Solving Systems of Fuzzy Diﬀerential Equation - Hikari · PDF file2088 A. Sadeghi et al...

International Mathematical Forum, Vol. 6, 2011, no. 42, 2087 - 2100

Solving Systems of Fuzzy Differential Equation

Amir Sadeghi 1 , Ahmad Izani Md. Ismail and Ali F. Jameel

School of Mathematical Sciences, Universiti Sains Malaysia

11800 USM, Penang, Malaysia

Abstract

In this paper, a solution procedure for the solution of the system of fuzzy differ-ential equations x(t) = α(A − In) [t(A − In) + In]−1 x(t), x(0) = x0 where A is realn × n matrix which does not have eigenvalues on R

− and x0 is a vector consistingof n fuzzy numbers is developed. In addition, the necessary and sufficient conditionfor the existence of fuzzy solution is presented. An example is provided to show theeffectiveness of the proposed theory.

Keywords: Fuzzy number, Fuzzy linear system, Fuzzy differential equations, Func-

tion of matrices

1 Introduction

In order to solve an n×n fuzzy linear system of equation, Friedman et al. in [14] proposed

a general model which replaces the original n × n fuzzy linear system by 2n × 2n crisp

function linear system. Authors such as Abbasbandy et al. in [1], [2], Allahviranloo et al.

in [3], [4], [5] and Dehghan and Hashemi [11] have extended the work of Friedman.

There have been application of fuzzy differential equation (FDEs) to model problems

where the degree of ambiguity is high [19]. FDEs can be studied by several approaches.

The Hukuhara differentiability for fuzzy number valued functions was the first approach

which has been utilized. Fuzzy differential equations were first formulated by Kaleva [18]

and Seikkala [22] in time dependent form. A very general formulation of a fuzzy first-order

initial value problem, has been given by Buckley and Feuring [9]. In their formulas, first

the crisp solution is found, fuzzified and then it is compared to see whether it satisfies

the FDEs. Also, Puri and Ralescu [21] have discussed several different methods. The

1Corresponding author ( E-mail address: [email protected])

2088 A. Sadeghi et al

existence and uniqueness of the solution of a fuzzy differential equation were investigated

in [21].

Pearson [20] using complex number representation has studied linear fuzzy differential

equations of the form x(t) = Ax(t) where A is a real n×n matrix and the initial condition

x(0) is a vector of n fuzzy numbers. Otadi et. al. [19] have investigated the solution of

a special system of linear homogenous FDEs of the form x(t) = Ax(t) + Bx(t), with

initial condition x(0) = x0 where A and B are real n× n matrix and the initial condition

x0 is a vector of fuzzy numbers. The embedding method was used to solve the system.

Fard [13] has extended this approach and has solved non-homogenous FDEs of the form

x(t) = Ax(t)+Bx(t)+f(t), with initial condition x(0) = x0 by using variational iteration

method.

We consider the initial problem x(t) = α(A − In) [t(A − In) + In]−1 x(t), x(0) = x0

where A is an n×n matrix that has non-negative eigenvalues in real axis. This initial value

problem was introduced by Davis and Higham in [10]. In our paper, x0 is fuzzified and

the embedding method is used to solve the fuzzified initial value problem. By using some

special functions of matrix, the fuzzy solution of fuzzy differential equation is obtained.

Furthermore, the necessary and sufficient condition for the existence of fuzzy solution x(t)

is presented.

The outline of the paper is as follows: In Section 2 we will present basic concepts.

In Section 3 we will discuss the necessary and sufficient condition for the existence of

solution of FDEs. A numerical example will be given in Section 4 and the conclusions

will be presented in Section 5.

2 Preliminaries

In this section, several basic concepts of fuzzy system, fuzzy differential equation and

function of matrices are recalled.

2.1 Basic definitions

A nonempty subset A of R is called convex if and only if (1 − ξ)x + ξy ∈ A for every

x, y ∈ A and ξ ∈ [0, 1] [18].

Definition 2.1: [18] A fuzzy number is a function u : R → [0, 1] satisfying the following

properties:

Systems of fuzzy differential equation 2089

1. u is normal, i.e. ∃x0 ∈ R with u(x0) = 1;

2. u is a convex fuzzy set i.e u(ξx + (1− ξ)y) ≥ min{u(x), u(y)}, ∀x, y ∈ R, ξ ∈ [0, 1];

3. u is upper semi-continuous on R;

4. {x ∈ R : u(x) > 0} is compact, where A denotes the closure of A.

