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

Ahmed S. Shehata; Kamal R. Raslan; Khalid K. Ali

doi:10.1515/nleng-2022-0378

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

Ahmed S. Shehata , Kamal R. Raslan and Khalid K. Ali

Published/Copyright: March 28, 2024

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From the journal Nonlinear Engineering Volume 13 Issue 1

Abstract

In this work, we explore the generalized discrete wave equation, which utilizes a specific 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 effects of the q-addition. To solve the generalized q-deformed wave equation, we investigate three different methods: the separation method, the reduced differential transform method, and the finite difference method. These methods offer distinct approaches for finding solutions to the equation. By comparing the results obtained from each method, we can evaluate their effectiveness and identify their respective strengths and limitations in solving the generalized q-deformed wave equation. The solutions obtained from this newly defined 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.

Keywords: q-calculus; q-deformed; q-translation; separation method; RDTM; FDM

1 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) ∂ 2 ∂ T 2 u ( X , T ) = c 2 ∂ 2 ∂ X 2 u ( X , T ) ,

where c is the constant speed of the wave propagation. D’Alembert is credited with the discovery of the one-dimensional 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

(2) X n = n a , n ∈ Z ,

we have the discrete wave equation as follows:

(3) ∂ 2 ∂ T 2 u ( X n , T ) = c 2 Δ X 2 u ( X n , T ) ,

where the definition of the operators for finite differences is given as follows:

(4) Δ X u ( X n , T ) = u ( X n + 1 , T ) − u ( X n , T ) a .

Taking the limit when a → 0 at Eq. (4), we obtain Eq. (1). By Eq. (2), as we are aware,

(5) X n + 1 − X n = a .

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 thermodynamic theory. So that it may have a dimension of inverse length. The parameter q in the nonextensive thermodynamics is dimensionless. Hence, q may be thought of as 1 ∕ ξ 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

(6) X n + 1 ⊖ q X n = a ,

(7) X n + 1 = X n ⊕ q a ,

where in [22,23], the definitions of the q-addition and q-subtraction are

(8) c ⊕ q d = c + d + q c d ,

(9) c ⊖ q d = c − d 1 + q d .

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).

(10) X n + 1 = ( 1 + q a ) X n + a .

Solving Eq. (10), we obtain

(11) X n = 1 q ( [ 1 + q a ] n − 1 ) ,

and upon q > 0 , we obtain

(12) lim n → ∞ X n = ∞ ,

and

(13) lim n → − ∞ X n = − 1 q .

When q < 0 , we obtain

(14) lim n → ∞ X n = 1 ∣ q ∣ ,

and

(15) lim n → − ∞ X n = − ∞ .

Here, we require ∣ q ∣ a < 1 . If X 0 = 0 , there is nonsymmetry in the discrete position. We have

(16) X − n = − X n ( 1 + q a ) n , n ≥ 1 .

For the discrete positions obeying Eq. (6), the difference operator becomes

(17) Δ X : q u ( X n , T ) = u ( X n + 1 , T ) − u ( X n , T ) X n + 1 ⊖ q X n = ( 1 + q X n ) × u ( X n + 1 , T ) − u ( X n , T ) X n + 1 − X n .

Therefore, we obtain the limit of continuity

(18) Δ X : q u ( X n , T ) → D X q = ∂ u ∂ X ( 1 + q X ) .

We may see here that under the q-translation X → X ⊕ δ X , the q-derivative D X q stays invariant. In [28], quantum theory with q-translation invariance was recently created. The q-deformed wave equation with Eq. (18) is obtained by q-translation symmetry expressed in terms of

(19) ∂ 2 ∂ T 2 u ( X , T ) = c 2 ( D X q ) 2 u ( X , T ) , where q ∈ ( 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).

3.1 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 ( 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:

(20) f ( X ) = u ( X , 0 ) , g ( X ) = u T ( X , 0 ) ,

and boundary conditions :

(21) u ( L , T ) = u ( 0 , T ) = 0 .

