1 Introduction

Despite the difficulty of studying nonlinear differential-difference equations (NDDEs), this class of equations has always been considered one of the most essential tools for researchers to describe various models in different fields of science (Parand and Delkhosh 2017; Houwe etal. 2020; Rasheed etal. 2023). The importance of NDDEs has dramatically increased with the development of new concepts, causing classical methods to be incapable of solving some of these new types of equations. The role of NPDEs and NDDEs is evident in describing many biological phenomena (Wang and Wen 2018; Parand etal. 2017b, c; Akram et al. 2023).

To achieve exact solutions in solving NPDEs and NDDEs, numerous established analytical techniques have been effective, for instance: the He-Elzaki transform method (Modanli etal. 2023), extended (\(G^{\prime }/G^{2}\))-expansion method (Akram and Zainab 2020), residual Power seriesn method (Abdulazeez et al. 2023; Tariq etal. 2023; Modanli etal. 2021), homotopy analysis approach (Abdulazeez etal. 2022; Sadaf and Akram 2021), the extended Sinh-Gordon equation expansion technique (Akram et al. 2022; Sadaf etal. 2022), the explicit finite difference method (Abdulazeez and Modanli 2022), the improved tan(\(\Psi (\zeta )/2\))-expansion method (Akram etal. 2022), the modified simple equation approach (Akram et al. 2023), modified auxiliary equation approach (Sadaf etal. 2023), generalized Kudryashov method (Akram etal. 2023), the generalized Pseudospectral approach (Delkhosh and Parand 2019; Delkhosh and Cheraghian 2022), etc.

This research aims to bring forth exact solutions using a recently developed generalized exponential function method (GERFM), This method is one of the effective methods for calculating exact wave solutions, which is used in the present article to solve the DDBE.

This method was already introduced by Ghanbari in 2018 (Ghanbari and Inc 2018), and after that year, he and other researchers have repeatedly used it for partial differential equations. For instance, Ghanbari and Kuo (2019) found exact wave solutions for the variable-coefficient (1 + 1)-dimensional Benjamin-Bona-Mahony and (2 + 1)-dimensional asymmetric Nizhnik-Novikov-Veselov equations by means of the GERFM. Also, Ghanbari etal. (2020) presented the generalized exponential function method to a novel extension of the nonlinear Schrödinger equation. Tarla etal. (2022) investigates the propagation of solitons in the Hamiltonian amplitude equation via GERFM.

The differential-difference Burgers’ equation (DDBE) that incorporates discrete terms along with continuous terms. This equation is used to model systems where both continuous and discrete processes play a role in the evolution of the variables. The differential-difference Burgers’ equation (DDBE) has applications in various fields, including hydrodynamics, fluid dynamics, nonlinear acoustic waves, mathematical physics, biological systems, numerical simulations, and materials science (Mohanty et al. 2022).

The general form of such an equation looks like this:

$$\begin{aligned} \frac{\textrm{d} {\mathfrak {u}}_k\left( t\right) }{\textrm{d}t}=\left( 1+{\mathfrak {u}}_k\left( t\right) \right) \left( {\mathfrak {u}}_{k+1} \left( t\right) -{\mathfrak {u}}_k\left( t\right) \right) . \end{aligned}$$
(1)

Where \({\mathfrak {u}}_k\left( t\right) =\mathfrak {u}\left( k,t\right)\), \(k\in {\mathbb {Z}}\), This equation in 2010 by Aslan (2012) was solved using the discrete (\(G^{\prime }/G\))-expansion method.

The generality of GERFM is described in Sect. 2, where we address the requirements of this approach and the categorization of the solutions. Section 3 presents the GERFM for solving the differential-difference Burgers’ equation and the associated traveling wave solutions utilizing 3D and contour diagrams. In the final part, we deliver a summary of our findings.

