Available online at www.ispacs.com/jfsva
Volume 2015, Issue 2, Year 2015 Article ID jfsva-00212, 12 Pages doi:10.5899/2015/jfsva-00212
Research Article
Modified fractional Euler method for solving Fuzzy sequential
Fractional Initial Value Problem under H-differentiability
H. Roshanfekr Varazgahi1∗, S. Abbasbandy1
(1)Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.
Copyright 2015 c⃝H. Roshanfekr Varazgahi and S. Abbasbandy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper, the solution to Fuzzy sequential Fractional Initial Value Problem [FFIVP] under Caputo type fuzzy frac-tional derivatives by a modified fracfrac-tional Euler method is presented. The Caputo-type fuzzy fracfrac-tional derivatives are defined based on Hukuhara difference and strongly generalized fuzzy differentiability. The modified fractional Euler method based on a generalized Taylors formula and a modified trapezoidal rule is used for solving initial value problem under fuzzy sequential fractional differential equation of order 0<β<1. Solving two examples of linear and nonlinear FFIVP illustrates the method.
Keywords: Fuzzy sequential fractional differential equations, Fractional Initial Value Problem, Caputo-type fuzzy fractional se-quential H-differentiability.
1 Introduction
Recently, applying fractional differential equations increased sharply and have attracted a considerable attention in mathematics and in applications. So that they’ve been used in modeling of many physical, chemical process and in engineering [7, 10, 11, 14, 16].
There has been two important published books in this way. The first one has been written by Podlubny and the second one by Kilbas et.al [18]. In addition, there has been lots of research papers to consider solutions of fractional differ-ential equations [1, 20, 21, 22, 23] and there has also been lots of references published.
In this contribution, we attempt to investigate the solutions of fractional differential equations with uncertainty called fuzzy fractional differential equations (FFDEs).
Recently, Agarwal, Arshad and Allahviranloo, Salahshour,[6, 8, 3, 30] proposed on the fractional differential equa-tions with uncertainty. They have considered the Riemann-Liouville’s differentiability for solving FFDEs that is a combination of Hukuhara difference and Riemann-Liouville derivative. Some basic paper exist which are written by Bede et. al [12] while shortcoming of applications of Hukuhara difference has been discussed. As the result, we will adopt a generalization of strongly generalized differentiability to fractional case.
At the end, sequential fractional differentiability (we adopt the convenient terminology of Miller and Ross) by using Hukuhara differences which is called Caputo sequential fractional H-differentiability of the same orderσk, will be suggested. Naturally, we arrive at solutions under Caputo sequential fractional H-differentiability which are different from those obtained in [30].
There fore, a direct procedure will be adopted to derive the definition which is constructed based on the combination of strongly generalized differentiability and Caputo sequential fractional H-derivative of the same orderσk.
Simultaneously, an analytical method will be intended for solving fuzzy sequential fractional differential equations FSFDEs. Since considering the solutions of FSFDEs is a new subject, first need to provided an analytical method for solving it, then numerical methods can be used.
Finally, Modified fractional Euler method will be adopted to solve FSFDEs. There exist numbers of useful papers about [2, 26, 27].
This paper is organized as follows:
In section 2, we recall some well-known definitions of fuzzy numbers and is given some needed concepts. In section 3, Caputo H-differentiability is introduced and some of its properties is considered.In section 4, Caputo sequential H-differentiability is introduced. Consequently, the Modified fractional Euler method are considered for fuzzy-valued under Caputo sequential H-derivative, and the solutions of FSFDEs are investigated by using the fuzzy Laplace trans-forms and their inverses in section 5. In section 6, some examples are solved to illustrate the method. Finally, conclusion is drawn in section 7.
2 Preliminaries
The basic definition of fuzzy numbers is given in [34].
We denote the set of all real numbers byRand the set of all fuzzy number onRis indicated byE.A fuzzy number is
a mappingu:R→[0,1]with the following properties:
(a)uis upper semi-continuous,
(b)uis fuzzy convex, i.e.,u(λx+ (1−λ)y)≥min{u(x),u(y)}for allx,y∈R,λ∈[0,1],
(c)uis normal, i.e.,∃x0∈Rfor whichu(x0) =1,
(d) suppu={x∈R|u(x)>0}is the support of theu, and its closure cl(supp u) is compact. An equivalent parametric definition is also given in [17, 24, 35] as follows:
Definition 2.1. A fuzzy number u in parametric form is a pair(u,u)of functions u(r), u(r),0≤r≤1, which satisfy the following requirements:
1. u(r)is a bounded non-decreasing left continuous function in(0,1], and right continuous at 0,
2. u(r)is a bounded non-increasing left continuous function in(0,1], and right continuous at 0,
3. u(r)≤u(r),0≤r≤1.
