Design of evolutionary optimized finite difference based numerical computing for dust density model of nonlinear Van-der Pol Mathieu’s oscillatory systems

Abstract

In this study, a new evolutionary optimized finite difference based computing paradigm is presented for dynamical analysis of dust density model for the ensemble of electrical charges and dust particles represented with nonlinear oscillatory system based on hybridization of Van-der Pol and Mathieu equation (VDP-ME). Strength of accurate and effective discretization ability of finite difference method (FDM) is exploited to transform VDP-ME to equivalent nonlinear system of algebraic equations. The residual error based fitness function of the transformed model is constructed by the competency of approximation theory in mean square sense. The optimization of the residual error of the system through hybrid meta-heuristic computing paradigm GA-SQP; genetic algorithm (GA) for viable global search aided with rapid fine tuning of sequential quadratic programming (SQP). The proposed GA-SQP-FDM is applied on variants of dust density model of VDP-ME by varying the rate of charged dust grain production, as well as, loss and comparison of results with state of art numerical procedure established the worth of the scheme in term of accuracy and convergence measures endorsed through statistical observations on large dataset.

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Mathematics and Computers in Simulation 181 (2021) 444–470
www.elsevier.com/locate/matcom
Original articles
Design of evolutionary optimized finite difference based numerical
computing for dust density model of nonlinear Van-der Pol Mathieu’s
oscillatory systems
Ihtesham Jadoona, Muhammad Asif Zahoor Rajab,c,∗, Muhammad Junaidc,
Ashfaq Ahmeda, Ata ur Rehmanc, Muhammad Shoaibd
aDepartment of Electrical and Computer Engineering, COMSATS University Islamabad, Wah Campus, Rawalpindi, Pakistan
bFuture Technology Research Center, National Yunlin University of Science and Technology, 123 University Road, Section 3,
Douliou, Yunlin 64002, Taiwan ROC
cDepartment of Electrical and Computer Engineering, COMSATS University Islamabad, Campus Attock, Attock, Pakistan
dDepartment of Mathematics, COMSATS University Islamabad, Campus Attock, Attock, Pakistan
Received 18 February 2020; received in revised form 21 September 2020; accepted 6 October 2020
Available online 10 October 2020
Abstract
In this study, a new evolutionary optimized finite difference based computing paradigm is presented for dynamical analysis
of dust density model for the ensemble of electrical charges and dust particles represented with nonlinear oscillatory system
based on hybridization of Van-der Pol and Mathieu equation (VDP-ME). Strength of accurate and effective discretization ability
of finite difference method (FDM) is exploited to transform VDP-ME to equivalent nonlinear system of algebraic equations.
The residual error based fitness function of the transformed model is constructed by the competency of approximation theory
in mean square sense. The optimization of the residual error of the system through hybrid meta-heuristic computing paradigm
GA-SQP; genetic algorithm (GA) for viable global search aided with rapid fine tuning of sequential quadratic programming
(SQP). The proposed GA-SQP-FDM is applied on variants of dust density model of VDP-ME by varying the rate of charged
dust grain production, as well as, loss and comparison of results with state of art numerical procedure established the worth
of the scheme in terms of accuracy and convergence measures endorsed through statistical observations on large dataset.
c
⃝2020 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights
reserved.
Keywords: Evolutionary computing; Dusty plasma; Finite difference method; Sequential quadratic programming; Integrated computing
1. Introduction
The research community shows growing interest in plasma studies due to their paramount importance in broad
applications arising in space science and astronomy [10,51]. Plasma is a quasi-neutral gas in which neutral
atoms and charge particles co-exist. The electromagnetic character becomes dominant due to a large range of
∗Corresponding author at: Future Technology Research Center, National Yunlin University of Science and Technology, 123 University Road,
Section 3, Douliou, Yunlin 64002, Taiwan ROC.
E-mail addresses: jadoon@ciit-attock.edu.pk (I. Jadoon), rajamaz@yuntech.edu.tw (M.A.Z. Raja), fa15-ree-017@ciit-attock.edu.pk
(M. Junaid), ashfaqahmed@ciitwah.edu.pk (A. Ahmed), dr.ataurrehman@ciit-attock.edu.pk (A.u. Rehman), dr.shoaib@cuiatk.edu.pk
(M. Shoaib).
https://doi.org/10.1016/j.matcom.2020.10.004
0378-4754/ c
⃝2020 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights
reserved.