2 Procedure analysis

In this section, the GERFM method is briefly explained. Consider the differential-difference equation in the following form:

$$\begin{aligned} \begin{aligned}&{\mathcal {N}}({\mathfrak {u}}_{m+p_1}\left( t\right) ,\dots ,{\mathfrak {u}}_{m+p_k}\left( t\right) ,\dots ,{{\mathfrak {u}}^{\alpha }}_{m+p_1}\left( t\right) ,\dots , \\&{{\mathfrak {u}}^{\alpha }}_{m+p_k}\left( t\right) ,\dots ,{{\mathfrak {u}}^{\left( r\alpha \right) }}_{m+p_1}\left( t\right) ,\dots ,{{\mathfrak {u}}^{\left( r\alpha \right) }}_{m+p_k}\left( t\right) )=0, \end{aligned} \end{aligned}$$
(2)

Using the following traveling wave transform

$$\begin{aligned} \begin{aligned}&{\mathfrak {u}}_{k+p_s}\left( t\right) ={\mathfrak {U}}_k\left( {\zeta }_k\right) , \quad s=1,2,\dots ,k,\\&{\zeta }_k=\textrm{d}k+ct+\zeta _0. \\ \end{aligned} \end{aligned}$$
(3)

Where d and c are constants to be determined later, then Eq. (2) reduces to an ODE of the following form:

$$\begin{aligned} \begin{aligned}&{\mathcal {Q}}({\mathfrak {U}}_{m+p_1}\left( {\zeta }_n\right) ,\dots ,{\mathfrak {U}}_{m+p_k}\left( {\zeta }_n\right) ,\dots ,{{\mathfrak {U}}'}_{m+p_1}\left( {\zeta }_n\right) ,\dots ,\\&{{\mathfrak {U}}'}_{m+p_k}\left( {\zeta }_n\right) ,\dots ,{{\mathfrak {U}}^{\left( r\right) }}_{m+p_1}\left( {\zeta }_n\right) , \dots ,{{\mathfrak {U}}^{\left( r\right) }}_{m+p_k} \left( {\zeta }_n\right) )=0.\\ \end{aligned} \end{aligned}$$
(4)

This technique is built using a hypothetical solution that can be displayed as follows:

$$\begin{aligned}{} & {} {\mathcal {Y}}\left( \zeta _k \right) ={{\mathcal {V}}}_0+\sum ^{a_0}_{\mathfrak {i}=1}{{{\mathcal {V}}}_{\mathfrak {i}}{{\mathbb {L}}}^{\mathfrak {i}}\left( \zeta _k \right) }+\sum ^{a_0}_{\mathfrak {i}=1}{\frac{{{\mathcal {Z}}}_{\mathfrak {i}}}{{{\mathbb {L}}}^{\mathfrak {i}}\left( \zeta _k \right) }}. \end{aligned}$$
(5)
$$\begin{aligned}{} & {} {\mathbb {L}}\left( \zeta _k \right) =\frac{r_1\exp \left( {\gamma }_1\zeta _k \right) +r_2\exp \left( {\gamma }_2\zeta _k \right) }{r_3\exp \left( {\gamma }_3\zeta _k \right) +r_4\exp \left( {\gamma }_4\zeta _k \right) }. \end{aligned}$$
(6)

In Eq. (5), \({{\mathcal {V}}}_0,{{\mathcal {V}}}_{\mathfrak {i}},{{\mathcal {Z}}}_{\mathfrak {i}}\left( 1\le \mathfrak {i}\le a_0\right)\), and in Eq. (6) \(r_{\mathfrak {i}},{\gamma }_{\mathfrak {i}},\left( 1\le \mathfrak {i}\le 4\right)\), Recall that we use the balance rules to determine the positive integer \(a_0\).