Moreover, we also can present ther-cut representation of fuzzy number as[u]r= [u(r),u(r)]for all 0≤r≤1. According to Zadeh,s extension principle, operation of addition onEis defined by
(u⊕v)(x) =sup y∈R
min{u(y),v(x−y)}, x∈R, (2.1)
and scalar multiplication of a fuzzy number is given by
(k⊙u)(x) =
{ u(x/k), k>
0,
e
0, k=0,
where ˜0∈E.
The Hausdorff distance between fuzzy numbers given byd:E×E−→R+∪{0},
d(u,v) = sup r∈[0,1]
max{|u(r)−v(r)|,|u(r)−v(α)|},
whereu= (u(r),u(r)),v= (v(r),v(r))⊂Ris utilized in [12]. Then, it is easy to see thatd is a metric inEand has
the following properties (see [29]):
(2) d(ku,kv) =|k|d(u,v), ∀k∈R,u,v∈E,
(3) d(u+v,w+e)≤d(u,w) +d(v,e), ∀u,v,w,e∈E,
(4) (d,E)is a complete metric space.
Theorem 2.1. [33]. Let f(x)be a fuzzy-valued function on[a,∞)and it is represented by(f(x;r),f(x;r)).For any fixed r∈[0,1], assume f(x;r)and f(x;r)are Riemann-integrable on [a,b]for every b≥a, and assume there are
two positive M(r)and M(r)such that∫ab|f(x;r)|dx≤M(r)and∫ab|f(x;r)|dx≤M(r)for every b≥a.Then f(x)
is improper fuzzy Riemann-integrable on[a,∞)and the improper fuzzy Riemann-integral is a fuzzy number. Further more, we have:
∫ ∞
a
f(x)dx=
(∫ ∞
a
f(x;r)dx,
∫ ∞
a
f(x;r)dx
)
.
Definition 2.2. Let x,y∈E.If there exists z∈Esuch that x=y+z, then z is called the H-difference of x and y, and it is denoted by x⊖y.
In this paper, the sign ”⊖” always stands for H-difference, and also note thatx⊖y̸=x+ (−1)y.
3 Riemann-Liouville H-differentiability
In this section, the concept of fuzzy Riemann-Liouville derivatives are considered using Hukuhara difference. We try to produce such definitions and statements similar to the non-fractional one in the fuzzy context [12].
We denoteCF[a,b] as a space of all fuzzy-valued functions which are continuous on [a,b].Also, we denote the space of all lebesque integrable fuzzy-value functions on the bounded interval[a,b]⊂R by LF1[a,b], we denote the space of fuzzy-value functions f(x) which have continuous H-derivative up to ordern−1 on [a,b]such that
f(n−1)(x)∈ACF([a,b])byACFn([a,b]).
Now, we define the fuzzy Riemann-Liouville integral of fuzzy-valued function as follows:
Definition 3.1. Let f(x)∈CF[a,b]∩LF1[a,b],the fuzzy Riemann-Liouville integral of fuzzy-valued function f is defined as following:
(
Iaβ+f
)
(x) = 1
Γ(β) ∫ x
a
f(t)dt
(x−t)1−β, x>a, 0<β ≤1. (3.2)
Since, f(x;r) = [f(x;r),f(x;r)], for all 0≤r≤1, then we can indicate the fuzzy Riemann-Liouville integral of fuzzy-valued function f based on the lower and upper functions as following:
Theorem 3.1. ([3]). Let f(x)∈CF[a,b]∩LF1[a,b],, the fuzzy Riemann-Liouville integral of fuzzy-valued function f is defined as following: (
Iaβ+f)(x;r) =[(Iaβ+f)(x;r),(Iaβ+f)(x;r)], 0≤r≤1, (3.3)
where
(
Iaβ+f)(x;r) = 1
Γ(β) ∫ x
a
f(t;r)dt
(x−t)1−β, 0≤r≤1, (3.4)
and
(
Iaβ+f)(x;r) = 1
Γ(β) ∫ x
a
f(t;r)dt
(x−t)1−β, 0≤r≤1. (3.5)
Now, we define fuzzy Riemann-Liouville fractional derivatives of orderβ∈C,(Re(β)>0)for fuzzy-valued function
f.
