New solutions for the generalized q -deformed wave equation with q -translation symmetry

: In this work, we explore the generalized discrete wave equation, which utilizes a speci ﬁ c irregular space interval. The introduction of this irregular space interval is motivated by its connection to the q -addition, a mathematical operation that arises in the nonextensive entropy theory. By taking the continuous limit, we obtain the wave equation with q -deformation, which captures the e ﬀ ects of the q -addition. To solve the generalized q -deformed wave equation, we investigate three di ﬀ erent methods: the separation method, the reduced di ﬀ erential transform method, and the ﬁ nite di ﬀ erence method. These methods o ﬀ er distinct approaches for ﬁ nding solutions to the equation. By comparing the results obtained from each method, we can evaluate their e ﬀ ectiveness and identify their respective strengths and limitations in solving the generalized q -deformed wave equation. The solutions obtained from this newly de ﬁ ned equation have potential applications in modeling physical systems with violated symmetries. The inclusion of the q -deformation allows for a more comprehensive description of such systems, which may exhibit nonextensive behavior or possess irregularities in their spatial intervals. By incorporating these features into the wave equation, we can improve our understanding and modeling capabilities of complex physical phenomena.


Introduction
The wave equation is a second-order linear hyperbolic partial differential equation that describes the propagation of a variety of waves, such as sound or water waves.It arises in different fields such as acoustics, electromagnetics, or fluid dynamics.In its simplest, the wave equation takes the following form: (1 where c is the constant speed of the wave propagation.D'Alembert is credited with the discovery of the onedimensional wave equation in 1747 [1], and he provided the string's motion model equation in one dimension in 1743 [2].The wave equation of three-dimensional is discovered by Euler 10 years later [3].The wave equation is a useful description that encompasses a broad spectrum of events and is generally used to simulate modest oscillations about the equilibrium, which is that the system will frequently be adequately approximated using the Hooke principle.The wave equation has various applications, not just in fluid dynamics but also in electromagnetic fields, optics, gravitational physics, heat transfer, etc. Fractional calculus has long been regarded as a branch of pure mathematics with no practical applications since its formulation is predicated on the idea of a noninteger order that can be integral or derivative.However, in recent years, the function of fractional calculus has evolved, and we find applications of this branch in many fields such as nanofluid flow [4], quantum mechanics [5], and electrical engineering [6].The q-derivative, known as the Jackson derivative, is a q-analog for the ordinary derivative that was first presented with Jackson in the fields of combinatorics and quantum calculus [7].Q-calculus has over the past 20 years, and q-calculus has evolved in the specialization subjects and acted as a link between physics and mathematics.Physicists make up the bulk of users of the q-calculus worldwide.The discipline has rapidly grown as a result of applications of fundamental hypergeometric series to a variety of topics including quantum theory [8], number theory [9], and mechanical statistics [10].Numerous findings from research on the theory of operators for q-calculus in the last few years have been used in a variety of fields, including the geometric function theory of complex analysis [11], problems in ordinary fractional calculus [12], optimal control [13], solutions of the q-difference equations [14], q-integral equations [15], and q-transform analysis [16].
The fractional q-calculus can be defined as a q-protraction for basic fractional calculus, and it has several applications in the mathematical sciences such as timescale [17].From this standpoint, some researchers began to develop the wave equation and put it as q-deformed equation.By adding a deformation parameter q, the q-deformed wave equation, also known as q-deformed quantum mechanics, is a branch of theoretical physics that extends traditional quantum mechanics [18,19].Usually, this q-deformation is related to noncommutative geometry and quantum groups.
In this article, we need to discuss the distorted wave equation.To act this, we must have a separate incarnation of the wave equation in which the space is discrete but the time is continuous.Discrete physics has been investigated in different areas [20,21].If we suppose that the discrete elements are denoted by we have the discrete wave equation as follows: where the definition of the operators for finite differences is given as follows: Taking the limit when → a 0 at Eq. ( 4), we obtain Eq. (1).By Eq. ( 2), as we are aware, n n 1 (5) This suggests that the wave equation of the form (1) is guaranteed by the uniform space interval.Put differently, we shall receive a different form if we examine a nonuniform discrete location of the wave formula.We examine the discrete wave equation in this study with certain nonuniform space intervals that appear in the nonextensive entropy theory [22,23] and are associated with q-addition or q-subtraction.Taking the continuous limit gives us the wave equation with q-deformation.
We solve Eq. ( 19) by two analytical methods: separation method (SM) [24] and reduced differential transform method (RDTM) [25].In addition, we solved it by a numerical method, namely finite difference method (FDM) [26].This article is organized as follows: in Section 2, we present the q-deformed wave equation; in Section 3, we introduce the analytical solutions for the problem by using SM and RDTM; in Section 4, we compute the numerical solution for the problem by using FDM; in Section 5, the discussion of our results is presented; and finally, in Section 6, the conclusion of the article is introduced.
2 The q-deformed wave equation The q-deformed wave equation is covered within this segment.It is based on the q-addition and q-subtraction found in nonextensive thermodynamics [22,23].We present the parameter q, which differs from the nonextensive thermo- dynamic theory.So that it may have a dimension of inverse length.The parameter q in the nonextensive thermody- namics is dimensionless.Hence, q may be thought of as ∕ξ 1 in the q-deformed wave equation, where ξ represents the length measure.Now, let us present the distinct position using a nonuniform time interval, where the value of the distance between consecutive locations is where in [22,23], the definitions of the q-addition and qsubtraction are The nonuniform lattice, which consists of discrete points and obeys Eq. ( 6), differs from the uniform lattice and may be considered an instance of a medium that is not homogenous in the continuous limit ( ) → a 0 .We believe that further examples of the discrete locations described by the various pseudo additions (deformations of the typical additions) may be provided by the continuous limit's nonhomogeneous medium.For instance, the α-addition was used in [27] to characterize the nonhomogeneous medium in which anomalous diffusion developed.
The relationship is provided by Eq. ( 6).
n n 1 (10) Solving Eq. ( 10), we obtain n n (11) and upon > q 0, we obtain n n (13) When < q 0, we obtain n n (14) and n n (15) Here, we require , there is nonsymmetry in the discrete position.We have For the discrete positions obeying Eq. ( 6), the difference operator becomes Therefore, we obtain the limit of continuity q n q : (18) We may see here that under the q-translation → ⊕ δ , the q-derivative D q stays invariant.In [28], quantum theory with q-translation invariance was recently created.The qdeformed wave equation with Eq. ( 18) is obtained by q-translation symmetry expressed in terms of , , w h e r e 0 , 1 .
3 Analytical solutions In this section, we investigate two different methods, SM and RDTM, to solve the generalized q-deformed wave Eq. ( 19).

