qSPACE (qTIME)DEFORMATION OF THE CONTINUITY EQUATION^{1} Department of Mathematics, ALQunfudhah University College, Umm AlQura University, KSA^{1} Department of Mathematics, Rada'a College of Education and Science, Albaidha University, Albaidha, Yemen 




Received 6 August 2021 Accepted 19 August2021 Published 31 August 2021 Corresponding Author Hakeem
A. Othman, hakim_albdoie@yahoo.com DOI 10.29121/granthaalayah.v9.i8.2021.4177 Funding:
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research received no specific grant from any funding agency in the public,
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The Author(s). This is an open access article distributed under the terms of
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credited. 
ABSTRACT 


A
qspace (qtime)deformation of the continuity equations are introduced using
the q derivative (or Jackson derivative). By quantum calculus, we solve such
equations. The free cases are discussed separately. 


Keywords: qDerivative,
q Space Deformation Continuity Equation, qTime Deformation Continuity
Equation, qCalculus 1. INTRODUCTION Generally speaking,
Continuity equations can comprise both "source" and
"sink" items, which permit them to give an account of amounts that
are almost preserved, such as the density of sorts of molecules which can be
produced or demolished by chemical interactions. That is to say Continuity equations underlie more specific transfer
equations like Navier–Stokes equations, Boltzmann transport equation, and the
convection spread equation. By the diversity theory, a comprehensive continuity equation can also
be shaped in a” variant form”: σ is the
generation of Q per unit volume per unit time, it is time; j is the
ﬂux of Q, ρ is the amount of the quantity Q per unit volume where
∇ is divergence When Q is a preserved quantity that cannot be formed or ruined (such
as energy), σ = 0 and the equations become: In the field of Mathematics,
a qanalog of a theory, sameness or manifestation is an act of generalizing,
counting another parameter q that returns the original theory, expression in
the limit as q → 1. Mathematicians are, typically, interested in
qanalogs that rise naturally, rather than in randomly devising qanalogs of
known results. The earliest qanalog dealt with in detail is the basic hyper
geometric series, which was presented in the 19th century. 


In this way, it is normal to ask what is the qSpace (qtime)deformation of the continuity equation.
The present paper has been arranged as follows:
Section 1 introduces the topic theory. Section 2 brieﬂy will give an exposition of some basic concept and preliminaries of the language of qcalculus. In section 3, we introduced free qSpace (free qtime)continuity equation, and we deduced some theorems. Finally, in Section 4, we study qSpace (qtime)deformation of the continuity equation.
In future works, we will try to extend the continuity equation in the quantum white noise setting, see [8 and 9]
2. PRELIMINARIES
We will give an exposition of some basic concept and preliminaries of the language of qcalculus (see [1, 2, 3, 4, 5, 6, and 7]). Therefore, natural number n has the following q deformation:
with
Sporadically we will write [∞]_{ q} for the limit of these numbers: 1/((1q)). The q factorials and q binomial coeffcients are deﬁned naturally as
with
For q ∈ (0,1) and analytic f:C⟶C deﬁne operators Z and D_ (q) as follows:
The probability distribution of a noncommutative random variable , where a is a bounded operator on some Hilbert space satisfying, the distribution probability of a noncommutative arbitrary changeable ,
for some
(1)
3. FREESPACE (FREETIME)CONTINUITY EQUATION
In this section, the freeSpace (FreeTime)continuity equation cases are discussed.
3.1. FREESPACECONTINUITY EQUATION
Let D_{t }given by
(2)
As a free analogue of the continuity equation
we will study the following equation
(3)
Theorem 1: The freespacecontinuity equation (3) gives
Where and are given.
Proof: Using (2) in equation (3), we get
(4)
Then
Therefore, we obtain
Which complete the proof.
3.2. FREETIMECONTINUITY EQUATION
Let D_{t }given by
. (5)
As a free analogue of the continuity equation
we will study the following equation
(6)
Theorem 2. The freetimecontinuity equation (6) gives
,
where and are given.
Proof: Using (5) in equation (6), we get
And then
Therefore, we obtain
which completes the proof
4. qSPACEDEFORMATION OF THE CONTINUITY EQUATION
Let q∈ (0,1), Recall that
(7)
As a qspacedeformation of the continuity equation
we will study the following equation
(8)
Theorem 3: For q∈ (0,1), the qspacedeformation of the continuity equation (8) gives
Where and are given.
Proof: From equation (7), we get
(9)
And, then
∙
∙
∙
.
Then, we obtain
∙
∙
∙
.
Therefore, we deduce that
As k→∞, we get
which completes the proof.
5. qTIMEDEFORMATION OF THE CONTINUITY EQUATION
Let q∈ (0,1). Recall that
(10)
as a qtimedeformation of the continuity equation
we will study the following equation
(11)
Theorem 4. For q∈ (0,1), the q timedeformation of the continuity equation (2) gives
Where and are given.
Proof: From equation (10), we get
(12)
And, then
∙
∙
∙
Then, we obtain
∙
∙
∙
Therefore, we deduce that
As k→∞, we get
which complete the proof.
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