I. Jadoon, M.A.Z. Raja, M. Junaid et al. Mathematics and Computers in Simulation 181 (2021) 444–470
Coulomb forces [1]. Plasma has various applications in latest technological trends such as in display screen
equipment [23], surface treatment or coating [33], in laser for population inversion [43] and Lightning [14],
scattering phenomena [48], fusion and astrophysics [53]. When dust component is included in plasma it is called
complex plasma or dusty plasma. Dusty plasma consists of electrical charges and dust particles of small size [49].
Dust particles change the properties of plasma [11,15,52,58]. Continuity and momentum equations are used to
drive the Van-der Pol and Mathieu’s equation (VDP-ME) for dust particles dynamics [32,57] and these hybrid
relation explores the stable and non-stable oscillations. The analysis of charged dust density model of VDP-ME
is conducted through numerical and analytical procedures [16,20,22,24]. The reported studies for the solution of
VDP-ME are available by exploitation of deterministic solver of both numerical and analytical procedures with their
own advantages, applications and limitations. However, the stochastic solver by exploitation of artificial intelligence
techniques has not yet being explored to analyze the dynamics of VDP-ME arising in the studies of dusty plasma.
Stochastic numerical computing paradigm are broadly exploited by the research community for analysis of linear
and nonlinear problems governed with ordinary and partial differential equations [5,25,30,34,44]. A brief vista of
recent applications include solution of random matrix theory models [39], nonlinear optics models [6], electrical
conducting solid models [29], pantograph model [35], atomic physics models [46], astrophysics models [3],
induction motor models [33], plasma physics models [37,40], thermodynamics studies [4], bioinformatics [41] and
fractional order systems [38], All these illustrative applications of stochastic numerical computing solver motivate
the authors to explore and exploit in these techniques to develop an alternate, accurate, viable, stable and robust
computing paradigm for the solution of dust density dynamics of VDP-ME. The salient features of the proposed
study are brief summarized as:
•A novel design of evolutionary optimized finite difference method is presented to study the dynamics of dust
density model representing nonlinear oscillatory system based on Van der Pol Mathieu’s equation (VDP-ME).
•Finite difference scheme is exploited to transform the VDP-ME into system of nonlinear equations and
competency of approximation theory in mean squared error sense is utilized to construct the merit function of
the transformed system.
•Strength of hybrid meta-heuristic computing paradigm based on global search efficacy of GAs and rapid
convergence ability of SQP is exploited as an optimization mechanism.
•Comparison with state of art numerical results show that the design scheme is effective, reliable and accurate
for dynamical analysis of dust density model by varying the rate of charge dust grain production and loss.
•Verification and validated of performance of proposed methodology GA-SQP-FDM is ascertained through
statistical observations on dataset generated through sufficiently large number of independent trails by means
of accuracy, convergence and complexity measures.
•Coherent structure, smooth implementation, explicability, applicability and stability are other illustrative perks.
Rest of study in different section is organized as follows: An overview of system model for ensemble of electrical
charges and dust particles is presented in Section 2. Design methodology based on evolutionary finite difference
method is presented in Section 3. Performance metrics defined in Section 4. Results of simulations on single and
multiple trials are presented in Section 5. While the conclusions along with future research studies are narrated in
Section 6.
2. System model for ensemble of electrical charges and dust particles
In the present work dust charging variability is modeled via turbid behavior of charged dust in the plasma [28].
Mathieu’s equations for parametric oscillations [12] is used to hybrid with Van der Pols equation. To facsimile
dust particle dynamics Van der Pol Mathieu Equation (VDP-ME) is used [32,57]. The following work is showing
the mathematical expression and derivatives with continuous mappings in Finite Difference Method (FDMs)
methodology of the models which approximate the solution of Van der Pol and Mathieu’s system.
VDP-ME was introduced to analyze the behavior of dust grain charge with the assumptions that, when the dust
components are uncharged, the combined effect is due to gravitational interaction and dust component is surrounded
by ionized or radioactive charge, it becomes highly charged and hence gravitational effect is significantly weak.
Mass mdof dust grain is constant with varying charge with time qd(t) = −Zd(t)e. Collusion less dusty plasma
unmagnetized consisting of electron mass mewith charge −eand ions mass miwith charge +Zie.
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Density evolution equation of the cold inertial dust fluid density ndand velocity of dust particle vdis given as,
∂nd
∂t+ndo
∂vd
∂z=αnd−1
3βn3
d.(1)
where, αis production of charged dust grain, βis loss of charged dust grain due to third body recombination. For
simplicity, dust is assumed to be homogeneously distributed, i.e., ∂n/∂z=0. The three body interaction is given
as,
X++e+z→X∗+z.(2)
Dust momentum equation (Derived by using ma =q E ) is shown as,
∂vd
∂t= − qd
md
∂φ
∂z(3)
The Poisson’s equation for electric potential ϕis given as,
∂2φ
∂z2= − e
εo
(Zini−ne−Zdnd).(4)
Differentiating Eq. (1) with respect to t, we get,
∂nd
∂t+∂
∂t(ndo
∂vd
∂z)=α∂
∂tnd−βn2d
∂nd
∂t.(5)
Differentiating Eq. (3) with respect to z, we get,
∂
∂z(∂vd
∂t)= − qd
md
∂2φ
∂2z.(6)
Putting the value of ∂/∂ z(∂ vd/∂ t)from Eq. (5) in Eq. (4), in order to diminish velocity nd, we found the equation
as,
∂2nd
∂t2−(α−βn2d)∂nd
∂t=ndo
qd
md
∂2φ
∂z2.(7)
Electrons and ions are assumed to be in local thermodynamics equilibrium, so neand niobey Boltzmann distribution
ne=neo exp (eφ/ KBTe)and ni=nio exp (−zieφ /KBTi), where KBdenotes Boltzmann constant, Teand Ti
is the temperatures of electron and ion respectively. Hence, by using Boltzmann distribution and Taylor series
representation, we approximate neand nias,
ne
neo ≈1+eφ
KBTe+1
2(eφ
KBTe
)2+ · ·· ,(8)
and
ni
nio ≈1+eφ
KBTi+1
2(eφ
KBTi
)2+ · ·· .(9)
Combining Eqs. (8) and (9) into Eq. (4), Taking only first order terms into account. We get,
∂2φ
∂z2= − 4πqdK2
K2+K2
D
nd.(10)
where KDDebye wave number is written as, KD=λ−1
De f f =λ−2
Di +λ−2
De for electron and ion Debye radii are
expressed as, λDe =KBTe/4πne0e2and λDi =KBTi/4πne0z2
ie2respectively. Since dust particle is very
small, i.e., zdnd,0≪zini,0,ne,0. Hence, we can approximate zini,0−ne,0≈0, realistic fact occurring naturally
in dusty plasmas. We see that the ions and inertia-less electrons disturb the transmission of acoustic dust waves via
a dynamical charge equilibrium [42]. In the long Debye-wavelength limit λ≪λDef f , namely λ≪λDe f f , hence,
ignoring the effect of variation on electric potential in Eq. (10).
Substituting the value of ∂2φ/∂ z2from Eq. (10) into Eq. (7), we get,
d2nd
dt 2−(α−βn2
d)dn d
dt = − 4πnd oq2
d0K2
md(K2+K2
D)nd.(11)
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Considering Analytical tractability, dust particle changing in time is shown as qd=qdo √1+scos Ωt, where
frequency Ωand parameter sare constants. so Eq. (11) for the dust density becomes,
d2nd
dt 2−(α−βn2
d)dn d
dt = − 4πnd oq2
d0K2
md(K2+K2
D)nd(1 +scos Ωt).(12)
Introducing ωpd , dust particle frequency; ωpd =4πnd oq2d o /md, and ωo,oscillatory frequency; ωo=ωpd K/
K2+K2
D. By substituting values of ωpd and ωosimultaneously into Eq. (12), we get,
d2nd
dt 2=(α−βn2
d)dn d
dt −ω2
ond(1 +scos Ωt).(13)
Let nd=yand dy/d t =x, then Eq. (13) may be written as,
y′=x,
x′=(α−βy2)y′−ω2(t)y.(14)
with the associated initial conditions are,
x(0)=1 and y(0)=1.
Eq. (14) represents the dust density model based on VDP-ME, which is used as a system model in this study.
3. Design methodology
In this section, the design scheme for analysis of Van-der Pol and Mathieu equation for ensemble of electrical
charges and dust particles is presented. Firstly, discretization procedure based on finite difference scheme is
presented for the system model that convert the ODEs to nonlinear systems of algebraic equations. Secondly,
optimization scheme based on GA-SQP is exploited for the solution of the systems by minimization of residual
error in mean squared sense and finally, performance metric used in the study are presented.
3.1. Discretization of system model through finite differences
Finite differences based discretization schemes are the basic and fundamental procedure for solving the ordinary
and partial differential equations by transforming the original system in to system of nonlinear equations.
Let the approximate solution of the VDP-ME to be calculated on equally space input grids in the interval
t∈[0,T], ti=ih,i=1,2,..., Nwhere Nis total number of grid points and his the step size, i.e., h=
1/N. The differential equations of the VDM-ME based system (14) for t1≤ti≤tN−1are represented as:
˙y=Fy(xi,yi,ti),(15)
˙x=Fx(xi,yi,ti),(16)
where the functions Fyand Fxare defined, respectively as:
Fy(xi,yi,ti)=x,(17)
Fx(xi,yi,ti)=(α−βy2)˙y−ω2(ti)y,(18)
with initial condition as:
y(t0)=x(t0)=1,
while yi=y(ti)=y(ih), xi=x(ti)=x(i h) for t1≤ti≤tN. The finite difference formulae based on 7 stencil are
used to discretize the derivative in terms of forward ∆, backward ∇and central operators and in case of ˙yare
given as:
˙y=1
h
∆[y(ti−1),y(ti),...,y(ti+5)]=1
h
∆[yi−1,yi,...,yi+5],
=−10yi−1−77 yi+150yi+1−100yi+2+50yi+3−15yi+4+2yi+5
60h,
(19)
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I. Jadoon, M.A.Z. Raja, M. Junaid et al. Mathematics and Computers in Simulation 181 (2021) 444–470
˙y=1
h[y(ti−3),y(ti−2),...,y(ti+3)]=1
h[yi−3,yi−2,...,yi+3]
=−yi−3+9yi−2−45yi−1+45yi+1−9yi+2+yi+3
60h
(20)
˙y=1
h∇[y(ti−6),y(ti−5),...,y(ti)]=1
h∇[yi−6,yi−5,...,yi]
=10yi−6−72 yi−5+225yi−4−400yi−3+450yi−2−360yi−1+147yi
60h
,(21)
similarly, for ˙xare given as:
˙x=1
h
∆[x(ti−1),x(ti),...,x(ti+5)]=1
h
∆[xi−1,xi,...,xi+5],
=−10xi−1−77xi+150xi+1−100xi+2+50xi+3−15xi+4+2xi+5
60h,
(22)
˙x=1
h[x(ti−3),x(ti−2),...,x(ti+3)]=1
h[xi−3,xi−2,...,xi+3]
=−xi−3+9xi−2−45xi−1+45xi+1−9xi+2+xi+3
60h
(23)
˙x=1
h∇[x(ti−6),x(ti−5),...,x(ti)]=1
h∇[xi−6,xi−5,...,xi],
=10xi−6−72xi−5+225xi−4−400xi−3+450xi−2−360xi−1+147xi
60h,
(24)
The residual error (RE) for each yand x, i.e., R Eyi,∆,REyi,Πand REyi,∇, in case of forward, central and
backward difference formulae are given, respectively, as:
REyi,∆=Fy(xi,yi,ti)=1
h
∆[yi−1,yi,...,yi+5],i=1,2 (25)
REyi,Π=Fy(xi,yi,ti)=1
h[yi−3,yi,...,yi+3],i=3,4,...,N−3 (26)
REyi,∇=Fy(xi,yi,ti)=1
h∇[yi−6,yi−5,...,yi],i=N−2,N−1,N(27)
While for RExi,∆,RExi,Πand RExi,∇, in case of forward, central and backward difference formulae are
given, respectively, as:
RExi,∆=Fx(xi,yi,ti)=1
h
∆[xi−1,xi,...,xi+5],i=1,2 (28)
RExi,Π=Fx(xi,yi,ti)=1
h[xi−3,xi,...,xi+3],i=3,4,...,N−3 (29)
RExi,∇=Fx(xi,yi,ti)=1
h∇[xi−6,xi−5,...,xi],i=N−2,N−1,N(30)
and then,
RE(y)=