Analytical solution for Eq. (19) by using SM
To find the solution for Eq. ( 19) using SM, we need to apply the following steps of Eq. (19).
Step (1): In this step, we suppose that the solution for Eq. ( 19) can be expressed as a multiply of two functions, say ( ) X and T ( ) T , and then substitute them in Eq. ( 19).
Step (2): From step 1, we obtain two partial differential equations and solve them as below.Now let us apply the above steps as follows: Consider a rod of length L with initial conditions: and boundary conditions : We look for a solution of the form Inserting Eq. ( 22) into Eq.( 19), we obtain Thus, we have But the boundary function gives us = A 0 and That provides Consequently, the general solution of Eq. ( 19) is Applying the initial condition, we have New solutions for the generalized q-deformed wave equation with q-translation symmetry  3 If the orthogonality relation is applied we have ) From Eq. (35) into Eq.(36), we obtain the analytical solution of the form: 3.2 Analytical solution for Eq. ( 19) by using RDTM The RDTM is applied to find the solution for Eq. ( 19) by using the following steps: Step (1): Several fundamental definitions and characteristics for RDTM have been examined in this stage.These may be found in [26,[29][30][31][32].
. The RDT of T ( ) u ˜, about T is determined by In Eq. (38) ( ) U k is the transformed function and T ( ) u ˜, is the original function.
Definition 3.2.The reduced differential inverse transform of ( ) U k is defined as follows: From Eqs (38) and (39), we obtain .
Step (2): In this step, we will present some important theorems for the RDTM that we use for solving Eq. (19).
Proof.See the study by Atici and Eloe [17].

Numerical results
In this section, we aim to present the numerical outcomes obtained using FDM for Eq. ( 19) and compare them with the solutions of Eqs (37) and ( 48) obtained through analytical methods.Our goal is to examine the behavior and accuracy of the numerical methods under different parameter values for , q, and T. To achieve this, we perform numerical simulations using various combinations of parameter values.We systematically vary the values of , q, and T to explore their impact on the solutions obtained through numerical methods.This analysis provides a comprehensive understanding of the behavior of the numerical methods and their suitability for solving Eq. ( 19) under different parameter values.It helps us identify the regimes where the numerical methods perform well and the scenarios where they may exhibit limitations or discrepancies compared to the analytical solution.Ultimately, this evaluation guides us in selecting appropriate numerical methods for similar equations in practical applications.Now, we assume that represents the exact solution at the grid point T ( ) , i n , while U represents the corresponding numerical solution.The approximation for the spatial derivative is given as follows: New solutions for the generalized q-deformed wave equation with q-translation symmetry  5 The approximation for the time second derivative with respect to T is Through the substitution of Eqs ( 49) and (50) into Eq.( 19), the system of difference equations at U i n , can be derived.Numerical results can be obtained by solving this system using the Mathematica software.

The model's numerical solutions for
Eq. ( 19) by using FDM with Eq. ( 37) In this analysis, we aim to compare the numerical results obtained for Eq. ( 19) with the corresponding analytical solution Eq. ( 37).Together, Table 1 and Figure 1 offer a comprehensive analysis of the comparisons made between the numerical and analytical results at = q 0.01, = c 1, , T = Δ 0.001, and T = 0.1 with different values of .Now, we the numerical results obtained for Eq. ( 19) using the analytical solution of Eq. ( 37) at = 0.5, , T = Δ 0.001, and T = 0.1 with different values of q.The detailed results can be found in Table 2 and Figure 2.
The comparison between the numerical results of Eq. ( 19) and the analytical solution of Eq. (37 , T = Δ 0.001, and = q 0.01 with different values of T are present in both Table 3 and Figure 3.