Definition 3.2. . Let f(x)∈CF[a,b]∩LF1[a,b],and x0∈(a,b)andΦ(x) =Γ(n−1β)
∫x a
f(t)dt
(x−t)β−n+1 (n= [β] +1;x>a). We say that f(x)is fuzzy Riemann-Liouville fractional differentiable of orderβ,(β>0)at x0, if there exists an
ele-ment(Dβ
a+f)(x0)∈E, such that for all h>0sufficiently small
(i) (Dβa+f)(x0) = lim h−→0
Φ(n−1)(x0+h)⊖Φ(n−1)(x0)
h =h−→0lim
Φ(n−1)(x0)⊖Φ(n−1)(x0−h)
or
(ii) (Dβa+f)(x0) = lim h−→0
Φ(n−1)(x0)⊖Φ(n−1)(x0+h) −h =h−→0lim
Φ(n−1)(x0−h)⊖Φ(n−1)(x0)
−h , (3.7)
For sake of simplicity, we say that a fuzzy-valued functionf isRL[(i)−β]-differentiable if it is differentiable as in the Definition 3.2 case(i), and isRL[(ii)−β]-differentiable if it is differentiable as in the Definition 3.2 case(ii).
Theorem 3.2. Let f(x)∈CF[a,b]∩LF1[a,b],and x0∈(a,b),(β>0), and(n= [β]+1)such that for all0≤r≤1, then
(i) If f(x)be aRL[(i)−β]-differentiable fuzzy-valued function, then:
(Dβ
a+f)(x0;r) =
[
Dβ
a+f(x0,r),D
β
a+f(x0,r)
]
(3.8)
(ii) If f(x)be aRL[(ii)−β]-differentiable fuzzy-valued function, then:
(Dβa+f)(x0;r) =
[
Dβa+f(x0;r),D
β
a+f(x0,r)
]
(3.9)
where
(Dβa+f)(x0;r) =
[
1
Γ(n−β)
dn dxn
∫ x
a
f(t;r)dt
(x−t)β−n+1
]
x=x0
, (3.10)
and
(Dβa+f)(x0;r) =
[
1
Γ(n−β)
dn dxn
∫ x
a
f(t,r)dt
(x−t)β−n+1
]
x=x0
, (3.11)
Remark 3.1. For caseβ =n∈N, the fuzzy Riemann-Liouville fractional derivative reduces to the generalized H-differentiability of order n [19]
Lemma 3.1. Let f(x)∈LF1(a,b), fm−α∈ACFm([a,b]).And0<β <1, then we have
(
Iaβ+Dβa+f
)
(x) =f(x)⊖ m
∑
j=1
fm−(m−jβ)(a)
Γ(β−j+1)(x−a)
β−j
,
for caseRL[(i)−β]-differentiability and we have
(
Iaβ+Dβa+f
) (x) =−
∑
mj=1
fm−(m−jβ )(a)
Γ(β−j+1)(x−a)
β−j
⊖(−f(x)),
for caseRL[(ii)−β]-differentiability, provided the mentioned Hukuhara differences exist.
Proof. See [3].
Property 3.1. (Composition with fractional Riemann-Liouville H-derivatives). Letα>0andβ >0be such that, n−1<α≤n, m−1<β≤m,(n,m∈N)andα+β<n,and let f(x)∈LF1(a,b)and fm−α∈ACFm([a,b]).Then we
have
(
Dαa+Dβa+f
)
(x) =(Dα+β
a+ f
) (x)⊖
m
∑
j=1
(
Dβ−jf)(a+)(x−a)
−j−α
Γ(1−j−α),
for case Dβa+f(x)isRL[(i)−α]-differentiability and we have
(
Dαa+Dβa+f
) (x) =−
(
m
∑
j=1
(
Dβ−jf)(a+)(x−a)
−j−α
Γ(1−j−α)
)
⊖(−Dα+β
a+ f
) (x),
4 Caputo sequential H-differentiability
In this section, by using the composition rule for Caputo H-differentiability, we can replace all sequential frac-tional H-derivatives(CDσm
a+ f
)
defined in terms of the Caputo fractional H-differentiability Eqs.(3.6),(3.7)by
CDσk
0+≡ CDαk
0+ C
Dαk−1
0+ ... CDα1
0+,
Dσk−1
0+ ≡D
αk−1
0+ D
αk−1
0+ ...D
α1
0+, where
σk=
k
∑
j=1
αj, (k=1, ...,m),
0<αj≤1, (j=1, ...,m), and
f(x)∈CFm[a,b]∩LF1[a,b].