We look for a solution of the form

(22) u ( X , T ) = X ( X ) T ( T ) .

Inserting Eq. (22) into Eq. (19), we obtain

(23) 1 c 2 T d 2 T d T 2 = 1 X ( D X q ) 2 X = − λ , λ > 0 .

Thus, we have

(24) T ( T ) = c 1 cos ( T c λ ) + c 2 sin ( T c λ ) ,

and

(25) X ( X ) = A cos ln ( 1 + q X ) λ q + B sin ln ( 1 + q X ) λ q .

But the boundary function gives us A = 0 and

(26) sin ln ( 1 + q L ) λ q = 0 .

That provides

(27) λ = q n π ln ( 1 + q L ) = λ n , n = 1 , 2 , …

Consequently, the general solution of Eq. (19) is

(28) u ( X , T ) = ∑ n = 1 ∞ B n sin ln ( 1 + q X ) ln ( 1 + q L ) n π × c 1 cos t c q n π ln ( 1 + q L ) + c 2 sin t c q n π ln ( 1 + q L ) .

Applying the initial condition, we have

(29) f ( X ) = ∑ n = 1 ∞ B n c 1 sin n π ln ( 1 + q X ) ln ( 1 + q L ) ,

and

(30) g ( X ) = ∑ n = 1 ∞ B n c 2 q n π ln ( 1 + q L ) sin n π ln ( 1 + q X ) ln ( 1 + q L ) .

If the orthogonality relation is applied

(31) ∫ 0 L sin ln ( 1 + q X ) ln ( 1 + q L ) n π sin ln ( 1 + q X ) ln ( 1 + q L ) m π × d X 1 + q X = ln ( 1 + q L ) 2 q δ n m ,

we have

(32) B n = 2 q c 1 ln ( 1 + q L ) ∫ 0 L f ( X ) × sin ln ( 1 + q X ) ln ( 1 + q L ) n π d X 1 + q X ,

and

(33) c 2 = 2 n π B n ∫ 0 L g ( X ) × sin n π ln ( 1 + q X ) ln ( 1 + q L ) d X 1 + q X .

Suppose the initial wave distributions f ( X ) and g ( X ) are constants, i.e., f ( X ) = u 0 , g ( X ) = ξ 0 .

Let us now examine the situation involving L = 1 and c = 1 . Next, we have

(34) B n = − 2 u 0 n π c 1 ( ( − 1 ) n − 1 ) ,

(35) c 2 = ξ 0 c 1 ln ( 1 + q L ) n q u 0 π .

Thus, we have

(36) u ( X , T ) = 4 u 0 π c 1 ∑ n = 1 ∞ 1 ( 2 n − 1 ) sin × ( 2 n − 1 ) π ln ( 1 + q X ) ln ( 1 + q ) c 1 cos ( 2 n − 1 ) π q T ln ( 1 + q ) + c 2 sin ( 2 n − 1 ) π q T ln ( 1 + q ) .

From Eq. (35) into Eq. (36), we obtain the analytical solution of the form:

(37) u ( X , T ) = 4 u 0 π c 1 ∑ n = 1 ∞ 1 ( 2 n − 1 ) sin × ( 2 n − 1 ) π ln ( 1 + q X ) ln ( 1 + q ) × c 1 cos ( 2 n − 1 ) π q T ln ( 1 + q ) + ξ 0 c 1 ln ( 1 + q L ) n q u 0 π sin ( 2 n − 1 ) π q T ln ( 1 + q ) .

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 [26,29–32].

Definition 3.1

Examine a function that ( n + 1 ) variables u ( X ˜ , T ) = u ( X 1 , X 2 , … , X n , T ) such that X ˜ ∈ R n , X ˜ = ( X 1 , X 2 , … , X n ) . The RDT of u ( X ˜ , T ) about T is determined by

(38) U k ( X ˜ ) = 1 k ! ∂ k ∂ T k u ( X ˜ , T ) T = 0 , k ∈ N .