2

i=1RE(yi,∆)2+
N−3

i=3RE(yi,Π)2+
N

i=N−2RE(yi,∇)2,(31)
RE(x)=



2

i=1RE(xi,∆)2+
N−3

i=3RE(xi,Π)2+
N

i=N−2RE(xi,∇)2,(32)
Accordingly, the fitness function for dust density model of VDP-ME is given as:
RE=1
2(RE(y)+RE(x))(33)
Now the requirement is to minimize the error function (33) with optimization mechanism such that REapproaches
0, then the parameters yand xwill approximate the solution of dust density model of VPD-ME.
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I. Jadoon, M.A.Z. Raja, M. Junaid et al. Mathematics and Computers in Simulation 181 (2021) 444–470
3.2. Optimization procedure
To study the dynamics of dust density model, the fitness function i.e., REdefined in (33), is optimized by
integrated computational heuristics based on global search with GAs and hybrid with SQP for local refinements,
GA-SQP. A brief overview of the techniques along with their flow charts and recent applications are narrated in
this section.
GAs is member of the class of evolutionary computing paradigms that is formed on mathematical modeling of
the theory of evolution in natural genetic process. First real application of GAs was associated with the research
work conducted by Holland [17,19] during early seventies of last century. The GAs have capability to handle the
complex linear as well as nonlinear global search optimization problems through exploitation of its reproduction
operators based on selection, crossover and mutation [13]. GAs is a promising heuristic applied to solve several
applications of practical interest [9,18,26,28] such as [47] optimization of nonlinear Troesch’s system [27], thermal
analysis of porous fin model [7], optimization in nonlinear stiff oscillatory problems [21], home health care routing
problem [50] function differential equation [2,45] and HIV infection model of CD4+T cells [55].
Hybridization process with efficient local search is normally conducted to improve the speed of convergence
of the GAs. Therefore, GA is hybrid with viable local search for rapid refinements of the parameters. The hybrid
approach GA-SQP is used for training of design parameter of the FDM, i.e., GA-SQP-FDM. The both GA and SQP
are implemented with built-in routines available in MATLAB optimization toolbox with settings of the parameters
and sequence of procedural step narrated in Fig. 1 and necessary details in pseudo-code as tabulated in Table 1.
4. Performance measures
Three performance measures are incorporated to analyze the GA-SQP-FDM for solving VDP-ME based on
the Mean Absolute Error (MAE), Root Mean Squared Error (RMSE) and Nash–Sutcliffe Efficiency (NSE).
Mathematical formulations of these performance indices for the solutions, y(t) and x(t) are defined as: [2,45,55]
M AEy=1
m
m