5 Generalized Taylors formula
Generalized Taylors formula under the Caputo-type fractional derivative was introduced in crisp context [14]. Here, we introduce it under the Caputo-type fuzzy fractional derivatives as follows:
Theorem 5.1. Let˜f(x)∈CF[0,b]∩LF[0,b]. and suppose that cDkβ˜f(x)∈CF[0,b] for k =0,1, ...,n+1 where
0<β <1,0≤x0≤x and0<x≤b.Then we have
[f(x)]α=[fα(x),f¯α(x)]
fα(x) =
n
∑
i=0
xiβ
Γ(iβ+1)
c
Diβfα(0) +
cD(n+1)βfα(x
0)
Γ(nβ+β+1)x
(n+1)β (5.12)
fα(x) =
n
∑
i=0
xiβ Γ(iβ+1)
c
Diβf¯α(0) +
cD(n+1)βf¯α(x
0)
Γ(nβ+β+1) x
(n+1)β (5.13)
wherecDβfα(0) =cDβfα(x),cDβfα(0) =cDβfα(x).
5.1 FSFDEs under the Caputo sequential H-differentiability
In this section, by using the composition rule for Caputo H-differentiability, we can replace all sequential fractional H-differentiability in FFDEs by the Caputo fractional H-differentiability of the same orderσk, are discussed.
First we consider the fuzzy sequential fractional differential equation of the same orderσk, with the initial conditions
(
Dσm
0+y
)
(x) =f[x,y(x)] (
Dσk−1
0+ y
)
(0) =bk∈E, k=1, ...,m,
(5.14)
With sequential fractional H-derivatives(Dσm
a+y
)
defined in terms of the Riemann-Liouville fractional H-differentiability Eqs(3.6),(3.7)by
Dσk
0+≡D
αk
0+D
αk−1
0+ ...D
α1
0+,
Dσk−1
0+ ≡D
αk−1
0+ D
αk−1
0+ ...D
α1
where
σk=
k
∑
j=1
αj, (k=1, ...,m),
0<αj≤1, (j=1, ...,m), and
f(x)∈CFm[a,b]∩LF1[a,b],
6 Solving fuzzy initial value problem of fractional order
In this section, the modified fractional Euler method for solving fuzzy sequential initial value problem of frac-tional order under the Caputo-type fuzzy fracfrac-tional derivative will be presented. The method is based on the fracfrac-tional Euler method-used as a prediction at each step-and the modified trapezoidal rule-used to make a correction to obtain the finite value at each step-which has been proposed by Odibat and Momani [15] in non-fuzzy context. To this end, consider the following FFIVP
Example 6.1. Let us consider the following FSFDE (fractional nuclear decay equation)
Dα0+
(
Dβ0+y
)
(x) =λ⊙y(x),0<α,β <1
Dα0+−1
(
Dβ0+y
)
(0,r) =b1∈E,
(
Dβ0+−1y
)
(0,r) =b2∈E.
whereα+β=0.5,α1=α,α2=βandm=2.Therefore,σ1=α,σ2=α+β andy(x)is the number of radionuclides present in a given radioactive,λ is a decay constant. We solve this example according two following cases forλ∈R.
Case I.Suppose thatλ∈R+= (0,+∞), UsingDβ0+y(x)isRL[(i)−α]−differentiability, we get the solution of FSFDE as following:
{
y(x;r) =b2(r)(xα−1)Eα+β,α[λxα+β]+b1(r)(xα+β−1)Eα+β,α+β[λxα+β], 0≤r≤1,
y(x;r) =b2(r)(xα−1)Eα+β,α
[
λxα+β]+b1(r)(xα+β−1)Eα+β,α+β
[
λxα+β], 0≤r≤1. (6.15)
Case II.Suppose thatλ ∈R−= (−∞,0), then usingD0β+y(x)isRL[(ii)−α]−differentiability, the solution will be obtained similar to Eq. (6.15).