In Eq. (38) U k ( X ˜ ) is the transformed function and u ( X ˜ , T ) is the original function.

Definition 3.2

The reduced differential inverse transform of U k ( X ˜ ) is defined as follows:

(39) u ( X ˜ , T ) = ∑ k = 0 ∞ U k ( X ˜ ) T k .

From Eqs (38) and (39), we obtain

(40) u ( X ˜ , T ) = ∑ k = 0 ∞ T k k ! ∂ k ∂ T k u ( X ˜ , T ) T = 0 .

Remark 3.3

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

Step (2): In this step, we will present some important theorems for the RDTM that we use for solving Eq. (19).

Theorem 3.4

If f ( X ˜ , T ) = ∂ n ∂ T n g ( X ˜ , T ) , then F k ( X ˜ ) = ( k + 1 ) ( k + 2 ) … ( k + n ) G k + n ( X ˜ ) .

Proof

See the study by Atici and Eloe [17].□

Theorem 3.5

f ( X ˜ , T ) = ∂ n ∂ X i n g ( X ˜ , T ) , then, F k ( X ˜ ) = ∂ n ∂ X i n G k ( X ˜ ) , i = 1 , 2 , … , n .

Proof

See the study by Atici and Eloe [17].□

Step (3): We apply the above theorems to find the solution for Eq. (19).

Now, we apply the above steps to (19). From Eq. (19), we can write

(41) ∂ 2 ∂ T 2 u ( X , T ) = c 2 ( 1 + q X ) 2 ∂ 2 u ( X , T ) ∂ X 2 + q ( 1 + q X ) ∂ u ( X , T ) ∂ X .

Taking the RDTM for Eq. (41), we have

(42) ( k + 1 ) ( k + 2 ) U k + 2 ( X ) = c 2 ( 1 + q X ) ( 1 + q X ) ∂ 2 ∂ X 2 U k ( X ) + q ∂ ∂ X U k ( X ) ,

where k = 0 , 1 , 2 , … , and taking the initial conditions, we obtain

(43) U 0 ( X ) = u ( X , 0 ) = f ( X ) = ∑ n = 1 2 − 2 u 0 n π [ ( − 1 ) n − 1 ] sin n π ln ( 1 + q X ) ln ( 1 + q L ) ,

(44) U 1 ( X ) = u T ( X , 0 ) = g ( X ) = ∑ n = 1 2 − 2 u 0 n π sin n π ln ( 1 + q X ) ln ( 1 + q L ) .

Set k = 0 in Eq. (42), then we obtain

(45) U 2 ( X ) = c 2 ( 1 + q X ) 2 × ( 1 + q X ) ∂ 2 ∂ X 2 U 0 ( X ) + q ∂ ∂ X U 0 ( X ) .

Hence,

(46) U 2 ( X ) = c 2 ( 1 + q X ) 2 × ( 1 + q X ) − 4 u 0 q 2 cos π ln ( 1 + q X ) ln ( 1 + q L ) ln ( 1 + q L ) + π sin π ln ( 1 + q X ) ln ( 1 + q L ) ( 1 + q X ) 2 ( ln ( 1 + q L ) ) 2 + q 4 q cos π ln ( 1 + q X ) ln ( 1 + q L ) ( 1 + q X ) ln ( 1 + q L ) ,

since we have

(47) u ( X , T ) = ∑ k = 0 ∞ U k ( X ) T k .