Following [5], the family of fuzzy numbers will be denoted by E. For 0 < r ≤ 1, [u]r

denotes {x ∈ R : u(x) > r} and[u]0 denotes {x ∈ R : u(x) > 0}. It should be noted that

for any 0 ≤ r ≤ 1, [u]r is a bounded closed interval. For u, v ∈ E and λ ∈ R, the sum

u + v and the product λu can be defined by [u + v]r = [u]r + [v]r, [λu]r = λ[u]r, r ∈ [0, 1]

, where [u]r + [v]r means the addition of two intervals of R and [u]r means the product

between a scalar and a subset of R.

Arithmetic operation of arbitrary fuzzy numbers u = (u(r), u(r)), v = (v(r), v(r)) and

λ ∈ R, can be defined as [14]:

u = v iff u(r) = v(r) and u(r) = v(r),

u + v = (u(r) + v(r), u(r) + v(r)),

u − v = (u(r) − v(r), u(r) − v(r)),

λu =

⎧⎨⎩(λu(r), λu(r)), λ ≥ 0,

(λu(r), λu(r)), λ < 0.

(2.1)

Note that crisp number σ is simply represented by u(r) = u(r) = σ, 0 ≤ r ≤ 1.

In the following we recall some definitions concerning fuzzy differential equations.

Definition 2.2: A mapping f : T → E for some interval T ⊆ R is called a fuzzy process.

Therefore, its r-level set can be written as follows [13],

[f(t)]r = [f r−(t), f r

+(t)]r, t ∈ T, r ∈ [0, 1]. (2.2)

Definition 2.3: [18] A function f : T → E is said to be Hukuhara differentiable at

t0 ∈ T , if there exists an element f ′(t0) ∈ E such that for all h > 0 sufficiently small. In

other words there exist f(t0 + h)Hf(t0), f(t0)Hf(t0 − h) such that

limh→0+

f(t0 + h)Hf(t0)

h= lim

h→0+

f(t0)Hf(t0 − h)

h= f ′(t0) (2.3)


Where w = uHv is Hukuhara difference which for u, v ∈ E is defined by u = v + w. If

f : T → E be Hukuhara differentiable then, the boundary function f r+(t) and f r

+(t) are

(Seikkala) differentiable [22] and

[f ′(t)]r = [(f r−(t))′, (f r

+(t))′]r, t ∈ T, r ∈ [0, 1]. (2.4)

2.2 Fuzzy linear systems

Definition 2.4: [14] The n × n system of linear equations

n∑j=1

aijxj = yi, i = 1, . . . , n. (2.5)

where the coefficient matrix A = (aij), 1 ≤ i, j ≤ n, is a crisp n × n matrix and yi ∈E, 1 ≤ i ≤ n, is called a fuzzy system of linear equations (FLE).

Two crisp n×n linear systems for all i can be extended to an 2n× 2n crisp linear system

as follows:

If aij ≥ 0 for i, j = 1, . . . , 2n, then we can put si,j = si+n,j+n = aij .

If aij < 0 for i, j = 1, . . . , n, then we can put si,j+n = si+n,j = aij.

In other words, S1, S2, A = S1 − S2, S1 + S2 = |A| = (|aij|), i, j = 1, . . . , n, and

x =

⎛⎜⎜⎜⎜⎝

x1

x2...

xn

⎞⎟⎟⎟⎟⎠ , x =

⎛⎜⎜⎜⎜⎝

x1

x2

...

xn

⎞⎟⎟⎟⎟⎠ , y =

⎛⎜⎜⎜⎜⎝

y1

y2...

yn

⎞⎟⎟⎟⎟⎠ , y =

⎛⎜⎜⎜⎜⎝

y1

y2...

yn

⎞⎟⎟⎟⎟⎠

By using matrix notation, an 2n × 2n linear system of equation can be obtained as

SX = Y . (2.6)

where S =(

S1 −S2−S2 S1

), X =

(xx

), and Y =

( y

y

).

Theorem 2.1: [14] The unique solution X of Equation (2.6) is a fuzzy vector for arbitrary

Y if and only if S−1 is nonnegative, i.e, (sij)−1 ≥ 0.