The model's numerical solution for
Eq. ( 19) by using FDM with Eq. (48) In this part, we discuss the numerical solution for Eq.(19) with Eq. (48) obtained using RDTM under different parameter values for , q, and T. In this analysis, we aim to compare the numerical results obtained for Eq. ( 19) with the corresponding analytical solution of Eq. ( 48).Together, Table 4 and Figure 4 offer a comprehensive analysis of the    , T = Δ 0.001, and T = 0.1 with different values of .Now, we compare the numerical results obtained for Eq. ( 19) using the analytical solution of Eq. ( 48) at = 0.5, , T = Δ 0.001, and T = 0.1 with different values of q.The detailed results can be found in Table 5 and Figure 5.
The comparison between the numerical results of Eq. (19) and the analytical solution of Eq. ( 48 , T = Δ 0.001, and = q 0.01 with different values of T are present in both Table 6 and Figure 6.

Discussion
The figures presented in study serve as visual representations that depict the behavior of waves described by the solutions under various conditions and parameters.In Figures 1 and 4, we utilize our methods, namely "SM" and "RDTM," to generate graphs for Eqs (37) and (48) using the following parameter values: = q 0.01, = c 1, = = u c 0.1 0 1 , T = Δ 0.001, and T = 0.1.These figures also include a com- parison between the numerical findings of Eq. ( 19) and the corresponding analytical solutions given in Eqs (37) and (48).Furthermore, in Figures 2 and 5, we compare the  , T = Δ 0.001, and = q 0.01.New solutions for the generalized q-deformed wave equation with q-translation symmetry  7 numerical results of Eq. ( 19) under different values of q, while keeping = 0.5, = c 1, = = u c 0.1 0 1 , T = Δ 0.001, and T = 0.1 constant.In addition, Figures 3 and 6  , T = Δ 0.001, and = q 0.01.These figures also demonstrate the variation in the wave curve as time progresses, depicting both increasing and decreasing trends.From a close examination of Figures 1 and 2, it is evident that the analytical and numerical solutions exhibit a high degree of similarity, indicating the accuracy of the employed methods.These figures provide compelling evidence supporting the validity of the methodologies employed in this study and affirm the accuracy of the generated Overall, the figures serve as vital visual aids, effectively illustrating the key findings of the research.

Conclusion
In conclusion, this article provides a comprehensive analysis of the q-deformed wave equation and offers valuable insights into its solutions.The article introduces the qdeformed wave equation, represented by Eq. ( 19).This equation serves as the foundation for the subsequent analysis.We present analytical solutions for Eq. ( 19) using SM and RDTM.These analytical solutions provide valuable mathematical expressions that describe the behavior of the wave equation under different conditions and parameter values.In addition, the article explores numerical solutions for Eq. ( 19) to complement the analytical findings.
In one approach, the article employs FDM in conjunction with the SM to obtain a numerical solution.This numerical solution offers a practical and computationally efficient means of approximating the behavior of the wave equation.Furthermore, the article demonstrates another numerical solution for Eq. ( 19) by utilizing the FDM in combination with the RDTM.This alternative numerical solution provides further insight into the wave equation's characteristics and behavior.Finally, the article enhances the understanding of the discussed concepts by presenting visual illustrations.These figures visually represent the waves described by the analytical and numerical solutions under various conditions and parameter values.The illustrations aid in interpreting the results and provide a more intuitive understanding of the wave equation's behavior.
Acknowledgements: The authors are grateful for the reviewer's valuable comments that improved the manuscript.
Funding information: The authors state that there is no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.Ahmed S. Shehata wrote the article and ran   , T = Δ 0.001, and = q 0.01.

Remark 3 . 3 .
Notice that the (RDTM) is close to the one dimensional (DTM) because the (RDTM) is considered as the standard (DTM) of T ( ) u ˜, regarding the variable T. But the matching recursive algebraic equation is the variable's function ( ) = ˜, , …, n 1 2

Figure 3 :
Figure 3: The graph shows the effect of changing T on the numerical solution of Eq. (19) at 0.5, = c 1, = = u c 0.1 0 1

Table 1 :
Comparison between numerical results for Eq.(19) and solution of Eq. (37) at different values of

Table 2 :
Comparison between numerical results for Eq.(19) and solution of Eq. (37) at different values of q

Table 3 :
Comparison between numerical results for Eq.(19) and solution of Eq. (37) at different values of T

Table 4 :
Comparison between numerical results for Eq.(19) and solution of Eq. (48) at different values of

Table 5 :
Comparison between numerical results of Eq. (19) and solution for Eq.(48) at different values of q

Table 6 :
Comparison between numerical results for Eq.(19) and solution for Eq.(48) at different values of T