i=1yi− ˆyi,M A Ex=1
m
m

i=1xi− ˆxi,(34)
R M S E y=



1
m
m

i=1yi− ˆyi2,R M S Ex=



1
m
m

i=1xi− ˆxi2,(35)
N S E y=1−m
i=1(yi− ˆyi)2
m
i=1(yi−yi)2,N S E x=1−m
i=1(xi− ˆxi)2
m
i=1(xi−xi)2,(36)
where mis the number of input grid points, yand xare the reference numerical solutions of Adams numerical
method (ANM) (ADM), x,and yare their average values, and computed solution ˆyand ˆx, respectively.
Desired optimal values for performance indices of M A E,R M S E and N S E are 0, 0 and 1, respectively.
Accordingly, an operator ENSE is defined below for NSE with desire optimal value is 0 as:
E N S E y=1−E N S E y,E N S E x=|1−E N S E x|(37)
The global performance operators G M A E ,G R M S E and G E N S E are defined as average values of M A E,
R M S E , and E N S E , respectively. Mathematical relations of these operators are given as:
G M A E y=1
L
L

i=11
m
m

k=1yi− ˆyii
G M A Ex=1
L
L

i=11
m
m

k=1xi− ˆxii
(38)
G R M S E y=1
L
L

i=1




1
m
m

k=1yi− ˆyi2
i
,G R M S E x=1
I
I

i=1




1
m
m

k=1xi− ˆxi2
i
(39)
G E N S E y=1
L
L

i=1m

k=1
(yi− ˆyi)2m

k=1
(yi−yi)2i
,G E N S Ex=1
L
L

i=1m

k=1
(xi− ˆxi)2m

k=1
(xi−xi)2i
(40)
Here, Lrepresents total independent trials.
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I. Jadoon, M.A.Z. Raja, M. Junaid et al. Mathematics and Computers in Simulation 181 (2021) 444–470
Fig. 1. Workflow sequence of proposed study in block structures.
450

I. Jadoon, M.A.Z. Raja, M. Junaid et al. Mathematics and Computers in Simulation 181 (2021) 444–470
Table 1
Pseudocode of GA-SQP for optimization.
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I. Jadoon, M.A.Z. Raja, M. Junaid et al. Mathematics and Computers in Simulation 181 (2021) 444–470
5. Simulation and results
In this section, results of simulations are presented for four scenarios of the VDP-ME for dust grain charge in
dusty plasmas for different input domains of time-dependent parameter as well as variation in the rate of charged
dust grain production αand charged dust grain loss rate β.
5.1. Scenario: I: VDP-ME for dust density by varying the rate of charged dust grain production α
The system model of the scenario as given in Eqs. (14) for t∈[0,5] with fixed values of β=0.0001 and
angular frequency ω=1 as:
˙y=x,
˙x=(α−0.0001y2)˙y−y,
y(0) =x(0) =1.
(41)
The cases 1, 2 and 3 are formulated for VDP-ME (41) of dust density with α=0.2, 0.6 and 0.9, respectively.
The fitness function for case 1 of said scenario by taking t∈[0,5] with step size h=0.1 with N=50 as:
RE=1
2