Substituting Eqs (43), (44), and (45) in Eq. (47), we obtain

(48) u ( X , T ) = ∑ n = 1 2 − 2 u 0 n π [ ( − 1 ) n − 1 ] sin ln ( 1 + q X ) ln ( 1 + q L ) n π + ∑ n = 1 2 − 2 u 0 n π sin n π ln ( 1 + q X ) ln ( 1 + q L ) T + c 2 ( 1 + q X ) 2 [ ( 1 + q X ) × − 4 u 0 q 2 cos [ π ln ( 1 + q X ) ln ( 1 + q L ) ] ln ( 1 + q L ) + π sin [ π ln ( 1 + q X ) ln ( 1 + q L ) ] ( 1 + q X ) 2 ( ln ( 1 + q L ) ) 2 + q 4 q cos [ π ln ( 1 + q X ) ln ( 1 + q L ) ] ( 1 + q X ) ln ( 1 + q L ) T 2 + …

4 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 X , q , and T . To achieve this, we perform numerical simulations using various combinations of parameter values. We systematically vary the values of X , 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 X represents the exact solution at the grid point ( X i , T n ) , while U represents the corresponding numerical solution. The approximation for the spatial derivative is given as follows:

(49) u X = 1 ( Δ X ) ( U i + 1 , n − U i , n ) , u X X = 1 ( Δ X ) 2 ( U i + 1 , n − 2 U i , n + U i − 1 , n ) .

The approximation for the time second derivative with respect to T is

(50) u T T = 1 ( Δ T ) 2 ( U i , n + 1 − 2 U i , n + U i , n − 1 ) .

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.

4.1 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 , u 0 = c 1 = 0.1 , Δ T = 0.001 , and T = 0.1 with different values of X .

Table 1

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

X	Numerical solutions	Analytical solutions	Absolute error
0.1	0.0645134	0.0645228	9.44656 × 1 0 ‒ 6
0.2	0.1057350	0.1057460	1.10179 × 1 0 ‒ 5
0.3	0.1168190	0.1168230	3.56080 × 1 0 ‒ 6
0.6	0.1104140	0.1104070	6.36948 × 1 0 ‒ 6
0.7	0.1168360	0.1168400	3.82593 × 1 0 ‒ 6
0.8	0.1053260	0.1053370	1.09519 × 1 0 ‒ 5
0.9	0.0640150	0.0640242	9.16746 × 1 0 ‒ 6

$Figure 1 The graph shows the comparison between the numerical and analytical solutions for Eq. (19) at q = 0.01 q=0.01 , c = 1 c=1 , u 0 = c 1 = 0.1 {u}_{0}={c}_{1}=0.1 , Δ T = 0.001 \Delta {\mathfrak{T}}=0.001 , T = 0.1 {\mathfrak{T}}=0.1 , and X ∈ ( 0 , 1 ) {\mathcal{X}}\in \left(0,1) with different values of X {\mathcal{X}} .$

Figure 1

The graph shows the comparison between the numerical and analytical solutions for Eq. (19) at q = 0.01 , c = 1 , u 0 = c 1 = 0.1 , Δ T = 0.001 , T = 0.1 , and X ∈ ( 0 , 1 ) with different values of X .

Now, we compare the numerical results obtained for Eq. (19) using the analytical solution of Eq. (37) at X = 0.5 , c = 1 , u 0 = c 1 = 0.1 , Δ T = 0.001 , and T = 0.1 with different values of q . The detailed results can be found in Table 2 and Figure 2.

Table 2

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

q	Numerical solutions	Analytical solutions	Absolute error
0.1	0.106136	0.106128	7.24815 × 1 0 ‒ 6
0.2	0.107324	0.107321	2.65994 × 1 0 ‒ 6
0.3	0.108513	0.108515	1.79445 × 1 0 ‒ 6
0.4	0.109651	0.109656	5.86652 × 1 0 ‒ 6
0.6	0.111629	0.111642	1.21940 × 1 0 ‒ 5
0.7	0.112429	0.112444	1.42418 × 1 0 ‒ 5
0.8	0.113088	0.113103	1.54722 × 1 0 ‒ 5
0.9	0.113600	0.113616	1.58667 × 1 0 ‒ 5