Now, in this research, we show that the following relations can be considered in Eq. (4) as follows:

$$\begin{aligned} \begin{aligned}&{\mathfrak {u}}_{k+1}(t)={\mathfrak {U}}\left( {\zeta }_{k+1}\right) ={\mathfrak {U}}\left( {\zeta }_k+d\right) ,\\&{\mathfrak {u}}_{k-1}(t)={\mathfrak {U}}\left( {\zeta }_{k-1}\right) ={\mathfrak {U}}\left( {\zeta }_k-d\right) .\\ \end{aligned} \end{aligned}$$
(7)

By inserting Eq. (5) into Eq. (4), we obtain a polynomial, which indicates that the analytical solutions for Eq. (2) are obtained.

3 Applications of GERFM for the DDBE

To solve Eq. (1), using the method of GERFM, as explained in the previous section, we first apply the transformation \({\mathfrak {u}_k}\left( t\right) ={\mathfrak {U}}\left( \zeta _k \right)\), \({\zeta }_k=\textrm{d}k+ct+\zeta _0\), where d and c are constants to be determined, and \(\zeta _0\) is an arbitrary phase constant, to arrive at the following ordinary differential equation:

$$\begin{aligned} c{\mathfrak {U}}^{\prime }(\zeta _k)=\left( 1+{\mathfrak {U}}\left( {\zeta }_k\right) \right) \left( {\mathfrak {U}}_{k+1}\left( {\zeta }_k\right) -{\mathfrak {U}}\left( {\zeta _k }\right) \right) . \end{aligned}$$
(8)

According to the method explained in the previous section, balancing the highest-order linear term with the highest nonlinear term in Eq. (8) yields \(a_0=1\). Therefore, we will have the solution of the equation in the following hypothetical form:

$$\begin{aligned} {\mathcal {Y}}\left( \zeta _k \right) ={{\mathcal {V}}}_0+{{\mathcal {V}}}_1{\mathbb {L}}\left( \zeta _k \right) +\frac{{{\mathcal {Z}}}_1}{{\mathbb {L}}\left( \zeta _k \right) }. \end{aligned}$$
(9)

Where the constant coefficients of \({{\mathcal {V}}}_0\), \({{\mathcal {V}}}_1\), \({{\mathcal {Z}}}_1\) are determined in the next step, and \({\mathbb {L}}\left( \zeta _k \right)\) is obtained according to Eq. (5).

Category I: catching \(\left[ r_1 \; r_2 \;r_3\; r_4 \right] =\left[ -1 \; 1\; 1 \; 1 \right]\) and \(\left[ {\gamma }_1 \; {\gamma }_1 \; {\gamma }_1 \; {\gamma }_1 \right] =\left[ 1 \; -1 \; 1 \; -1 \right]\) in Eq. (5) render:

$$\begin{aligned} {\mathbb {L}}\left( \zeta _k \right) =-{\textrm{tanh} (\zeta _k }). \end{aligned}$$
(10)

By putting Eq. (9) into Eq. (8) and collecting coefficients of \({\mathbb {L}}^{i}\left( \zeta _k \right)\) for \(i=0,1,2...\) and setting them to zero, we derive a system of algebraic equations. By solving this system using MAPLE software, we can determine the unknown parameters. Therefore, the many various anthologies are obtained as follows:

Set 1:

$$\begin{aligned} c= -{{\mathcal {V}}}_1, \quad {{\mathcal {V}}}_0=-(2 {{\mathcal {V}}}_1+1)tanh(2d), \quad {{\mathcal {Z}}}_1={{\mathcal {V}}}_1. \end{aligned}$$

Where \(d, {{\mathcal {V}}}_1\) are constant. By substituting the values of the above obtained parameters into Eq. (8), the following traveling wave solution is obtained:

$$\begin{aligned} {\mathcal {Y}}\left( \zeta _k \right) =-(2 {{\mathcal {V}}}_1+1)tanh(2d)-{{\mathcal {V}}}_1 tanh(\zeta _k)-\frac{{{\mathcal {V}}}_1}{tanh(\zeta _k)}. \end{aligned}$$
(11)

Where \(\zeta _k=kd -{{\mathcal {V}}}_1 t +\zeta _0.\)

The graphical diagrams related to this traveling wave solution Eq. (11) is presented in Fig. 1.