2

i=1RE(xi,∆)2+
47

i=3RE(xi,Π)2+
50

i=48 RE(xi,∇)2
+



2

i=1RE(yi,∆)2+
47

i=3RE(yi,Π)2+
50

i=48 RE(yi,∇)2







,(42)
where REterms are defined in Eqs. (31)–(33) for
Fyi=x,Fxi=y(0) =x(0) =1(0.2−0.0001y2)˙y−y(43)
with forward ∆, central , and backward ∇difference operator are define in Eq. (25)–(27), respectively. Similarly,
the systems with their fitness functions are formulated for cases 2 and 3.
5.2. Scenario: 2: VDP-ME for dust density by varying the rate of charged dust grain loss β
The system model, given in Eqs. (14) for t∈[0,5] with fixed values of α=0.1 and angular frequency ω=1,
for the said scenario is given as:
˙y=x,
˙x=(0.1−βy2)˙y−y,
y(0) =x(0) =1.
(44)
Three respective cases of the system (44) are formulated by taking β=0.1, 0.2 and 0.3. The fitness function for
case 1 is similar as given in Eq. (42) of last scenario with
Fyi=x,Fxi=(0.1−0.1y2)˙y−y
y(0) =x(0) =1(45)
Accordingly, the systems with their fitness functions are formulated for cases 2 and 3.
5.3. Scenario 3: VDP-ME for dust density by varying the both rates of charged dust grain production αand loss
β
The system model for said scenario as given in Eqs. (14) for t∈[0,5] with fixed value of angular frequency
ω=1, while varying parameter αand βas:
˙y=x,
˙x=(α−βy2)˙y−y,
y(0) =x(0) =1.
(46)
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The cases 1, 2 and 3 of the system (46) are formulated for (α=0.2, β=0.4), (α=0.6, β=0.5) and (α=0.9,
β=0.6), respectively, while the fitness function for case 1 is given in Eq. (46) of the scenario 1 with
Fyi=x,Fxi=(0.2−0.4y2)˙y−y
y(0) =x(0) =1(47)
Accordingly, the systems and their fitness functions are formulated for the rest of the cases
5.4. Scenario 4: Analysis of dust density model of VDP-ME for larger input domain with variation of αand loss β
The system model as given in Eqs. (14) is analyze for larger input intervals t∈[0,25] and t∈[0,50] by varying
the values of αand βby with fixed angular frequency ω=1 as:
˙y=x,
˙x=(α−βy2)˙y−y,
y(0) =x(0) =1.
(48)
There cases of the said scenarios are taken as reported in [32], for respective values of α=β=0.01, α=0.01
and β=0.1, and α=1 and β=0.001.
The fitness function for case 1 is similar to Eq. (42) of the scenario 1 with
Fyi=x,Fxi=(0.01 −0.01y2)˙y−y
y(0) =x(0) =1(49)
Similarly, the systems with their fitness functions are formulated for the cases 2 and 3 of the scenarios.
5.5. Analysis on a single run
The optimization of the fitness functions for all three cases of each scenario is performed as per procedure listed
in last section for GA-SQP-FDM. Iterative updates of the fitness functions, i.e., learning curves (LCs), are plotted
for GAs and GA-SQP for each case of first three scenarios are presented in Fig. 2, while LCs for all three cases
of 4th scenario are provided in Fig. 3 for input domain t∈[0,25] and Fig. 4 for input interval t∈[0,50]. It is
seen that steady results of GAs after executing sufficient generations are effectively as well as rapidly tuned by the
process of hybridization for each case of all four scenarios of dust density model based on VDM-ME.
The solutions determined by the proposed scheme GA-SQP-FDM are plotted graphically in Figs. 5 and 6for
charge dust grain density y(t), and its change x(t) for each case of all three scenarios, respectively, for VDM-ME
of dust density model. The results of reference numerical solver based on Adams numerical method (ANM) are
calculated using ‘NDSolve’ routine of Mathematica software package for differential equation and are provided in
Figs. 5 and 6for each variation. The results of ANM are used a standard to compare the performance of the proposed
methodology due to non-availability of exact solutions. The solution presented in Figs. 5 and 6are consistently
overlapping the reference solution all three cases of each scenario of dust density dynamics.
In order to access the level of accuracy achieve the proposed procedure GA-SQP-FDM, the absolute error (AE)
are calculated from ANM results and are presented in Figs. 5 and 6for y(t), and x(t) for each case all three scenarios,
respectively, of VDP-ME of dust density model. The values of the performance indices based on MAE, RMSE and
ENSE are also calculated and are provided in Fig. 7. The performance indices based on AE, MAE, RMSE and ENSE
show that the accuracy of the order 4 to 8 decimal places is achieved generally by the proposed GA-SQP-FDM,
which established that the given scheme is accurate for solving the VDP-ME based dust density models.
Complexity of solver based on GA-SQP-FDM is measured through calculations of execution time, genera-
tions/iterations completed during the process of optimization with GA-SQP and number of functions evaluated
for tuning of optimization variables. All three complexity measures are determined for GA-SQP-FDM algorithm
for all three cases of each scenario of dust density dynamical model of plasma physics and results are listed in
Table 7 on the basis of 200 independent trials. The values of execution time, iteration and functions evaluated are
40 ±2, 2200 and 47000 ±9000, respectively.
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Fig. 2. Learning curves for GA in blue lines for the mean behavior and black lines for the best while with green lines for GA-SQP. The
sub-figs (a–c), (d–f) and (g–i), and sub-figs (j–l), (m–o) and (p–r) are for the cases of scenarios 1, 2 and 3, respectively, of dust density
dynamics. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 3. Results of proposed scheme for all three cases of each scenario for dust density model of VDP-MD in case of t∈[0,25].
The oscillatory behavior of the VDP-ME based dust density model is analyzed for solving the systems with
reported variation in both rate of charged dust grain production αand loss βfor larger input interval, i.e., scenario
4 with t∈[0,25] and t∈[0,50] [57]. Results of proposed GA-SQP-FDM for t∈[0,25] and t∈[0,50] are
455

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Fig. 4. Results of proposed scheme for all three cases of each scenario for dust density model of VDP-MD in case of t∈[0,50].
provided in Figs. 3 and 4in terms of learning curves, solution of y(t) and x(t) along with their parametric plots
(PPs) based illustrations, respectively.
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Fig. 5. Comparison of the results of proposed scheme GA-SQP-FDE for each case of all three scenarios of dust density model of VDP-ME
in case of charge dust grain density y(t).
5.6. Analysis on multiple independent trials
The reliable inferences on the performance of the proposed GA-SQP-FDM are presented here through results of
statistics and their analyses.
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Fig. 6. Comparison of the results of proposed scheme GA-SQP-FDE for each case of all three scenarios of dust density model of VDP-ME
in case of change of dust grain density x(t).
The proposed GA-SQP-FDM is executed for 200 independent trails for dynamical analysis of dust density model
of VDP-ME. Results on the basis of fitness attained for different trials are presented in Fig. 8 for each case of all
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Fig. 7. Comparison through different performance indices for all three cases of each scenario of dust density model of VDP-ME.
three scenarios of dust density dynamics. The values of performance indices of MAE, RMSE and ENSE are also
calculated for each trial and results are presented in Figs. 9 and 10 for dust density parameter y(t) and its change
x(t), respectively for all three cases of each scenario based on VDE-ME. The fitness, MAE, RMSE and ENSE are
generally lie around 10−14 to 10−18, 10−04 to 10−05 , 10−04 and 10−07 to 10−08, respectively. The behavior of the all
the performance measures is consistent, i.e., similar trend of oscillation in the results, with values reasonably close
to the desire gauges, established the reliability of GA-SQP-FDM for solving dust density model of VDM-ME.
The measure of central tendency and variations, i.e., mean, minimum (MIN), maximum (MAX) and standard
deviation (SD) values of AE, are used to analyze the performance further. Results on the basis of 200 independent
trials for the dust density dynamics are tabulated in Tables 2–4 for all three cases of scenarios 1, 2 and 3, respectively.
The mean values for cases 1, 2, and 3 are around 10−04 to 10−07, 10−04 to 10−08 and 10−04 to 10−07 , respectively,
for y(t) and 10−05 to 10−07, 10−04 to 10−07 and 10−04 to 10−07 , respectively, for x(t) of scenario 1. Accordingly,
reasonably accurate results are obtained from GA-SQP-FDM for rest of the cases. The small values of mean and
SD endorsed the stable performance of the GA-SQP-FDE for solving VDP-ME for each variation.
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Fig. 8. Comparison through fitness attained in different independent trails for all three cases of each scenario of dust density model of
VDP-ME.
The analysis of the GA-SQP-FDM is conducted on global performance indices of G M A E x,G M A Ey,
G R M S E x,G R M S E y,G E N S E xand G E N S E y. The results of these measures on the basis of y(t) and x(t) are
presented in Tables 5 and 6, respectively, along with the values of mean fitness for all three cases of each scenario
460