$Figure 2 The graph shows the effect of changing q q on the numerical solution of Eq. (19) at X = 0.5 {\mathcal{X}}=0.5 , c = 1 c=1 , u 0 = c 1 = 0.1 {u}_{0}={c}_{1}=0.1 , Δ T = 0.001 \Delta {\mathfrak{T}}=0.001 , and T = 0.1 {\mathfrak{T}}=0.1 .$

Figure 2

The graph shows the effect of changing q on the numerical solution of Eq. (19) at X = 0.5 , c = 1 , u 0 = c 1 = 0.1 , Δ T = 0.001 , and T = 0.1 .

The comparison between the numerical results of Eq. (19) and the analytical solution of Eq. (37) at X = 0.5 , c = 1 , u 0 = c 1 = 0.1 , Δ T = 0.001 , and q = 0.01 with different values of T are present in both Table 3 and Figure 3.

Table 3

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

T	Numerical solutions	Analytical solutions	Absolute error
0.1	0.10513100	0.10512000	1.12223 × 1 0 ‒ 5
0.2	0.13582200	0.13579500	2.74716 × 1 0 ‒ 5
0.3	0.14630800	0.14629200	1.50468 × 1 0 ‒ 5
0.4	0.11357400	0.11360600	3.20644 × 1 0 ‒ 5
0.5	0.04280840	0.04288240	7.39881 × 1 0 ‒ 5
0.6	‒ 0.0347419	‒ 0.0346828	5.91431 × 1 0 ‒ 5
0.7	‒ 0.0850336	‒ 0.0850521	1.85761 × 1 0 ‒ 5
0.8	‒ 0.0965413	‒ 0.0966404	9.91319 × 1 0 ‒ 5
0.9	‒ 0.0868499	‒ 0.0869524	1.02468 × 1 0 ‒ 4

$Figure 3 The graph shows the effect of changing T {\mathfrak{T}} on the numerical solution of Eq. (19) at X = 0.5 {\mathcal{X}}=0.5 , c = 1 c=1 , u 0 = c 1 = 0.1 {u}_{0}={c}_{1}=0.1 , Δ T = 0.001 \Delta {\mathfrak{T}}=0.001 , and q = 0.01 q=0.01 .$

Figure 3

The graph shows the effect of changing T on the numerical solution of Eq. (19) at X = 0.5 , c = 1 , u 0 = c 1 = 0.1 , Δ T = 0.001 , and q = 0.01 .

4.2 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 X , 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 comparisons made between the numerical and analytical results at q = 0.01 , c = 1 , u 0 = c 1 = 0.1 , Δ T = 0.001 , and T = 0.1 with different values of X .

Table 4

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

X	Numerical solutions	Analytical solutions	Absolute error
0.1	0.0336924	0.0338638	1.71341 × 1 0 ‒ 4
0.2	0.0645670	0.0648576	2.90631 × 1 0 ‒ 4
0.3	0.0899229	0.0902478	3.24817 × 1 0 ‒ 4
0.4	0.1072830	0.1075540	2.70994 × 1 0 ‒ 4
0.6	0.1107590	0.1107860	2.69568 × 1 0 ‒ 5
0.7	0.0955352	0.0954651	7.00654 × 1 0 ‒ 5
0.8	0.0701578	0.0700535	1.04290 × 1 0 ‒ 4
0.9	0.0371277	0.0370550	7.27697 × 1 0 ‒ 5

$Figure 4 The graph shows the comparison between the numerical and analytical solutions of Eq. (19) at q = 0.01 q=0.01 , c = 1 c=1 , u 0 = c 1 = 0.1 {u}_{0}={c}_{1}=0.1 , Δ T = 0.001 \Delta {\mathfrak{T}}=0.001 , and T = 0.1 {\mathfrak{T}}=0.1 .$

Figure 4

The graph shows the comparison between the numerical and analytical solutions of Eq. (19) at q = 0.01 , c = 1 , u 0 = c 1 = 0.1 , Δ T = 0.001 , and T = 0.1 .