Theorem 2.2: [14] The matrix S is nonsingular if and if only the matrices S1 − S2 and

S1 + S2 are both singular.


2.3 Function of matrix

Suppose f(z) is an analytic function on a closed contour Γ which encircles spec(A), where

spec(A) denote the set of eigenvalues of matrix A. A function of matrix can be defined

by using Cauchy integral definition

f(A) =1

2πi

∮Γ

f(z)(zI − A)−1dz (2.7)

The entries of (zI − A)−1 are analytic on Γ and f(A) is analytic in a neighborhood of

spec(A) [15]. Matrix exponential, matrix logarithm and matrix pth root are examples of

function of matrices and are defined as follows.

Definition 2.5: (Exponential) [15] The exponential of A ∈ Cn×n, denoted by eA or

exp(A), is defined by exp(A) =∑∞

i=0Ai

i!.

Definition 2.6: (Principal logarithm) [17] Suppose A is an n × n matrix. A unique

matrix X which satisfies the nonlinear matrix equation eX − A = 0 such that the eigen-

values of X lie in the strip {z : −π < Im(z) < π} is called the principal logarithm of A,

and is denoted by X = log A.

Definition 2.7: (principal pth root) [24] Let A be an n × n matrix. A unique matrix

X that satisfy the nonlinear matrix equation Xp − A = 0 such that all of eigenvalues of

X lie in the segment {z : −π/p < arg(z) < π/p}, is called the principal pth root of A and

is denoted by X = A1/p.

3 System of fuzzy ordinary differential equation

In this section, the homogenous system of fuzzy differential equation⎧⎨⎩x(t) = α(A − In) [t(A − In) + In]−1 x(t),

x(0) = x0

(3.1)

where x(t) = dxdt

, will be solved. In the following theorem the general solution of the

nonhomogeneous equation is presented.

Theorem 3.1: Let A ∈ Cn×n have non-negative real eigenvalues and α ∈ R. The


non-homogeneous initial value ordinary differential equation problem⎧⎨⎩x(t) = α(A − In) [t(A − In) + In]−1 x(t) + f(t, x),

x(0) = x0

(3.2)

has a unique solution

x(t) = [t(A − In) + In]α x0 +

∫ t

0

[(t(A − In) + In)(s(A − In) + In)−1

]αf(s, x)ds (3.3)

Proof: For proving this theorem we utilize the following facts [16]

1.∫ t

0(A − In) [s(A − In) + In]−1 ds = log [t(A − In) + In],

2. log A commutes with [t(A − In) + In]−1.

By considering equation (3.2), we can multiply the factor e−α log[t(A−In)+In] to both sides

of equation (3.2) as follows

(x(t) − α(A − In) [t(A − In) + In]−1 x(t)

)e−α log[t(A−In)+In] = f(t, x)e−α log[t(A−In)+In]

(3.4)

Then we have

d

dt

(e−α log[t(A−In)+In]x(t)

)= f(t, x)e−α log[t(A−In)+In] (3.5)

Integrating both sides, we have∫ s

0

d

dt

(e−α log[t(A−In)+In]x(t)

)dt =

∫ s

0

f(t, x)e−α log[t(A−In)+In]dt (3.6)

Hence, it can be obtained that

e−α log[s(A−In)+In]x(s) − x(0) =

∫ s

0

f(t, x)e−α log[t(A−In)+In]dt (3.7)

x(s) = eα log[s(A−In)+In]x(0) +

∫ s

0

f(t, x)eα log[s(A−In)+In]e−α log[t(A−In)+In]dt (3.8)

Therefore, it can be obtained

x(t) = [t(A − In) + In]α x0 +

∫ t

0

[(t(A − In) + In)(s(A − In) + In)−1

]αf(s, x)ds.


�For t = 1 we can obtain

x(1) = Aαx0 + Aα

∫ 1

0

[s(A − In) + In]−α f(s, x)ds.