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Fig. 9. Comparison through values of performance indices attained in different independent trails for all three cases of each scenario of dust
density model of VDP-ME for y(t).
of dust density dynamics. The values of mean fitness, G M A E ,G R M S E and G E N S E are around 10−08 to 10−21,
10−02 to 10−05, 10−02 to 10−05 and 10−04 to 10−09 , respectively, for both dynamics of dust density y(t) and x(t)
parameters. The reasonably close to the ideal values of these global performance indices are generally obtained
which established the consistent correctness of the GA-SQP-FDM for solving dusty plasma systems.
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Fig. 10. Comparison through values of performance indices attained in different independent trails for all three cases of each scenario of
dust density model of VDP-ME for x(t).
The analysis on the performance is conducted further with the help of probability plots, histogram illustrations
and boxplots. Results of fitness for 200 independent trials of GA-SQP-FDM on the basis of histograms, boxplots
and cumulative distribution function (CDF) are shown in Fig. 11 for all three cases of each scenario of VDP-ME.
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Fig. 11. Analysis of the performance for GA-SQP-FDM through statistical observations on the basis of fitness attained in independent trials.
While these illustrations for dust density parameter y(t) and x(t) are provided Figs. 12 and 13, respectively, by
means of all three performance indices of M A E,R M S E and E N S E . Histogram plots show that the fitness is
oscillating between 10−17 to 10−19 for scenario 1 and the results presented in boxplots show that even the worst
outlier attained the fitness value in the range of 10−17 for scenario 2 while the results of CDF show that almost sure
event to achieve fitness is around 10−15. The similarly histogram plots for both studies of dust density parameter
y(t) and x(t) show that most of the independent runs achieved the values of MAE around 10−05, median values of
RMSE around 10−05 and sure event to attain the ENSE values around 10−09. Thus, the accurate and convergence
performance of GA-SQP-FDE is ascertained for solving the dust density model of VDP-ME.
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Fig. 12. Analysis of the performance for GA-SQP-FDM for independent trials through statistical observations on the basis of MAE, RMSE
and ENSE measures on y(t).
6. Conclusions with future research directions
A new stochastic computing paradigm GA-SQP-FDM is designed effectively for problems arising in dusty
plasma studies represented with dust density model of nonlinear Van-der Pol Mathieu’s oscillatory systems using
the enrich strength of discretization of finite difference method optimized with global search efficacy of GAs and
rapid convergence ability of SQP. The proposed GA-SQP-FDM is evaluated on variants of dust density model of
VDP-ME by varying the rate of charged dust grain production and loss for both small and larger input domains
and comparison of results with state of art numerical procedure based on Adams method show that accuracy of
the order 10−04 to 10−08 attained consistently for each case. Worth of the designed GA-SQP-FDM is endorsed by
statistical observations on sufficient large dataset generated through execution of 200 independent trials to solve
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Fig. 13. Analysis of the performance for GA-SQP-FDM for independent trials through statistical observations on the basis of MAE, RMSE
and ENSE measures on x(t).
the dust density model of VPD-ME in terms of mean and standard deviation of absolute error, mean absolute error
(MAE), root mean squared error (RMSE), Nash–Sutcliffe efficiency (NSE), probability plots, histogram illustrations,
boxplots and complexity measures.
In future, one may explore the proposed GA-SQP-FDM to analyze the fractional order systems based on
nonlinear Painlev´
e, Riccati and Bratu’s, Van-der Pol and Bagley–Torvik equations. Alternate optimization solvers
based methodologies such as particle swarm optimization, differential evolution, backtracking search optimization,
firework, firefly, follower pollination, hill climbing, moth flame optimization or ant–bee colony optimized finite
different computing paradigms can be exploited to dust density model of VDM-ME as well as other applications
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Table 2
Comparison of the result using measure of central tendency and variation parameters for each case of scenario 1 of dust density model of
VDP-ME.
Index tCase-1 Case-2 Case-3
MIN MEAN SD MIN MEAN SD MIN MEAN SD
y(t)
0.0 0 0 0 0 0 0 0 0 0
0.5 1.06E−04 1.06E−04 4.94E−09 1.26E−04 1.26E−04 6.92E−09 1.36E−04 1.36E−04 6.56E−08
1.0 5.39E−05 5.39E−05 4.14E−09 2.32E−05 2.32E−05 6.06E−09 1.23E−05 1.29E−05 5.15E−08
1.5 3.77E−05 3.77E−05 4.20E−09 3.59E−06 3.60E−06 4.65E−09 8.53E−06 8.91E−06 3.42E−08
2.0 1.83E−05 1.83E−05 5.07E−09 1.81E−06 1.81E−06 3.43E−09 2.53E−06 2.62E−06 1.77E−08
2.5 1.33E−06 1.34E−06 5.77E−09 2.46E−05 2.46E−05 3.44E−09 5.71E−05 5.73E−05 2.01E−08
3.0 2.37E−05 2.37E−05 5.80E−09 9.57E−05 9.57E−05 4.56E−09 8.04E−05 8.05E−05 3.53E−08
3.5 2.25E−04 2.25E−04 4.89E−09 3.96E−04 3.96E−04 5.17E−09 1.03E−04 1.03E−04 4.14E−08
4.0 8.29E−05 8.30E−05 3.90E−09 2.77E−05 2.77E−05 4.74E−09 9.75E−08 3.33E−07 2.48E−08
4.5 2.31E−05 2.32E−05 4.03E−09 7.97E−08 8.32E−08 3.94E−09 8.20E−06 8.47E−06 2.54E−08
5.0 2.93E−07 3.17E−07 5.07E−09 9.86E−07 1.00E−06 3.32E−09 2.68E−06 2.77E−06 1.64E−08
x(t)
0.0 0 0 0 0 0 0 0 0 0
0.5 1.12E−05 1.12E−05 5.25E−09 1.04E−04 1.04E−04 3.37E−09 2.11E−04 2.11E−04 2.10E−08
1.0 8.96E−05 8.96E−05 5.57E−09 1.41E−04 1.41E−04 4.17E−09 1.64E−04 1.64E−04 3.83E−08
1.5 6.51E−06 6.52E−06 5.37E−09 3.20E−05 3.20E−05 4.82E−09 2.73E−05 2.78E−05 4.18E−08