Now, we compare the numerical results obtained for Eq. (19) using the analytical solution of Eq. (48) at X = 0.5 , c = 1 , u 0 = c 1 = 0.1 , Δ T = 0.001 , and T = 0.1 with different values of q . The detailed results can be found in Table 5 and Figure 5.

Table 5

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

q	Numerical solutions	Analytical solutions	Absolute error
0.1	0.114199	0.114357	1.57443 × 1 0 ‒ 4
0.2	0.113561	0.113719	1.58237 × 1 0 ‒ 4
0.3	0.112798	0.112956	1.58827 × 1 0 ‒ 4
0.4	0.111938	0.112097	1.59427 × 1 0 ‒ 4
0.6	0.110010	0.110172	1.61375 × 1 0 ‒ 4
0.7	0.108972	0.109135	1.63025 × 1 0 ‒ 4
0.8	0.107899	0.108065	1.65296 × 1 0 ‒ 4
0.9	0.106800	0.106968	1.68299 × 1 0 ‒ 4

$Figure 5 The graph shows the effect of changing q q on the numerical solution of Eq. (19) at X = 0.5 {\mathcal{X}}=0.5 , c = 1 c=1 , u 0 = c 1 = 0.1 {u}_{0}={c}_{1}=0.1 , Δ T = 0.001 \Delta {\mathfrak{T}}=0.001 , and T = 0.1 {\mathfrak{T}}=0.1 .$

Figure 5

The graph shows the effect of changing q on the numerical solution of Eq. (19) at X = 0.5 , c = 1 , u 0 = c 1 = 0.1 , Δ T = 0.001 , and T = 0.1 .

The comparison between the numerical results of Eq. (19) and the analytical solution of Eq. (48) at X = 0.6 , c = 1 , u 0 = c 1 = 0.1 , Δ T = 0.001 , and q = 0.01 with different values of T are present in both Table 6 and Figure 6.

Table 6

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

T	Numerical solutions	Analytical solutions	Absolute error
0.1	0.11075900	0.11078600	2.69568 × 1 0 ‒ 5
0.2	0.08851770	0.08916930	6.51641 × 1 0 ‒ 4
0.3	0.05422030	0.05792000	3.69966 × 1 0 ‒ 3
0.4	0.00786685	0.02013110	1.22642 × 1 0 ‒ 2
0.5	‒ 0.050542 8	‒ 0.0201439	3.03989 × 1 0 ‒ 2
0.6	‒ 0.1210090	‒ 0.0584145	6.25941 × 1 0 ‒ 2
0.7	‒ 0.2035310	‒ 0.0903933	1.13137 × 1 0 ‒ 1

$Figure 6 The graph shows the effect of changing T {\mathfrak{T}} on the numerical solution of Eq. (19) at X = 0.6 {\mathcal{X}}=0.6 , c = 1 c=1 , u 0 = c 1 = 0.1 {u}_{0}={c}_{1}=0.1 , Δ T = 0.001 \Delta {\mathfrak{T}}=0.001 , and q = 0.01 q=0.01 .$

Figure 6

The graph shows the effect of changing T on the numerical solution of Eq. (19) at X = 0.6 , c = 1 , u 0 = c 1 = 0.1 , Δ T = 0.001 , and q = 0.01 .

5 Discussion

The figures presented in this 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 0 = c 1 = 0.1 , Δ T = 0.001 , and T = 0.1 . These figures also include a comparison 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 numerical results of Eq. (19) under different values of q , while keeping X = 0.5 , c = 1 , u 0 = c 1 = 0.1 , Δ T = 0.001 , and T = 0.1 constant. In addition, Figures 3 and 6 display the numerical results of Eq. (19) at X = 0.5 , c = 1 , u 0 = c 1 = 0.1 , Δ T = 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 solutions. Overall, the figures serve as vital visual aids, effectively illustrating the key findings of the research.