It should be emphasized that for homogenous case (f(x, t) = 0) Theorem 3.1 can be

replaced by Theorem 2 in [10]. As we mentioned in Section 2 the derivative x(t) of a

fuzzy process x(t) can be defined as

x(t, r) = (x(t, r), x(t, r)), 0 ≤ r ≤ 1 (3.9)

provided that is equation defines a fuzzy number [22]. A is an n × n real matrix and a

vector made up of n fuzzy numbers described the initial condition x0. We study a specific

linear problem, where the parameters known and the initial condition of the system is

fuzzy. It is well known that there is no inverse element for an arbitrary fuzzy number

u ∈ E. In other words, there exists no element v ∈ E such that u + v = 0. In fact, for all

non-crisp fuzzy number u ∈ E we have u + (−u) = 0 [13,19]. Thus, the system of linear

fuzzy differential equation (3.1) cannot be equivalently replaced by the system of linear

fuzzy equations equations

⎧⎨⎩x(t, r) = α(A − In) [t(A − In) + In]−1 x(t, r),

x(0, r) = x0

(3.10)

which had been considered. Consider the system of linear fuzzy differential equations

(3.1). We are going to transform its n × n system with coefficient matrix A into 2n × 2n

system with coefficient matrix S. Define matrix S = (sij) where can be determined as

follows

If aij ≥ 0 for i, j = 1, . . . , 2n, then we can put si,j = si+n,j+n = aij .

If aij < 0 for i, j = 1, . . . , 2n, then we can put si,j+n = si+n,j = −aij .

while all the remaining sij are taken zero. Using matrix notation the following system

can be obtained

⎧⎨⎩x(t, r) = α(S − I2n) [t(S − I2n) + I2n]−1 x(t, r)

x(0, r) = x0

(3.11)


where S and I2n are 2n × 2n matrices, and

x(t, r) =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

x1(t, r)...

xn(t, r)

x1(t, r)...

xn(t, r)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, x(t, r) =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

x1(t, r)...

xn(t, r)

x1(t, r)...

xn(t, r)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, x0 =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

x01...

x0n

x01

...

x0n

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (3.12)

Thus, the solution of fuzzy differential equation is

x(t, r) = [t(S − I2n) + I2n]α x0 (3.13)

However, x(t, r) may still not be the appropriate fuzzy vector. The following theorem

provides necessary and sufficient conditions for x(t, r) be a fuzzy vector. First we will

show the matrix t(S − I2n) + I2n must be positive definite.

Theorem 3.2: [15] An arbitrary matrix is positive definite if and only if all its eigenvalues

are positive.

Theorem 3.3: If t((S1 − S2) − In) + In and t((S1 + S2) − In) + In for t > 0 are positive

definite matrix then t(S − I2n) + I2n is a symmetric positive definite matrix.

Proof: Clearly t(S − I2n)+ I2n is a symmetric matrix. Suppose that t((S1 −S2)− In)+ In

and t((S1 + S2) − In) + In are positive definite matrix. Let λ > 0 be a eigenvalue of

t(S − I2n) + I2n =

(tS1 + (1 − t)In tS2

tS2 tS1 + (1 − t)In

). (3.14)

Then, there exist x1,x2 = 0 such that(tS1 + (1 − t)In tS2

tS2 tS1 + (1 − t)In

)(x1

x2

)= λ

(x1

x2

)(3.15)

Thus, we have ⎧⎨⎩[tS1 + (1 − t)In]x1 + tS2x1 = λx1

tS2x1 + [tS1 + (1 − t)In]x2 = λx2

(3.16)


By addition and subtraction of (3.16), it can be obtained⎧⎨⎩tS1(x1 + x2) − tS2(x1 + x2) + (1 − t)(x1 + x2) = λ(x1 + x2)

tS1(x1 − x2) + tS2(x1 − x2) + (1 − t)(x1 − x2) = λ(x1 − x2)(3.17)

or ⎧⎨⎩t(S1 − S2)(x1 + x2) + (1 − t)(x1 + x2) = λ(x1 + x2)

t(S1 + S2)(x1 − x2) + (1 − t)(x1 − x2) = λ(x1 − x2)(3.18)

Clearly, λ is the eigenvalue of (S1 − S2) and (S1 + S2), which proves the theorem. �

Now, it is known that Aα can be obtained by using Aα = exp(α log A), where the log

is the principal logarithm. For α = 1/p, when p an integer, we have A1/p. It should be

mentioned that by using commutativity relation, it can be written

(Aα)p = exp(α log A)p = exp(pα log A) = exp(log A) = A

Therefore, Aα is some pth root of A. To determine which root it is, we need to find

its spectrum. The eigenvalues of Aα are of the form e1p

log λ, where λ is an eigenvalue of

A. Now log λ = x + iy with y ∈ (−π, π), and so e1p

log λ = ex/peiy/p lies in the segment

{z : −π/p < arg(z) < π}. The spectrum of Aα is therefore precisely that of A1/p, so these

two matrices are one and the same [16]. It is known that if positive definite matrix A has

positive entries then Aα has positive entries. Therefore, by using this point we have the

fuzzy solution for fuzzy differential equation.

