘Extension to Cauer type
networks’ can be extended to other transfer functions The
follow-ing one is now considered:
H1ð Þ ¼ Cs 1
s
x hþ 1
s
x lþ 1
1
x l
2
þ 1
2
1
x h
2
þ 1
The impulse response of HðsÞ is defined by
h1ð Þ ¼t 1
2pj
Z cþj1
For the computation of integral(A1.2), pathC¼c0[ ::: [c7of Fig A11is considered with c> xl
This path bypasses the negative axis around the branching point
z¼ xl and z¼ xh for t> 0 It thus avoids the complex plane domain for which the transfer function H1ðsÞ is not defined, i.e the segment½xh; xl
On pathC, the radii of sub-pathc1 and c7tend towards infin-ity, and the radius of sub-pathc4tends towards 0 Using Cauchy’s theorem with c> xl:
h1ð Þ ¼t 1
2pj
Z cþj1 cj1 H1ð Þes tsds¼ 1
2pj
Z
C c 0
H1ð Þes tsds
poles
in C
Res H 1ð Þes ts
Since
operator H1ðsÞ being strictly proper, by Jordan’s lemma integrals on the large circular arcs of radius R, R? 1 can be neglected:
Z
c1þ c7
Let s¼ xej p, x21;xh on c2 and thus ds¼ ej pdx Let also
s¼ xej p, x2 ½xh; 1½ onc6and thus ds¼ ej pdx Then
Z
c2þc 6
H1ð Þes tsds¼ xlm
xhm1
Z x h
1
xejpþxh
xejpþxl
ð Þm extejpdx
þ xlm
xhm1
x h
ðxejpþxhÞm1 ðxejpþxlÞm extejpdx¼ Jc
2 þ c6ðtÞ
ðA1:6Þ
Fig A1.1 Integration path considered.