6 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 q-deformed 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 some programs for the article. Kamal R. Raslan presented the idea and reviewed the results. Khalid K. Ali explained the results and reviewed the article in its final form.
Conflict of interest: There is no conflict of interest between the authors and anyone.
Informed consent: Not applicable.
Authorization for the use of experimental animals: Not applicable.
The place where the search was conducted: The research was conducted at Al-Azhar University.
Raw datasets, data management plans, etc.: Not applicable.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

[1] D’Alembert JLeR. Réeflexions sur la cause generale des vents. D Paris. 1747;9:182–224. Search in Google Scholar

[2] D’Alembert JLeR. Traite de dynamique. Gauthier-Villars et Cie éditions. 1743;9:1702–2003. Search in Google Scholar

[3] Euler L. Remarques sur les memoires préecéedents de M. Bernoulli. ibd. 1755;9(1753):196–222. Search in Google Scholar

[4] Blackledge J. Quantum mechanics and control using fractional calculus: a study of the Shutter problem for fractional quantum fields. Appl Mech. 2022;3:413–63. 10.3390/applmech3020026Search in Google Scholar

[5] Ali MF, Sharma M, Jain R. An application of fractional calculus in electrical engineering. Adv Eng Tech and Appl. 2016;5:41–5. 10.18576/aeta/050204Search in Google Scholar

[6] Lavagno A, Gervino G. Quantum mechanics in q-deformed calculus. J Ph Con Series. 2009;174:1–9. 10.1088/1742-6596/174/1/012071Search in Google Scholar

[7] Mussardo G, Giudici G, Viti J. The coprime quantum chain. J Stat Mech. 2017;033104:1–55. 10.1088/1742-5468/aa5bb4Search in Google Scholar

[8] Johal RS. q-calculus and entropy in nonextensive statistical physics. Ins Hep. 1998;58:4147–51. 10.1103/PhysRevE.58.4147Search in Google Scholar

[9] Jackson HF. q-Difference equations. A J Math. 1990;32:305–14. 10.2307/2370183Search in Google Scholar

[10] Nasir J, Qaisar S, Butt SI, Khan KA, Mabela RM. Some Simpson’s Riemann-Liouville fractional integral inequalities with applications to special functions. J Funct Spaces. 2022;2022:1–12. Search in Google Scholar

[11] Nasir J, Qaisar S, Butt SI, Khan KA, Mabela RM. Some Simpson’s Riemann-Liouville fractional integral inequalities with applications to special functions. J Funct Spaces. 2022;2022:2113742. 10.1155/2022/2113742Search in Google Scholar

[12] Anderson DR, Avery RI. Fractional-order boundary value problem with Sturm-Liouville boundary conditions. Electron J Differ Equ. 2015;29:1–10. 10.1186/s13662-015-0413-ySearch in Google Scholar

[13] Wang Y. Solutions of complex difference and q-difference equations. Adv Diff Equ. 2016;98:1–22. 10.1186/s13662-016-0790-xSearch in Google Scholar

[14] Maayah B, Moussaoui A, Bushnaq S, Arqub OA. The multistep Laplace optimized decomposition method for solving fractional-order coronavirus disease model (COVID-19) via the Caputo fractional approach. Dem Math. 2022;55:963–77. 10.1515/dema-2022-0183Search in Google Scholar

[15] Jleli M, Mursaleen M, Samet B. Q-integral equations of fractional orders. Elec J Diff Equ. 2016;17:1–14. 10.1186/s13663-016-0497-4Search in Google Scholar

[16] Brown JC. Calculation of a constant Q spectral transform. J Acoust Soc Am. 1991;89:425–34. 10.1121/1.400476Search in Google Scholar

