The Q-Time Deformed Wave Equation

For q ∈ ( 0 , 1 ) , we introduce the q -time deformed wave equation; using the q -derivative, the solution of the q -time deformed wave equation is given. Also, we introduce the free-time wave equation which is a limit case q ⟶ 0 of the q -time deformed wave equation.


Introduction
e wave equation is a second-order partial di erential equation; it is used in many branches of sciences. It is plays essential part in physics. e equation, is standard example of wave equation. ere are two real characteristic slopes at each point (x, t). On the whole, this equation is famous by the wave equation, in one dimension, and describes the propagation (bi-directional) of waves with nite speed ±c. e one-dimensional wave equation (1) can be solved exactly by the Jean-Baptiste and Rond d'Alembert method, using a Fourier transform method, or via separation of variables. On the contrary, in the domain of combinatorics and quantum calculus, the q-derivative is a q-analogue of the ordinary derivative, presented by Jackson Frank [1,2]. e q-di erentiation is parallel to ordinary derivatives, with inquisitive di erences. In approximation theory, the uses of q-calculus are new domain in most recent 25 years. A few di erent scientists have proposed the exponential-type operators.
is study is arranged as follows. In Section 2, we introduce the notion of the q-calculus. In Section 3, we introduce the q-time deformed wave equation for q ∈ (0, 1) and the solution of such equation is given. In Section 4, we study the free-time wave equation.

q-Time Deformed Wave Equation
Let q ∈ (0, 1). As a q-analogous of equation (11), we introduce the following equation: where For t > 0, we study equation (11) which will be called qtime deformed wave equation.
is the solution of equation (11), with any constants A and B, where k > 0. (2) u is given by which is the solution of equation (11) with any constants A and B, where k < 0 and j 2 � − 1.
Proof. Let 0 < q < 1. en, we obtain Now, let en, equation (11) is equivalent to Hence, where k is constant. We deduce that We apply equation (16) in (21); we obtain We obtain that By recurrence relation, we obtain Now, by simplification, we obtain 2 Journal of Mathematics where Now, let f be given by en, we have as n ⟶ ∞; the last term becomes Hence, equations (25) and (29) give So, we obtain and this implies which gives erefore, en, we obtain is gives en, we obtain Note that, equation (38) can be found as follows: So, we take Finally, we obtain Now, for k > 0, let Journal of Mathematics en, we obtain which verifies equation (42). In this case, we obtain Now, for k < 0, let en, we obtain which verifies equation (42). In this case, we obtain Finally, we get, for k > 0, For k < 0, we obtain with any constants A and B. is completes the proof. □

Free-Time Wave Equation
Now, we will study the free-time wave equation: where Note that, since the limit (when q ⟶ 0) of D q,t is D 0,t , the free-time wave equation is the limit case (when q ⟶ 0) of the q-time deformed wave equation (11).

Theorem 2. (1) We have
is the solution of equation (51), with any constants a 0 , A and B, where k < 0.
which is the solution of equation (51), with any constants a 0 , A, and B, where f(0) � 0 and k > 0.
Proof. Let u(t, x) be given by en, equation (51) is equivalent to Hence, we obtain From equation (57), we obtain Equation (58) has a solution. If k > 0, we obtain and if k < 0, we obtain with any A and B. On the contrary, we have en, we obtain en, equation (59) becomes is gives which implies that is gives For k > 0, we obtain , we should take f(0) � 0. Hence, we obtain with any constants a 0 , A, and B, where k < 0. On the contrary, we obtain with any constants a 0 , A, and B, where f(0) � 0 and k > 0. is completes the proof.

Conclusion
In this study, the q-time deformed wave equation as well as the free-time wave equation are studied. We expect to study the quantum white noise [16][17][18][19][20] case which is now attractive in mathematical physics area.

Data Availability
No data were used to support this study.

Conflicts of Interest
All authors declare that they have no conflicts of interest.