2.0 1.71E−05 1.71E−05 4.91E−09 6.90E−06 6.96E−06 5.58E−09 2.44E−07 8.23E−07 4.64E−08
2.5 1.07E−05 1.07E−05 4.20E−09 1.16E−04 1.16E−04 5.65E−09 1.63E−04 1.63E−04 4.60E−08
3.0 5.37E−05 5.37E−05 4.38E−09 3.61E−04 3.61E−04 4.69E−09 4.27E−04 4.27E−04 3.28E−08
3.5 9.28E−05 9.29E−05 5.45E−09 2.04E−04 2.04E−04 3.43E−09 3.55E−05 3.56E−05 1.62E−08
4.0 1.58E−04 1.58E−04 5.90E−09 9.46E−05 9.46E−05 3.56E−09 2.63E−05 2.64E−05 2.37E−08
4.5 2.24E−05 2.24E−05 5.56E−09 6.53E−06 6.54E−06 3.86E−09 1.47E−05 1.50E−05 2.78E−08
5.0 3.94E−07 4.18E−07 4.85E−09 2.35E−07 2.40E−07 4.08E−09 5.43E−08 1.29E−07 1.58E−08
Table 3
Comparison of the result using measure of central tendency and variation parameters for each case of scenario 2 of dust density model of
VDP-ME.
Index tCase-1 Case-2 Case-3
MIN MEAN SD MIN MEAN SD MIN MEAN SD
y(t)
0.0 0 0 0 0 0 0 0 0 0
0.5 6.56E−05 6.57E−05 4.92E−09 8.85E−06 9.16E−06 8.40E−08 4.04E−06 1.17E−02 2.63E−02
1.0 7.65E−05 7.65E−05 4.77E−09 6.28E−05 6.32E−05 6.24E−08 3.61E−05 1.04E−03 2.05E−03
1.5 7.62E−05 7.62E−05 5.04E−09 8.92E−05 8.98E−05 1.09E−07 1.07E−04 1.54E−02 3.43E−02
2.0 6.67E−05 6.67E−05 5.77E−09 1.07E−04 1.08E−04 1.93E−07 1.97E−04 3.70E−02 8.28E−02
2.5 7.63E−06 7.65E−06 6.40E−09 1.26E−05 1.40E−05 2.63E−07 9.36E−05 5.84E−02 1.32E−01
3.0 6.77E−07 6.95E−07 6.80E−09 4.98E−06 6.40E−06 2.95E−07 1.82E−06 7.48E−02 1.69E−01
3.5 2.15E−05 2.15E−05 6.74E−09 5.03E−06 5.74E−06 2.65E−07 2.85E−07 7.70E−02 1.74E−01
4.0 5.57E−05 5.57E−05 6.55E−09 1.01E−04 1.02E−04 1.88E−07 1.45E−05 5.70E−02 1.28E−01
4.5 2.66E−05 2.67E−05 6.75E−09 8.36E−05 8.45E−05 1.98E−07 1.27E−04 7.94E−03 1.66E−02
5.0 4.50E−08 6.63E−08 7.54E−09 3.55E−06 5.72E−06 3.85E−07 2.66E−04 7.26E−02 1.62E−01
x(t)
0.0 0 0 0 0 0 0 0 0 0
0.5 6.69E−05 6.70E−05 4.95E−09 8.79E−05 8.85E−05 1.10E−07 9.07E−05 1.45E−02 3.23E−02
1.0 1.26E−05 1.27E−05 5.51E−09 8.30E−05 8.39E−05 1.63E−07 1.44E−04 2.72E−02 6.10E−02
1.5 1.98E−05 1.99E−05 5.83E−09 6.47E−05 6.57E−05 1.94E−07 1.22E−04 3.87E−02 8.69E−02
2.0 5.78E−05 5.78E−05 5.85E−09 3.38E−05 3.46E−05 1.90E−07 7.43E−05 4.50E−02 1.01E−01
2.5 2.01E−05 2.01E−05 5.63E−09 2.23E−06 2.77E−06 1.45E−07 3.34E−06 4.07E−02 9.14E−02
3.0 1.21E−06 1.23E−06 5.81E−09 1.41E−05 1.46E−05 1.14E−07 2.14E−05 2.18E−02 4.86E−02
3.5 1.77E−05 1.77E−05 6.47E−09 7.81E−05 7.92E−05 2.09E−07 2.04E−04 1.57E−02 3.43E−02
4.0 1.40E−05 1.40E−05 7.33E−09 1.99E−04 2.01E−04 3.57E−07 6.32E−04 6.89E−02 1.53E−01
4.5 4.23E−06 4.25E−06 7.94E−09 8.31E−05 8.57E−05 4.81E−07 4.67E−04 1.30E−01 2.92E−01
5.0 7.85E−08 9.94E−08 8.05E−09 5.61E−09 7.61E−07 4.50E−07 6.85E−05 1.86E−01 4.19E−01
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Table 4
Comparison of the result using measure of central tendency and variation parameters for each case of scenario 3 of dust density model of
VDP-ME.
Index tCase-1 Case-2 Case-3
MIN MEAN SD MIN MEAN SD MIN MEAN SD
y(t)
0.0 0 0 0 0 0 0 0 0 0
0.5 1.40E−04 1.40E−04 4.85E−08 1.13E−04 1.15E−04 3.09E−07 2.71E−05 3.13E−05 1.35E−06
1.0 1.49E−05 1.50E−05 5.04E−08 1.18E−05 1.24E−05 2.82E−07 6.85E−06 1.06E−05 9.12E−07
1.5 5.94E−06 5.97E−06 5.45E−08 7.10E−06 7.29E−06 3.03E−07 7.53E−05 7.62E−05 7.30E−07
2.0 3.89E−05 3.93E−05 6.20E−08 1.48E−04 1.51E−04 3.92E−07 1.90E−06 6.26E−06 9.36E−07
2.5 1.39E−04 1.40E−04 6.65E−08 1.83E−05 2.13E−05 5.10E−07 1.25E−04 1.42E−04 2.07E−06
3.0 4.54E−05 4.55E−05 5.53E−08 1.96E−04 1.97E−04 4.29E−07 2.72E−05 3.08E−05 1.71E−06
3.5 1.00E−04 1.01E−04 3.16E−08 3.52E−05 3.62E−05 1.66E−07 3.93E−05 4.12E−05 4.22E−07
4.0 4.68E−05 4.69E−05 3.36E−08 5.32E−05 5.33E−05 1.89E−07 7.20E−07 4.65E−06 4.81E−07
4.5 2.78E−05 2.78E−05 4.79E−08 6.58E−05 6.60E−05 2.86E−07 6.90E−07 5.62E−06 6.78E−07
5.0 4.65E−06 4.69E−06 6.37E−08 6.76E−09 1.34E−06 2.78E−07 1.05E−06 1.08E−05 1.13E−06
x(t)
0.0 0 0 0 0 0 0 0 0 0
0.5 1.50E−04 1.50E−04 5.83E−08 4.95E−05 4.98E−05 3.44E−07 9.03E−06 1.78E−05 1.04E−06
1.0 2.08E−04 2.08E−04 4.64E−08 2.35E−04 2.36E−04 3.19E−07 1.17E−04 1.20E−04 1.36E−06
1.5 8.80E−07 9.60E−07 4.01E−08 1.20E−05 1.37E−05 3.00E−07 3.80E−05 4.11E−05 1.35E−06
2.0 1.60E−04 1.60E−04 4.04E−08 1.16E−03 1.16E−03 3.57E−07 2.38E−05 3.68E−05 1.89E−06
2.5 5.18E−05 5.19E−05 4.07E−08 4.91E−05 5.05E−05 3.19E−07 2.87E−04 2.97E−04 1.84E−06
3.0 1.20E−04 1.21E−04 7.09E−08 1.12E−03 1.12E−03 6.24E−07 2.56E−05 2.95E−05 3.24E−06
3.5 2.54E−04 2.54E−04 8.09E−08 4.50E−04 4.51E−04 6.03E−07 5.06E−05 5.40E−05 1.92E−06
4.0 1.18E−04 1.18E−04 5.40E−08 4.18E−05 4.35E−05 2.90E−07 1.75E−07 2.37E−06 5.06E−07
4.5 2.16E−05 2.18E−05 3.88E−08 5.56E−05 5.70E−05 2.28E−07 3.45E−07 2.75E−06 4.70E−07
5.0 1.22E−06 1.40E−06 3.77E−08 7.68E−06 9.57E−06 3.31E−07 8.81E−06 1.83E−05 1.10E−06
Table 5
Comparison through global performance indices for each case of all three scenario of VDP-ME for dust density parameter y(t).
Scenario Case Mean fitness G M A E yG R M S E yG E N SE y
Values SD Values SD Values SD Values SD
11 2.91E−19 2.23E−20 4.97E−05 2.74E−09 8.40E−05 2.15E−09 5.65E−09 2.89E−13
2 5.66E−18 1.01E−19 5.34E−05 2.94E−09 1.08E−03 2.41E−09 1.03E−08 4.60E−13
3 5.06E−17 3.33E−18 4.67E−05 6.37E−09 9.36E−05 2.03E−08 8.48E−09 3.69E−12
21 1.64E−21 2.20E−21 5.65E−05 3.85E−09 8.50E−05 3.23E−09 4.11E−09 3.13E−13
2 2.49E−17 2.39E−18 6.79E−05 1.05E−07 9.76E−05 1.03E−07 1.87E−09 3.93E−12
3 2.16E−08 4.86E−08 3.82E−02 8.56E−02 4.68E−02 1.05E−01 8.74E−04 1.97E−03
31 8.43E−17 4.14E−18 5.44E−05 8.57E−09 1.03E−03 1.21E−08 7.45E−09 1.75E−12
2 1.49E−15 1.06E−16 7.04E−05 4.90E−08 1.18E−03 6.04E−08 7.60E−09 7.75E−12
3 5.52E−15 4.17E−16 6.09E−05 1.76E−07 1.10 E-03 1.68E−07 5.45E−09 1.67E−11
of interest [8,31,36,38,54,56]. Additionally, design of fractional backward finite difference schemes based on
Grunwald–Letnikov definition, Diethelms quadrature based approach and Lubich’s method, for discretization and
their optimization with soft computing framework look promising to be explored.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could
have appeared to influence the work reported in this paper.
467