4 Numerical example

Consider the equation x(t) = α(A − In) [t(A − In) + In]−1 x(t) when

A =

(4 0

−1 2

)(4.1)

and the initial values are x01 ≈ 2 and x02 ≈ 2. This can be done by setting, for example,

x(0; r) =

(x01

x02

)=

((1 + r, 3 − r)

(2r, 6 − 4r)

)(4.2)


We have the following equation

x(t) = α

(3 0

−1 1

)[t

(3 0

−1 1

)+

(1 0

0 1

)]−1

x(t) (4.3)

Therefore, by using the embedding method (4.3) can be transformed to⎛⎜⎜⎜⎝

x1(t, r)x2(t, r)x1(t, r)x2(t, r)

⎞⎟⎟⎟⎠ = α

⎛⎜⎜⎜⎝

3 0 0 00 1 1 00 0 3 01 0 0 1

⎞⎟⎟⎟⎠⎡⎢⎢⎢⎣t

⎛⎜⎜⎜⎝

3 0 0 00 1 1 00 0 3 01 0 0 1

⎞⎟⎟⎟⎠+

⎛⎜⎜⎜⎝

1 0 0 00 1 0 00 0 1 00 0 0 1

⎞⎟⎟⎟⎠⎤⎥⎥⎥⎦

−1⎛⎜⎜⎜⎝

x1(t, r)x2(t, r)x1(t, r)x2(t, r)

⎞⎟⎟⎟⎠ (4.4)

Now, by integrating from this system of linear differential equation we can obtain⎛⎜⎜⎜⎝

x1(t, r)x2(t, r)x1(t, r)x2(t, r)

⎞⎟⎟⎟⎠ = exp

⎛⎜⎜⎜⎝log

⎡⎢⎢⎢⎣t

⎛⎜⎜⎜⎝

3 0 0 00 1 1 00 0 3 01 0 0 1

⎞⎟⎟⎟⎠+

⎛⎜⎜⎜⎝

1 0 0 00 1 0 00 0 1 00 0 0 1

⎞⎟⎟⎟⎠⎤⎥⎥⎥⎦

α⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝

x1(0, r)x2(0, r)x1(0, r)x2(0, r)

⎞⎟⎟⎟⎠ (4.5)

By utilizing the property of matrix exponential function Ap = exp(p log A) we have⎛⎜⎜⎜⎝

x1(t, r)x2(t, r)x1(t, r)x2(t, r)

⎞⎟⎟⎟⎠ =

⎡⎢⎢⎢⎣t

⎛⎜⎜⎜⎝

3 0 0 00 1 1 00 0 3 01 0 0 1

⎞⎟⎟⎟⎠+

⎛⎜⎜⎜⎝

1 0 0 00 1 0 00 0 1 00 0 0 1

⎞⎟⎟⎟⎠⎤⎥⎥⎥⎦

α⎛⎜⎜⎜⎝

1 + r

2r

3 − r

6 − 4r

⎞⎟⎟⎟⎠ (4.6)

For specific case t = 1 the (4.6) becomes to⎛⎜⎜⎜⎝

x1(1, r)x2(1, r)x1(1, r)x2(1, r)

⎞⎟⎟⎟⎠ =

⎛⎜⎜⎜⎝

4 0 0 00 2 1 00 0 4 01 0 0 2

⎞⎟⎟⎟⎠

α⎛⎜⎜⎜⎝

1 + r

2r

3 − r

6 − 4r

⎞⎟⎟⎟⎠.

It remain to compute αth root of a matrix. For t = 0.1, 0.2, . . . , 1 the approximate

solution for α = 1/7 and α = 3/5 are obtained in Tables 1 to 4.

5 Conclusion

A technique for approximating the solution of a time dependent system of fuzzy linear differentialequations is presented. The necessary and sufficient condition for the existence of a fuzzy solutionis proposed. The original fuzzy system of differential equations with matrix coefficient A istransformed by a 2n× 2n crisp linear system of differential equations with matrix coefficient S.We have illustrated the proposed technique by an example.