Set 2:

$$\begin{aligned} c= -{{\mathcal {V}}}_1, \quad {{\mathcal {V}}}_0=-1- {{\mathcal {V}}}_1 coth(d), \quad {{\mathcal {Z}}}_1=0. \end{aligned}$$

Where \(d, {{\mathcal {V}}}_1\) are constant. By substituting the values of the above obtained parameters into Eq. (8), the following traveling wave solution is obtained:

$$\begin{aligned} {\mathcal {Y}}\left( \zeta _k \right) =-1- {{\mathcal {V}}}_1 coth(d)-{{\mathcal {V}}}_1 tanh(\zeta _k). \end{aligned}$$
(12)

Where \(\zeta _k=kd -{{\mathcal {V}}}_1 t +\zeta _0.\)

The graphical diagrams related to this traveling wave solution Eq. (12) is presented in Fig. 2.

Set 3:

$$\begin{aligned} c= -{{\mathcal {Z}}}_1, \quad {{\mathcal {V}}}_0=-1- {{\mathcal {Z}}}_1 coth(d), \quad {{\mathcal {V}}}_1=0. \end{aligned}$$

Where \(d, {{\mathcal {Z}}}_1\) are constant. By substituting the values of the above obtained parameters into Eq. (8), the following traveling wave solution is obtained:

$$\begin{aligned} {\mathcal {Y}}\left( \zeta _k \right) =-1- {{\mathcal {Z}}}_1 coth(d)-\frac{{{\mathcal {Z}}}_1}{tanh(\zeta _k)}. \end{aligned}$$
(13)

Where \(\zeta _k=kd -{{\mathcal {Z}}}_1 t +\zeta _0.\)

The graphical diagrams related to this traveling wave solution Eq. (13) is presented in Fig. 3.

Category II: catching \(\left[ r_1 \; r_2 \; r_3 \; r_4 \right] =\left[ i \; -i \; 1 \; 1 \right]\) and \(\left[ {\gamma }_1 \;{\gamma }_1 \; {\gamma }_1 \; {\gamma }_1 \right] =\left[ -i \; i \; -i \; i \right]\) in Eq. (5) render:

$$\begin{aligned} {\mathbb {L}}\left( \zeta _k \right) =-tan (\zeta _k). \end{aligned}$$
(14)

the many various anthologies are obtained as follows:

Set 1:

$$\begin{aligned} c= {{\mathcal {V}}}_1, \quad {{\mathcal {V}}}_0=\frac{-\left( \tan ^{2}\left( d \right) \right) {{\mathcal {V}}}_1-\tan \! \left( d \right) +{{\mathcal {V}}}_1}{\tan \! \left( d \right) }, \quad {{\mathcal {Z}}}_1=-{{\mathcal {V}}}_1. \end{aligned}$$

Where \(d, {{\mathcal {V}}}_1\) are constant. By substituting the values of the above obtained parameters into Eq. (8), the following traveling wave solution is obtained:

$$\begin{aligned} {\mathcal {Y}}\left( \zeta _k \right) =\frac{-\left( \tan ^{2}\left( d \right) \right) {{\mathcal {V}}}_1-\tan \! \left( d \right) +{{\mathcal {V}}}_1}{\tan \! \left( d \right) }-{{\mathcal {V}}}_1 tan(\zeta _k)+\frac{{{\mathcal {V}}}_1}{tan(\zeta _k)}. \end{aligned}$$
(15)

Where \(\zeta _k=kd +{{\mathcal {V}}}_1 t +\zeta _0.\)

The graphical diagrams related to this traveling wave solution Eq. (15) is presented in Fig. 4.