I. Jadoon, M.A.Z. Raja, M. Junaid et al. Mathematics and Computers in Simulation 181 (2021) 444–470
Table 6
Comparison through global performance indices for each case of all three scenario of VDP-ME for dust density parameter x(t).
Scenario Case Mean fitness G M A ExG RM S E xG E N S Ex
Values SD Values SD Values SD Values SD
11 2.91E−19 2.23E−20 5.80E−05 1.10E−09 1.03E−04 1.19E−09 1.35E−08 3.11E−13
2 5.66E−18 1.01E−19 7.76E−05 6.80E−10 1.15E−04 1.12E−09 1.90E−08 3.69E−13
3 5.06E−17 3.33E−18 9.65E−05 3.07E−09 1.53E−04 6.82E−09 3.78E−08 3.36E−12
21 1.64E−21 2.20E−21 5.11E−05 3.12E−09 7.72E−05 2.95E−09 5.21E−09 3.98E−13
2 2.49E−17 2.39E−18 9.95E−05 1.72E−07 1.61E−04 1.45E−07 9.20E−09 1.66E−11
3 2.16E−08 4.86E−08 5.00E−02 1.12E−01 6.86E−02 1.54E−01 5.49E−03 1.24E−02
31 8.43E−17 4.14E−18 1.37E−04 9.54E−09 2.17E−04 3.92E−09 4.84E−08 1.75E−12
2 1.49E−15 1.06E−16 2.12E−04 1.25E−08 3.93E−04 6.84E−08 1.12E−07 3.91E−11
3 5.52E−15 4.17E−16 2.39E−04 7.57E−08 4.78E−04 2.16E−07 1.30E−07 1.17E−10
Table 7
Comparison of complexity parameter for GA-SQP-FDE for each case of all three scenario of dust density
model of VDP-ME.
Scenario Case Execution time Generation/Iterations Function evaluations
Mean SD Mean SD Mean SD
11 37.60 1.18 2200.00 0.00 532194.07 7763.59
2 36.53 1.25 2200.00 0.00 524731.54 8790.49
3 37.26 1.82 2200.00 0.00 529318.04 13052.63
21 39.50 2.37 2200.00 0.00 544162.66 15034.91
2 44.98 0.90 2200.00 0.00 550539.94 5208.34
3 43.38 1.11 2200.00 0.00 555916.24 8629.46
31 37.18 1.15 2200.00 0.00 529645.48 8149.88
2 34.62 1.23 2200.00 0.00 510047.58 9456.10
3 41.11 1.93 2200.00 0.00 489999.12 7321.39
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