Research Journal of Applied Sciences, Engineering and Technology 4(18): 3257-3259, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: December 30, 2011
Accepted: January 31, 2012
Published: September 15, 2012
Probabilistic Solution of Rational Difference Equations System with
Random Parameters
Seifedine Kadry and Chibli Joumaa
American University of the Middle East, Kuwait
Abstract: In this article, we study the periodicity of the solutions of the of rational difference equations system
of type xn =
a
b
(p $ 1), then we propose new “exact” procedure to find the probability density
, yn =
yn − p
xn + p − 2
function of the solution, where a, b, x0 = N and y0 = M are independent random variables.
Keywords: Linearization, random system, rational difference equations system, periodicity, probability density
function, transformation of random variables technique
Ozban (2007) studied the positive solutions of the system
of rational difference equations:
INTRODUCTION
Stochastic systems of difference equations usually
appear in the investigation of systems with discrete time
or in the numerical solution of systems with continuous
time. A lot of difference systems have variable structures
subject to stochastic abrupt changes, which may result
from abrupt phenomena such as stochastic failures and
repairs of the components, changes in the
interconnections of subsystems, sudden environment
changes, etc. Recently, there has been great interest in
studying difference equation systems. One of the reasons
for this is the necessity for some techniques that can be
used in investigating equations arising in mathematical
models describing real life situations in population
biology, economics, probability theory, genetics,
psychology etc. There are many papers related to the
difference equations system for example:
Cinar (2004) studied the solutions of the system of
difference equations:
xn + 1
xn =
Clark and Kulenovic (2002) investigated the global
asymptotic stability:
xn + 1 =
xn+1 = A +
yn
yn
,y = A+
, n = 0, 1, ..., p, q
x n − p n +1
xn− p
xn
yn
, yn + 1 =
a + cyn
b + dxn
Yang and Xu (2007) propose a method to deal with
the mean square exponential stability of impulsive
stochastic difference equations.
In this paper, we investigate the analytic solution of the
stochastic difference equations system:
xn =
yn
1
, yn + 1 =
=
yn
xn − 1 yn − 1
Papaschinopoulos and Schinas (1998) studied the
oscillatory behavior, the boundedness of the solutions,
and the global asymptotic stability of the positive
equilibrium of the system of nonlinear difference
equations:
byn − 3
a
, yn =
xn − q y n − q
yn − 3
a
yn − p
, yn =
b
xn + p − 2
(1)
where a, b, x0 = N and y0 = M are independent random
variables.
PERIODICITY OF THE SOLUTIONS
In this section, we study the periodicity of the
solutions of system (I).
Theorem 1: All solutions of (I) are periodic with
period 2.
Proof:
Corresponding Author: Seifedine Kadry, American University of the Middle East, Kuwait.
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Res. J. Appl. Sci. Eng. Technol., 4(18): 3257-3259, 2012
a
⎧
⎪ xn +2 = y
n− p+2
⎪
⎪
⎨
b
⎪y
=
=
⎪ n+ 2 xn+ p
⎪
⎩
transformation of random variables to get the p.d. f of xn
and yn.
a
a
= xn
b
b
xn
b
b
= yn
a
a
yn
=
Transformation of random variables technique
(TRVT): Definition: Let X be a continuous random
variable with generic probability density function f(x)
defined over the supportc1<x<c2. And, let Y = u(X) be an
invertible function of X with inverse function X =
v(Y).Then, using the Transformation of Random Variable
Technique (TRVT) defined by Kadry and Younes (2005),
Kadry (2007), El-Tawil et al. (2009) and Kadry and
Smaili (2007), the probability density function of Y is:
Therefore, the proof was completed.
Theorem 2: For , P = 1 all solutions of (I) are:
a n +1
b n+1
, y 2 n +1 = n
n
b M
a N
n+1
a N
b n +1 M
= n +1 , y 2 n + 2 =
b
a n +1
x 2 n +1 =
x2n+ 2
p.d.f (y) = p.d.f (v (y)).|v’(y)|.
defined over the supportu (c1) <y<u (c2).
To simplify our technique, let us consider the
following stochastic equation:
Proof: By induction, suppose the result holds for n - 1:
an
bn
,
y
=
b n −1 M 2 n −1 a n −1 N
bn M
an N
= n , y2n =
an
b
x2 n −1 =
x2 n
z=
x2 n+ 2 =
a
a n+1
b
b n+1
= n , y2 n+1 =
= n
y2n b M
x2n a N
a
y 2 n +1
=
xα
(α , β ∈ Z )
where, x and y are two independent random variables. To
find the p.d.f of z, firstly we linearize the denominator
then we find the p.d.f of the product of two random
variables:
We suppose * = 1/ x" then we apply the TRVT
technique to get the p.d.f of *:
For n:
x 2 n +1 =
yβ
a n+1 N
b
b n+1 M
,
y
=
=
2n+ 2
x2n+1
b n+1
a n +1
f δ (δ ) = −
1
δ 2α
α
δ −1+α f x ( x )
Hence, the proof is completed by induction.