For special case, let us considerβ=0 and 0<α<1,λ=1 (and assuming, of course,b2=˜0),b1(r) = (1+r,3−r), then we get the solution forCase Ias following:
cDαy˜(x) =f(x,y˜),
˜
y(0) =y˜0
0≤x≤b,0<α<1,
(6.16)
The FFIVP (6.16) can be considered equivalent by the following initial value problems if ˜y(x)isc[(i)−α]− differen-tiability we get:
cDαyr(x) = [f(x,y˜)]r=F(x,yr,yr),
cDαyr(x) = [f(x,y˜)]r=G(x,yr,yr),
y(0,r) =y0(r),y(0,r) =y0(r)
x∈[0,b], α∈(0,1)
and if ˜y(x)isc[(ii)−α]−differentiability we get:
cDαyr(x) =G(x,yr,yr),
cDαyr(x) =F(x,yr,yr),
y(0,r) =y0(r),y(0,r) =y0(r)
x∈[0,b], α∈(0,1)
(6.18)
letg(x)be a crisp continuous function and (⌈α⌉−times differentiable) in the independent variable x over the interval
of differentiation (integration) [0,b].let the interval[0,b]be subdivided into N subintervals[xj,xj+1]of step sizeh=
b N
using the nodesxj=jhfor j=0,1, ...,N.consider the following Riemann-liouville integral
Iαg(x) = 1
Γ(α) ∫x
0
g(t)
(x−t)1−αdt x,α∈R+
making use of the modified trapezoidal rule,Iαg(b) =T(g,h,α)−o(h2)where
T(g,h,α) = ((N−1)α+1−(N−α−1)Nα) hαg(0)
Γ(α+2)+
hαg(b)
Γ(α+2)+∑
N−1
j=1((N−j+1)α+1−2(N−j)α+1+ (N−
j−1)α+1)hαg(xj)
Γ(α+2)
The initial value problems (6.16) and (6.17) can be equivalent to the following integral equations
yr(x) =IαF(x,yr,yr) +yr(0)
yr(x) =IαG(x,yr,yr) +yr(0)
x∈[0,b]α∈(o,1)
(6.19)
By substitutingx=x1into 18 and approximation of theIαF(x,yr,yr),IαG(x,yr,yr)by the modified trapezoidal rule withh=x1−x0, we have
yr 1=α
hαF(x0,yr0,yr0)
Γ(α+2) +
hαF(x1,yr1,yr1)
Γ(α+2) +y
r 0
yr
1=α
hαG(x0,yr0,yr0)
Γ(α+2) +
hαG(x1,yy1,yr1)
Γ(α+2) +y
r 0 α∈(0,1)
(6.20)
Whereyjdenotesy(xj).now, we are going to estimateyr1,yr1by the fractional Euler method.consider the initial value problems (6.16) and (6.17), suppose ˜y,cDαy,˜cD2αy˜∈CF[0,b]and use the generalized Taylors formula to expand ˜y(x) aboutx0=0 and neglect the second order term ( involvingh2α)), the formula for the fractional Euler method is as follows:
yrj+1=yrj+
hαF(xj,yrj,yrj)
Γ(α+1)
yr
j+1=y r
j+
hαG(xj,yrj,yrj)
Γ(α+1)
α∈(0,1)
(6.21)
Therefore, by estimatingyrj,yr
yr1=αh
αF(x
0,yr0,yr0)
Γ(α+2) +
hαF(x1,yr0+
hαF(x0,yr0,yr0)
Γ(α+1) ,y
r 0+
hαG(x0,yr0,yr0)
Γ(α+1) )
Γ(α+2) +y
r 0
yr
1=α
hαG(x0,yr0,yr0)
Γ(α+2) +
hαG(x1,yr0+
hαF(x0,yr0,yr0)
Γ(α+1) ,y
r 0+
hαG(x0,yr0,yr0)
Γ(α+1) )
Γ(α+2) +y
r 0 α∈(0,1)
(6.22)
Finally, the modified fractional Euler method forα∈(0,1),xj∈[0,b]with gridxj= jh: j=0,1, ...,N:h=
b N can