[17] Atici FM, Eloe PW. Fractional q-Calculus on a time scale. J Nonl Math Phy. 2007;14:341–52. 10.2991/jnmp.2007.14.3.4Search in Google Scholar

[18] Boumali A, Bouzenada A, Zare S. Thermal properties of the q-deformed spin-one DKP oscillator. Phy A Stat Mech App. 2023;628:1–11. 10.1016/j.physa.2023.129134Search in Google Scholar

[19] Sobhani H, Hassanabadi H, Chung WS. DKP equation in the q-deformed quantum mechanics. Few Body Syst. 2023;64:1–18. 10.1007/s00601-023-01800-5Search in Google Scholar

[20] Cadzow JA. Discrete calculus of variations. Int J Control. 1970;11:393–407. 10.1080/00207177008905922Search in Google Scholar

[21] Friesl M, Slavík A, Stehlík P. Discrete-space partial dynamic equations on time scales and applications to stochastic processes. Appl Math Lett. 2014;37:86–90. 10.1016/j.aml.2014.06.002Search in Google Scholar

[22] Tsallis C. Possible generalization of Boltzmann-Gibbs statistics. J Stat Phys. 1988;52:479–87. 10.1007/BF01016429Search in Google Scholar

[23] Nivanen L, Méhauté ALe, Wang QA. Generalized algebra within a nonextensive statistics. Rep Math Phys. 2003;52:437–44. 10.1016/S0034-4877(03)80040-XSearch in Google Scholar

[24] Chung WS, Hassanabadi H, Kriz J. The q-deformed heat equation and q-deformed diffusion equation with q-translation symmetry. R Mexic de Física. 2022;68:1–6. 10.31349/RevMexFis.68.060602Search in Google Scholar

[25] Keskin Y, Oturanç G. Reduced differential transform method for partial differential equations. Int J Nonl Sci Numer Simul. 2009;10:741–9. 10.1515/IJNSNS.2009.10.6.741Search in Google Scholar

[26] Holmberg A, Lind MN, Böhme C. Numerical analysis of the two dimensional wave equation using weighted finite differences for homogeneous and heterogeneous media. U Universit. 2020;15:1–38. Search in Google Scholar

[27] Chung WS, Hassanabadi H. Deformed classical mechanics with α-deformed translation symmetry and anomalous diffusion. Mod Phys Lett B. 2019;33:1–11. 10.1142/S0217984919503688Search in Google Scholar

[28] Chung WS, Hassanabadi H. q-deformed quantum mechanics based on the q-addition. Fortschr Phys. 2019;67:1–7. 10.1002/prop.201800111Search in Google Scholar

[29] Gupta PK. Approximate analytical solutions of fractional Benney-Lin equation by reduced differential transform method and the homotopy perturbation method. Comput Math Appl. 2011;61(9):2829–42. 10.1016/j.camwa.2011.03.057Search in Google Scholar

[30] Abazari R, Abazari M. Numerical simulation of generalized Hirota-Satsuma coupled KdV equation by RDTM and comparison with DTM. Commun Nonl Sci Numer Simul. 2012;17(2):619–29. 10.1016/j.cnsns.2011.05.022Search in Google Scholar

[31] Yıldırım K, İbiş B, Bayram M. New solutions of the non linear Fisher type equations by the reduced differential transform. Nonl Sci Lett A. 2012;3(1):29–36. 10.14419/ijamr.v1i3.112Search in Google Scholar

[32] Haghbin A, Hesam S. Reduced differential transform method for solving seventh order Sawada Kotera equations. J Math Comput Sci (JMCS). 2012;5(1):53–9. 10.22436/jmcs.05.01.06Search in Google Scholar

Received: 2023-11-22

Revised: 2024-01-15

Accepted: 2024-02-04

Published Online: 2024-03-28

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/nleng-2022-0378

Keywords for this article

q-calculus; q-deformed; q-translation; separation method; RDTM; FDM

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