ANALYTIC STOCHASTIC SOLUTIONS
In this section, we develop an analytic technique to
find the stochastic solutions of (cf. theorem 2):
n +1
Theorem 3: Let X be a continuous random variable with
distribution function f(X) which is defined on the interval
[a, b], where, 0 < b < 4. Similarly, let Y be a random
variable of the continuous type with distribution function
g(Y) which is defined on the interval [c, d] , where, 0 < c
< d < 4. The p.d.f of z = XY, h(z), is:
n +1
x 2 n +1 =
a
b
, y 2 n +1 = n
n
b M
a N
x2n+ 2 =
a n +1 N
b n +1 M
,
y
=
2n+ 2
b n+1
a n +1
Once the linearization step has been done, it’s
required to find the p.d.f of the product of two random
variables. To do that, we developed the following
theorem:
Case 1: ad < bc:
The solution of a stochastic system of difference
equations is obtained when evaluating the statistical
characteristic of the solution process like the mean,
standard deviation, high order moments and the most
important characteristic "the probability density function"
(p.d.f.). Our proposed technique is based on the
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⎧ z/c ⎛
⎪∫α g ⎜⎝
⎪
⎪
⎛
h( z) = ⎨∫zz//dc g ⎜
⎝
⎪
⎪b ⎛
⎪∫z / d g ⎜
⎝
⎩
1
z⎞
⎟ f ( X ) dx , ac < z < ad
X⎠
X
1
z⎞
⎟ f ( X ) dx , ad < z < bc
X⎠
X
1
z⎞
⎟ f ( X ) dx , bc < z < bd
⎠
X
X
Res. J. Appl. Sci. Eng. Technol., 4(18): 3257-3259, 2012
Case 2: ad = bc:
⎧ z/c ⎛
⎪∫a g ⎜⎝
⎪
h( z ) = ⎨
⎪ b g ⎛⎜
∫z / d
⎪
⎝
⎩
⎧ z/c ⎛ z ⎞
1
⎪∫a g ⎜⎝ X ⎟⎠ f ( X ) X dx , ac < z < ad
⎪
⎪
1
⎛ z⎞
h( z) = ⎨∫zz//cc g ⎜ ⎟ f ( X ) dx , ad < z < bc
⎝
⎠
X
X
⎪
⎪b ⎛ z⎞
1
⎪∫z / d g ⎜ ⎟ f ( X ) dx , bc < z < bd
⎝ X⎠
X
⎩
1
z⎞
⎟ f ( X ) dx , ac < z < ad
⎠
X
X
1
z⎞
⎟ f ( X ) dx , ad < z < bd
X⎠
X
REFERENCES
Case 3: ad > bc:
⎧ z/c ⎛ z ⎞
1
⎪∫a g ⎜⎝ X ⎟⎠ f ( X ) X dx , ac < z < bc
⎪
⎪ ⎛ z⎞
1
h( z ) = ⎨∫ab g ⎜ ⎟ f ( X ) dx , bc < z < ad
⎝
⎠
X
X
⎪
⎪b ⎛ z⎞
1
⎪∫z / d g ⎜ ⎟ f ( X ) dx , ad < z < bd
⎝
⎠
X
X
⎩
Proof: Only the case of ad < bc is considered. The other
cases are proven analogously. Using kadry (2007), the
transformation w = X and z = XY is bijective. The
Jacobian of this transformation become:
j=
1
0
− z / w2
1/ w
= 1/ w
The joint p.d.f of w and z is:
f W , z (W , z ) = f ( w) f ( z / w)
1
w
Integrating with respect to w over the appropriate
intervals and replacing w with X in the final result yields
the marginal p.d.f of z:
Clark, D. and M.R.S. Kulenovic, 2002. A coupled system
of rational difference equations. Comput. Mathemat.
Appl., 43: 849-867.
El-Tawil, K., S. Kadry and A. Abou jaoude, 2009.
International Conference on Numerical Analysis and
Applied Mathematics, Volume 1 and Volume 2. AIP.
Kadry, S. and R. Younes, 2005. Etude Probabiliste d’un
Systeme Macanique a Parametres Incertains par une
Technique Baswe sur la Methode de transformation.
Proceeding of CANCAM, CANADA.
Kadry, S., 2007. On the generalization of probabilistic
transformation method. Appl. Math. Comput., 190:
1284-1289.
Kadry, S. and K. Smaili, 2007. One dimensional
transformation method in reliability analysis. J. Basic
Appl. Sci., 3(2).
Ozban, A.Y., 2007. On the system of rational difference
equations. Appl. Math.. Comput., 188: 833-837.
Papaschinopoulos, G. and C.J. Schinas, 1998. On a
system of two nonlinear difference equations. J.
Math. Anal. Appl., 219: 415-426.
Yang, Z. and D. Xu, 2007. Mean square exponential
stability of impulsive stochastic difference. Appl
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