Every physical theory is formulated in terms of mathematical objects. It is thus necessary to establish a set of rules to map physical concepts and objects into mathematical objects that we use to represent them. Often times this mapping is evident, as in classical mechanics, while for other theories, such as quantum mechanics, the mathematical objects are not so intuitive. The background of our study is based on the dynamics of closed quantum systems, with intrinsic components such as states, observables, measurements and evolution. Quantum geometry on the other hand can be traced to the early days of quantum mechanics. Specifically, if we consider Heisenberg’s commutation relations  
it becomes obvious that the geometry of classical phase space is completely lost. When coordinates such as x (position) and p (momentum) on a phase space cease to commute, then there can be no such space! Moreover, we discover that this operator algebra    forms some kind of noncommutative geometric space. This is in contrast to algebraic geometry  , which is built on a correspondence betweenspaces and commutative algebras. This correspondence in particular associates with any given space, the algebra of functions on it, and geometric notions are then expressed in a purely algebraic format. This principle turns out to be the most logical starting point for a generalized geometry such as quantum geometry. While for algebraic geometry, the spaces are affine schemes, a correspondence that is closer to differential geometry is given by the Gelfand-Naimark theorem . In this case, the spaces are topological spaces and the algebras are commutative C*-algebras.
In 2014, Maliki et al. , discussed the notion of q-deformed calculus in quantum geometry. Here they showed that the mathematical study of noncommutative geometry is intimately related to the so-called q-calculus, which is a generalization of the Newton-Leibnitz classical calculus.
In this work, we summarize some important q-calculus results which will enable us to study non-commutative differential equations, specifically we shall employ the reduced q-differential transform method (RqDTM) to solve partial q-differential equations.
1.1. The q-Differential Operator
For , we define the q-differential operator as;
Note that as . We make the following remarks.
1) At the beginning of the 20th century, F.H. Jackson  studied this modified derivative and many of its consequences.
2) This q-derivative can be applied to function not containing 0 in their domain of definition. Then it reduces to the ordinary derivative when q goes to 1.
3) One can easily check that the q-derivative operator is linear, i.e.,
(a) (b) (3)
1.2. Basic Notions of q-Calculus
The mathematical study of non-commutative geometry is intimately related to the so-called q-calculus. We begin with the q-differentials of some standard functions, then we give a brief introduction to q-numbers, q-factorials and q-integrals.
1.2.1. q-Derivative of Some Standard Functions
Following the procedure for computing the q-derivative from first principles, we now obtain important results for the q-derivative of the following standard functions such , and .
1.2.2. q-Derivative of the Function
By definition, we have
**Note: for typesetting use Dq(sinx) below, similarly for D(sinx).
Using the fact that
When , we obtain the classical derivative of function, i.e.
1.2.3. q-Derivative of the Exponential Function
We have by definition;
Since , letting we have;
From the above expression it is clear that when , we obtain the classical exponential function. We remark that the q-derivative of other standard functions can be obtained similarly.
1.2.4. q-Numbers and q-Factorials
We adopt basically the notations in . Thus the set of positive integers is denoted by . Furthermore, throughout this paper K denotes a field of characteristic 0 and denotes the field of rational functions in one parameter q over . is our base field in the q-deformed setting, while K is the base field in the classical setting. We define respectively the q-integers, q-factorials and q-binomials as follows:
For example, given then .
The properties of the q-integers, q-factorials and q-binomials and their proofs are presented in . We also have the following important results on the q-differential operator. For , and with as defined previously;
1.2.6. The q-Integral Operator
A function is a q-antiderivative of if . It is denoted by and called the Jackson integral . We make the following remarks.
· Similar to classical integral calculus, any given function has multiple q-antiderivatives.
· Though the q-antiderivative of a function might not be unique, it can prove that if , a function has up to an additive constant oneq-antiderivative which is continuous at .
· In  it is shown that;
1.2.7. Properties of the q-Integral
1) The Jackson integral gives a q-antiderivative which is continuous at , and is unique up to additive constant.
2) Generally, given is another function and denotes its q-derivative, we have formally
giving a q-analogue of the Riemann–Stieltjes integral . As an example of the foregoing, let , . We have
3) The integration by parts formula of Newton-Leibnitz calculus is interpreted in the present non-commutative context as;
1.3. Partial q-Derivative of a Multivariable Function
We define the partial q-derivative of a multivariable real continuous function with respect to a variable by;
where and, .
We adopt subsequently the identity for the kth order q-derivative with respect to .
Our objective in this section is to solve partial q-differential equations using a novel q-differential transform method introduced in .
1) Reduced q-Differential Transform Method (RqDTM)
Given that all q-differentials of exist in some neighborhood of , then let
where the t-dimensional spectrum function is the transformed function. Subsequently, the lowercase represents the original function while the uppercase stands for the transformed function. We have the following important definition.
The q-differential inverse transform of is defined by;
Substituting Equations (20) in (21) we obtain
In the subsequent theorems, we set so that . From the linearity of the q-derivative, we can establish the following fact; given then , being a constant. We have the following important theorem.
3) Theorem: Given then where
Proof: By definition (20), we have;
4) Theorem: Given then .
5) Theorem: Given then
Taking the RqDTM of the given partial q-differential equation, we have
The initial condition becomes
Starting with , the values of are computed successively as follows;
When , we have
Following the same procedure, it is easy to compute an expression for .
Formally, we have the required solution of the partial q-differential equation to be
Let us now consider the classical version of the given partial q-differential equation, namely
The above is a first order quasilinear partial differential equation easily solved by the method of characteristics. The associated auxiliary equations are;
These provide us with two possible integrals given by;
being arbitrary constants of integration. Using the initial condition, we have at . Hence, and . It then follows that .
Hence the required solution is
Using MathCAD14, a numerical algebra software  to expand the expression for u in terms of powers of t, we obtain
We make the interesting observation that when we set in (31) we obtain exactly the solution to the classical PDE.
7) Example. As an example of a second order partial q-differential equation we consider the q-diffusion Cauchy problem
Again we employ the RqDTM transform. Here in the initial data, is the q-exponential function defined by , with q-derivative . The RqDTM transform of the q-diffusion equation, gives the following recursion;
The initial data given is then written
Now, substituting (38) into (37), we obtain the following values successively
In view of (28), the differential inverse transform of gives
which is the analytic solution of the problem (36). Now consider the classical diffusion equation is written
By the method of separation of variables, it is easy to show that the solution is given by
Comparing with the q-solution , we see that the two solutions are in perfect agreement when q = 1, and satisfying their appropriate initial conditions.
In this research article, we introduced the concept of q-calculus in quantum geometry. This involves the study of the basic rules governing q-calculus as compared with the classical Newton-Leibnitz calculus. Our main objective is to employ the results obtained to solve partial q-differential equations. To this end we introduced the reduced q-differential transform method which provides the solution in the form of a convergent power series with easily computable components. With the help of a few examples, we were able to show that the proposed iteration technique is very effective and convenient. It turns out that when q = 1, the solution coincides with the classical version of the given initial value problem. In conclusion, q-calculus is a non-commutative calculus that generalizes the Newton-Leibnitz classical calculus. The Reduced Differential Transform method for solving differential equations was introduced and extended in this work to solve partial q-differential equations, which represent some form of dynamics in non-commutative spaces.