Set 2:

$$\begin{aligned} c= \tan \! \left( d \right) \left( {{\mathcal {V}}}_0 +1\right) , \quad {{\mathcal {V}}}_1=\tan \! \left( d \right) \left( {{\mathcal {V}}}_0 +1\right) , \quad {{\mathcal {Z}}}_1=0. \end{aligned}$$

Where \(d, {{\mathcal {V}}}_0\) are constant. By substituting the values of the above obtained parameters into Eq. (8), the following traveling wave solution is obtained:

$$\begin{aligned} {\mathcal {Y}}\left( \zeta _k \right) ={{\mathcal {V}}}_0 -\tan \! \left( d \right) \left( {{\mathcal {V}}}_0 +1\right) tan(\zeta _k). \end{aligned}$$
(16)

Where \(\zeta _k=kd +\tan \! \left( d \right) \left( {{\mathcal {V}}}_0 +1\right) t +\zeta _0.\)

The graphical diagrams related to this traveling wave solution Eq. (16) is presented in Fig. 5.

Set 3:

$$\begin{aligned} c= \tan \! \left( d \right) \left( {{\mathcal {V}}}_0 +1\right) , \quad {{\mathcal {V}}}_1=0, \quad {{\mathcal {Z}}}_1=-\tan \! \left( d \right) \left( {{\mathcal {V}}}_0 +1\right) . \end{aligned}$$

Where \(d, {{\mathcal {V}}}_0\) are constant. By substituting the values of the above obtained parameters into Eq. (8), the following traveling wave solution is obtained:

$$\begin{aligned} {\mathcal {Y}}\left( \zeta _k \right) ={{\mathcal {V}}}_0+\frac{\tan \! \left( d \right) \left( {{\mathcal {V}}}_0 +1\right) }{tan(\zeta _k)}. \end{aligned}$$
(17)

Where \(\zeta _k=kd +\tan \! \left( d \right) \left( {{\mathcal {V}}}_0 +1\right) t +\zeta _0.\)

The graphical diagrams related to this traveling wave solution Eq. (17) is presented in Fig. 6.

Category III: catching \(\left[ r_1 \; r_2 \; r_3 \; r_4 \right] =\left[ 1 \;1\; -1 \; 1 \right]\) and \(\left[ {\gamma }_1 \; {\gamma }_1 \; {\gamma }_1 \; {\gamma }_1 \right] =\left[ 1 \; -1 \; 1\; -1 \right]\) in Eq. (5) render:

$$\begin{aligned} {\mathbb {L}}\left( \zeta _k \right) =-coth(\zeta _k). \end{aligned}$$
(18)

The many various anthologies are obtained as follows:

Set 1:

$$\begin{aligned} c= -{{\mathcal {V}}}_1, \quad {{\mathcal {V}}}_0=-(2 {{\mathcal {V}}}_1+1)tanh(2d), \quad {{\mathcal {Z}}}_1={{\mathcal {V}}}_1. \end{aligned}$$

Where \(d, {{\mathcal {V}}}_1\) are constant. By substituting the values of the above obtained parameters into Eq. (8), the following traveling wave solution is obtained:

$$\begin{aligned} {\mathcal {Y}}\left( \zeta _k \right) =-(2 {{\mathcal {V}}}_1+1)tanh(2d)-{{\mathcal {V}}}_1 coth(\zeta _k)-\frac{{{\mathcal {V}}}_1}{coth(\zeta _k)}. \end{aligned}$$
(19)

Where \(\zeta _k=kd -{{\mathcal {V}}}_1 t +\zeta _0.\)

The graphical diagrams related to this traveling wave solution Eq. (19) is presented in Fig. 7.