be expressed as follows:
yrj= ((j−1)α+1−(j−α−1)jα)h αF(x
0,yr0,yr0)
Γ(α+2) +
hα
Γ(α+2)
j−1
∑
i=1
((j−i+1)α+1−2(j−i)α+1+(j−i−1)α+1)F(x
i,yri,yri)+
hαF(xj,yrj−1+
hαF(xj−1,yrj−1,yrj−1)
Γ(α+1) ,yrj−1+
hαG(xj−1,yrj−1,yrj−1)
Γ(α+1) )
Γ(α+2) +y
r 0
yr
j= ((j−1)
α+1−(j−α−1)jα)hαG(x0,y
r 0,yr0)
Γ(α+2) +
hα
Γ(α+2)
j−1
i=1
((j−i+1)α+1−2(j−i)α+1+(j−i−1)α+1)G(xi,yri,yri)+
hαG(xj,yrj−1+
hαF(xj−1,yrj−1,yrj−1)
Γ(α+1) ,yrj−1+
hαG(xj−1,yrj−1,yrj−1)
Γ(α+1) )
Γ(α+2) +y
r 0
Example 6.2. Consider the following Fuzzy Fractional Initial Value Problem
cD 3
4 ˜y(x) =−˜y(x)
˜
y(0) = (0.5,1,1.5)
x∈[0,1]
(6.23)
Table 1: The approximate solution to the FFIVP (6.2)-yr(x).
x r
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1.500 1.450 1.400 1.350 1.300 1.250 1.200 1.150 1.100 1.050 1.000 0.1 3.654 3.559 3.464 3.369 3.273 3.176 3.080 2.982 2.884 2.786 2.686 0.2 1.067 1.031 0.995 0.960 0.924 0.889 0.857 0.818 0.782 0.746 0.711 0.3 0.948 0.916 0.884 0.853 0.821 0.790 0.758 0.727 0.695 0.663 0.632 0.4 0.857 0.828 0.800 0.771 0.743 0.714 0.686 0.657 0.628 0.600 0.571 0.5 0.783 0.757 0.731 0.705 0.679 0.653 0.627 0.601 0.575 0.548 0.522 0.6 0.722 0.698 0.674 0.650 0.626 0.602 0.578 0.554 0.529 0.505 0.481 0.7 0.670 0.647 0.625 0.603 0.580 0.558 0.536 0.514 0.491 0.469 0.446 0.8 0.624 0.603 0.582 0.562 0.541 0.521 0.499 0.479 0.458 0.437 0.416 0.9 0.584 0.565 0.545 0.526 0.506 0.487 0.468 0.448 0.429 0.409 0.389 1 0.549 0.534 0.513 0.494 0.476 0.458 0.439 0.421 0.403 0.384 0.366
Table 2: The approximate solution to the FFIVP (6.2)-yr(x).
x r
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.500 0.550 0.600 0.650 0.700 0.750 0.800 0.850 0.900 0.950 1.000 0.1 0.414 0.455 0.496 0.538 0.579 0.620 0.662 0.703 0.745 0.786 0.827 0.2 0.355 0.391 0.426 0.462 0.497 0.533 0.569 0.604 0.640 0.675 0.711 0.3 0.316 0.347 0.379 0.410 0.442 0.474 0.505 0.537 0.568 0.600 0.632 0.4 0.285 0.314 0.343 0.371 0.400 0.428 0.457 0.485 0.514 0.543 0.571 0.5 0.261 0.287 0.313 0.339 0.365 0.392 0.418 0.444 0.470 0.496 0.522 0.6 0.240 0.264 0.289 0.313 0.337 0.361 0.385 0.409 0.433 0.457 0.481 0.7 0.223 0.245 0.268 0.290 0.312 0.335 0.357 0.379 0.402 0.424 0.446 0.8 0.208 0.229 0.249 0.270 0.291 0.312 0.333 0.353 0.374 0.395 0.416 0.9 0.194 0.214 0.233 0.253 0.272 0.292 0.311 0.331 0.350 0.370 0.389 1 0.183 0.201 0.219 0.238 0.256 0.274 0.293 0.311 0.329 0.348 0.366
7 Conclusion
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