Table 1: The approximate solution x1(t, r) for α = 1/7

t x1(t, r) x1(t, r)

0.1 1.038191865525826+ 1.038191865525826r 3.114575596577478 -1.038191865525826r

0.2 1.069448800053393+ 1.069448800053393r 3.208346400160180 -1.069448800053393r

0.3 1.096028741644688+ 1.096028741644688r 3.288086224934064 -1.096028741644688r

0.4 1.119225318154100+ 1.119225318154100r 3.357675954462300 -1.119225318154100r

0.5 1.139852281047597 +1.139852281047597r 3.419556843142790 -1.139852281047597r

0.6 1.158456468185973+ 1.158456468185973r 3.475369404557918 -1.158456468185973r

0.7 1.175423928888369+1.175423928888369r 3.526271786665106 -1.175423928888369r

0.8 1.191037840204096 + 1.191037840204096r 3.573113520612287 -1.191037840204096r

0.9 1.205512380406764+ 1.205512380406764r 3.616537141220292 -1.205512380406764r

1 1.219013654204475 + 1.219013654204475r 3.657040962613426 -1.219013654204475r

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t x2(t, r) x2(t, r)

0.1 -0.036724513845537+ 2.039659217206115r 6.070011633157630 -4.067076929797052r

0.2 -0.064591055694530+ 2.074306544412256r 6.136798225644061 -4.127082736926335r

0.3 -0.086755314178293 + 2.105302169111083r 6.200232755095525 -4.181685900162734r

0.4 -0.104975821347715+2.133474814960485r 6.260456683084501 -4.231957689471731r

0.5 -0.120327387570823+2.159377174524371r 6.317695006812016 -4.278645219858468r

0.6 -0.133511502198869+ 2.183401434173077r 6.372188966254070 -4.322299034279863r

0.7 -0.145009157972619+ 2.205838699804119r 6.424170555448864 -4.363341013617364

0.8 -0.155163139424513+ 2.226912540983679r 6.473853437051687 -4.402104035492521r

0.9 -0.164225458143115+ 2.246799302670413r 6.521430630487088 -4.438856785959789r

1 -0.172386210795995 + 2.265641097612956r 6.567075011777543 -4.473820124960581r

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t x1(t, r) x1(t, r)

0.1 1.170485428222120+ 1.170485428222120r 3.511456284666359 -1.170485428222120r

0.2 1.325781606935995+ 1.325781606935995r 3.977344820807986 -1.325781606935995r

0.3 1.469779414993713+ 1.469779414993713r 4.409338244981139 -1.469779414993713r

0.4 1.604920810670347+ 1.604920810670347r 4.814762432011040 -1.604920810670347r

0.5 1.732862107887867+ 1.732862107887867r 5.198586323663600 -1.732862107887867r

0.6 1.854790311984838+ 1.854790311984838r 5.564370935954514-1.854790311984838r

0.7 1.971591859370261+ 1.971591859370261r 5.914775578110783 -1.971591859370261r

0.8 2.083950266104649+ 2.083950266104649r 6.251850798313946 -2.083950266104649r

0.9 2.192406265011812+ 2.192406265011812r 6.577218795035435 -2.192406265011812r

1 2.297396709994070+ 2.297396709994070r 6.892190129982209 -2.297396709994070r

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Received: March, 2011



t x2(t, r) x2(t, r)

0.1 -0.167448862950502+ 2.173521993493736r 6.297300829880538 -4.291227699337306r

0.2 -0.315271477809251+ 2.336291736062739r 6.588513237775883 -4.567492979522394r

0.3 -0.448940980157391+ 2.490617849830036r 6.873265575946921 -4.831588706274276r

0.4 -0.571823348888832 + 2.638018272451861r 7.151623685503808-5.085428761940779r

0.5 -0.686156410893113+ 2.779567804882619r 7.423828200123705 -5.330416806134200r

0.6 -0.793513057573265+ 2.916067566396412r 7.690185289091549 -5.567630780268402r

0.7 -0.895046142718621+ 3.048137576021903r 7.951017871107548 -5.797926437804266r

0.8 -0.991628850355592+ 3.176271681853705r 8.206643245086990 -6.022000413588879r

0.9 -1.083940275027148+ 3.300872254996476r 8.457363064953229 -6.240431084983902r

1 -1.172520215225508+ 3.422273204762633r 8.703459327320553 -6.453706337783430r

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