Set 2:

$$\begin{aligned} c= -{{\mathcal {V}}}_1, \quad {{\mathcal {V}}}_0=-1- {{\mathcal {V}}}_1 coth(d), \quad {{\mathcal {Z}}}_1=0. \end{aligned}$$

Where \(d, {{\mathcal {V}}}_1\) are constant. By substituting the values of the above obtained parameters into Eq. (8), the following traveling wave solution is obtained:

$$\begin{aligned} {\mathcal {Y}}\left( \zeta _k \right) =-1- {{\mathcal {V}}}_1 coth(d)-{{\mathcal {V}}}_1 coth(\zeta _k). \end{aligned}$$
(20)

Where \(\zeta _k=kd -{{\mathcal {V}}}_1 t +\zeta _0.\)

The graphical diagrams related to this traveling wave solution Eq. (20) is presented in Fig. 8.

Set 3:

$$\begin{aligned} c= -{{\mathcal {Z}}}_1, \quad {{\mathcal {V}}}_0=-1- {{\mathcal {Z}}}_1 coth(d), \quad {{\mathcal {V}}}_1=0. \end{aligned}$$

Where \(d, {{\mathcal {Z}}}_1\) are constant. By substituting the values of the above obtained parameters into Eq. (8), the following traveling wave solution is obtained:

$$\begin{aligned} {\mathcal {Y}}\left( \zeta _k \right) =-1- {{\mathcal {Z}}}_1 coth(d)-\frac{{{\mathcal {Z}}}_1}{coth(\zeta _k)}. \end{aligned}$$
(21)

Where \(\zeta _k=kd -{{\mathcal {Z}}}_1 t +\zeta _0.\)

The graphical diagrams related to this traveling wave solution Eq. (21) is presented in Fig. 9.

Category IV: catching \(\left[ r_1 \; r_2 \; r_3 \;r_4 \right] =\left[ -1\; 0\; 1 \; 1 \right]\) and \(\left[ {\gamma }_1 \; {\gamma }_1 \; {\gamma }_1 \; {\gamma }_1 \right] =\left[ 0 \; 0 \; 0 \; 1 \right]\)in Eq. (5) render:

$$\begin{aligned} {\mathbb {L}}\left( \zeta _k \right) =-{\textrm{cot} \left( {\zeta }_k\right) }. \end{aligned}$$
(22)

The many various anthologies are obtained as follows:

Set 1:

$$\begin{aligned} c=- {{\mathcal {V}}}_1, \quad {{\mathcal {V}}}_0=\frac{\left( \tan ^{2}\left( d \right) \right) {{\mathcal {V}}}_1-\tan \! \left( d \right) -{{\mathcal {V}}}_1}{\tan \! \left( d \right) }, \quad {{\mathcal {Z}}}_1=-{{\mathcal {V}}}_1. \end{aligned}$$

Where \(d, {{\mathcal {V}}}_1\) are constant. By substituting the values of the above obtained parameters into Eq. (8), the following traveling wave solution is obtained:

$$\begin{aligned} {\mathcal {Y}}\left( \zeta _k \right) =\frac{\left( \tan ^{2}\left( d \right) \right) {{\mathcal {V}}}_1-\tan \! \left( d \right) -{{\mathcal {V}}}_1}{\tan \! \left( d \right) }-{{\mathcal {V}}}_1 cot(\zeta _k)+\frac{{{\mathcal {V}}}_1}{cot(\zeta _k)}. \end{aligned}$$
(23)

Where \(\zeta _k=kd -{{\mathcal {V}}}_1 t +\zeta _0.\)

The graphical diagrams related to this traveling wave solution Eq. (23) is presented in Fig. 10.

Set 2:

$$\begin{aligned} c=-{{\mathcal {V}}}_1, \quad {{\mathcal {V}}}_0=-{{\mathcal {V}}}_1 cot(d) - 1, \quad {{\mathcal {Z}}}_1=0. \end{aligned}$$

Where \(d, {{\mathcal {V}}}_1\) are constant. By substituting the values of the above obtained parameters into Eq. (8), the following traveling wave solution is obtained:

$$\begin{aligned} {\mathcal {Y}}\left( \zeta _k \right) =-{{\mathcal {V}}}_1 cot(d) - 1-{{\mathcal {V}}}_1 cot(\zeta _k). \end{aligned}$$
(24)

Where \(\zeta _k=kd -{{\mathcal {V}}}_1 t +\zeta _0.\)

The graphical diagrams related to this traveling wave solution Eq. (24) is presented in Fig. 11.

Set 3:

$$\begin{aligned} c={{\mathcal {Z}}}_1, \quad {{\mathcal {V}}}_0={{\mathcal {Z}}}_1 cot(d) - 1, \quad {{\mathcal {V}}}_1=0. \end{aligned}$$

Where \(d, {{\mathcal {Z}}}_1\) are constant. By substituting the values of the above obtained parameters into Eq. (8), the following traveling wave solution is obtained:

$$\begin{aligned} {\mathcal {Y}}\left( \zeta _k \right) ={{\mathcal {Z}}}_1 cot(d) - 1-\frac{{{\mathcal {Z}}}_1}{cot(\zeta _k)}. \end{aligned}$$
(25)

Where \(\zeta _k=kd +{{\mathcal {Z}}}_1 t +\zeta _0.\)

The graphical diagrams related to this traveling wave solution Eq. (25) is presented in Fig. 12.

Fig. 1
figure 1

The graphical 3D and contour diagrams related to the solution of Eq. (11) with parameters \(d=1, V_1 = 2, \zeta _0=0.05\)

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Fig. 2
figure 2

The graphical 3D and contour diagrams related to the solution of Eq. (12) with parameters \(d=1, V_1 = 2, \zeta _0=0.05\)

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Fig. 3
figure 3

The graphical 3D and contour diagrams related to the solution of Eq. (13) with parameters \(d=1, Z_1 = 3, \zeta _0=0.05\)

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Fig. 4
figure 4

The graphical 3D and contour diagrams related to the solution of Eq. (15) with parameters \(d=1, V_1 = -2, \zeta _0=0.05\)

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Fig. 5
figure 5

The graphical 3D and contour diagrams related to the solution of Eq. (16) with parameters \(d=1, V_0 = -2, \zeta _0=0.05\)

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Fig. 6
figure 6

The graphical 3D and contour diagrams related to the solution of Eq. (17) with parameters \(d=1, V_0 = -2, \zeta _0=0.05\)

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Fig. 7
figure 7

The graphical 3D and contour diagrams related to the solution of Eq. (19) with parameters \(d=1, V_1 = 2, \zeta _0=0.05\)

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Fig. 8
figure 8

The graphical 3D and contour diagrams related to the solution of Eq. (20) with parameters \(d=1, V_1 = 2, \zeta _0=0.05\)

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Fig. 9
figure 9

The graphical 3D and contour diagrams related to the solution of Eq. (21) with parameters \(d=1, Z_1 = 3, \zeta _0=0.05\)

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Fig. 10
figure 10

The graphical 3D and contour diagrams related to the solution of Eq. (23) with parameters \(d=1, V_1 = 2, \zeta _0=0.05\)

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Fig. 11
figure 11

The graphical 3D and contour diagrams related to the solution of Eq. (24) with parameters \(d=1, V_1 = 2, \zeta _0=0.05\)

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Fig. 12
figure 12

The graphical 3D and contour diagrams related to the solution of Eq. (25) with parameters \(d=1, Z_1 = 3, \zeta _0=0.05\)

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4 Conclusion

In this article, we obtained several solutions for the differential-difference Burgers’ equation using generalized exponential functions, such as hyperbolic and trigonometric function solutions. Also, the singular soliton, multi-soliton, and kink-soliton solutions were obtained, each having a different form that is used to describe physical phenomena. We displayed the 3-D and contour graphs associated with the solutions. Based on the findings of this research, the proposed strategy offers promising solutions to this problem.