Modeling Fractional Dust-Acoustic Shock Waves in a Complex Plasma Using Novel Techniques

Abstract

This work investigates how fractionality affects the dynamical behavior of dust-acoustic shock waves that arise and propagate in a depleted-electron complex plasma. This model consists of inertial negatively charged dust grains and inertialess nonextensive distributed ions. Initially, the fluid model equations that govern the propagation of nonlinear dust-acoustic shock waves are reduced to the integer Burgers-type equations using the reductive perturbation method. Thereafter, the integer Burgers-type equations are converted to the fractional cases using a suitable transformation. For analyzing this fractional family, both the Tantawy technique and the new iterative method are implemented within the Caputo sense framework. These methods can produce highly accurate analytical approximations without necessitating stringent assumptions or intricate computational processes, in contrast to other similar methods. Numerical examples and the calculation of the absolute error demonstrate the efficacy of the suggested methodologies, emphasizing their superior precision and swift convergence.

1. Introduction

Fractional calculus (FC) makes significant contributions to various scientific disciplines, including nonlinear structures in plasma physics [1,2,3,4], fluid mechanics [5,6], chemistry, diffusive transport, heat transfer [7], heat flow [8], modeling biological phenomena [9], electrical engineering [10,11], electromagnetic theory [12], and prediction of groundwater flow [13]. The FC has successfully modeled various physical and engineering phenomena, providing deep explanations for many previously unexplained phenomena and their properties that were not understood using traditional calculus. Hence, the idea of re-simulating various physical, chemical, biological, and engineering problems from the perspective of fractional calculus arose to gain a deeper understanding of the phenomena that occur in these environments [14,15,16,17].

As is well known, solving integer differential equations is easier than solving nonlinear fractional differential equations (FDEs), primarily since most FDEs do not support exact solutions. Therefore, several challenges and difficulties arise in analyzing these types of FDEs. To address these difficulties, various methods have been developed to solve this type of FDE. Some of the analytical and numerical methods that have been successful in analyzing (non)linear differential and integral equations in both integer and fractional forms are the finite difference method (FDM) [18,19], finite element method (FEM) [20], the discontinuous Galerkin method [21], the variational iteration method (VIM) [2,3,22], the Adomian decomposition method (ADM) [23,24], the homotopy perturbation method (HPM) [25,26], the homology analysis method (HAM) [27,28,29], the residual power series method (RPSM) [30,31], the new iteration method (NIM) [32,33], and many others. Each of the above methods has its advantages and disadvantages.

This study examined two types of fractional evolutionary wave equations (EWEs), specifically, the nonlinear one- and two-dimensional fractional Burgers equations (FBEs), which are widely used in various interdisciplinary fields, especially in plasma physics, water waves, and fluid mechanics. In the current investigation, the following three models will be analyzed:

Model (I): 

For a collisionless unmagnetized complex plasma having dust kinematic viscosity [34], the fluid model equations that are governed the dynamics of nonlinear dust-acoustic (DA) shock waves can be reduced to the following one-dimensional integer Burgers equation (BE) using the reductive perturbation method (RPM)

$${\mathsf{\partial }}_t\Phi +A\Phi {\mathsf{\partial }}_x\Phi -B{\mathsf{\partial }}_x^2\Phi =0,$$

(1)

where the coefficients on the nonlinear term A and the dissipative term B are functions of various relevant plasma model parameters. Equation (1) supports a hierarchy of multi-shock wave solutions, including the following single-shock wave solution:

$$\Phi \left(x,t\right)={\Phi }_{max}\left(1-tanh\left({\displaystyle \frac{x-u_{0}t}{\delta }}\right)\right),$$

(2)

where ${\Phi }_{max}=u_0/A$ and $\delta =2B/u_0$ represent the peak amplitude and width of shock waves, whereas $u_0$ denotes the shock wave speed.

To investigate the memory impact on the profile of shock waves described by Equation (1), we follow El-Wakil’s investigations, as illustrated in detail in Refs. [2,3]. Consequently, the one-dimensional integer BE (1) can be converted to the following homogenous one-dimensional fractional BE (FBE)

$$D_t^p\Phi +A\Phi {\mathsf{\partial }}_x\Phi -B{\mathsf{\partial }}_x^2\Phi =0,$$

(3)

with the IC

$$\Phi \left(x,0\right)\equiv {\Phi }_0={\Phi }_{max}\left(1-tanh\left({\displaystyle \frac{x}{\delta }}\right)\right).$$

(4)

Model (II): 

If the perturbations are considered in two dimensions, in this case, the fluid model equations that govern a physical model can be reduced to a two-dimensional BE [35]

$${\mathsf{\partial }}_t\Phi +A\Phi \left({\mathsf{\partial }}_x\Phi +{\mathsf{\partial }}_y\Phi \right)-B\left({\mathsf{\partial }}_x^2\Phi +{\mathsf{\partial }}_y^2\Phi \right)=0,$$

(5)

where $\Phi \equiv \Phi \left(x,y,t\right)$.

The exact one-shock wave solution for Equation (5) reads

$$\Phi \left(x,y,t\right)={\displaystyle \frac{\lambda }{A+Aexp\left(\frac{\lambda \left(x+y-\lambda t\right)}{2B}\right)}}.$$

(6)

By considering the memory impact, we follow El-Wakil’s investigations as discussed in detail in Refs. [2,3], where, ultimately, the two-dimensional integer BE (5) can be converted to the following two-dimensional FBE.

$$D_t^p\Phi +\Phi \left({\mathsf{\partial }}_x\Phi +{\mathsf{\partial }}_y\Phi \right)-B\left({\mathsf{\partial }}_x^2\Phi +{\mathsf{\partial }}_y^2\Phi \right)=0,\forall 0<p\le 1,$$

(7)

with the IC

$$\Phi \left(x,y,0\right)\equiv {\Phi }_0={\displaystyle \frac{\lambda }{A+Aexp\left(\frac{\lambda \left(x+y\right)}{2B}\right)}}.$$

(8)

The Burgers-type equation is the most elementary nonlinear model for characterizing diffusive waves in plasma physics, fluid dynamics, and various other disciplines. This family is a dissipative term widely used in modeling many physical problems, such as shock wave propagation in plasma physics, waves in fluid-filled viscous elastic tubes, sound waves in a viscous medium, magnetohydrodynamic waves in a medium with finite electrical conductivity, one-dimensional turbulence, hydrodynamic waves, elastic waves, gas dynamics, and many other fields [36,37,38,39,40,41].

Various types of space/time-fractional Burgers-type equations were analyzed using a range of numerical and analytical methods, including cubic B-spline finite elements [42], Newton’s polynomial approach [43], the HAM [44], the fractional Riccati expansion approach [45], the VIM [46], a linear FDM [47], etc. However, in this investigation, two crucial and effective modern methods, namely, the Tantawy technique (TT) [48,49,50,51,52,53,54,55,56,57] and the NIM [58,59,60,61], are applied to analyze and model DA shock waves in a depleted-electron collisionless, unmagnetized dusty plasma consisting of negatively charged dust particles and non-Maxwellian ions, subject to the nonextensive distribution. The TT is a recent and highly effective approach for studying various types of FDEs, offering superior flexibility compared to alternative methods. In early 2025, this technique was introduced and initially applied to analyze certain nonlinear physical FDEs, including the fractional KdV–Burgers and Burgers equations [48]. Although both the NIM and RPSM are among the most widely used methods for analyzing various types of FDEs, the TT offers greater flexibility in its application and lower computational cost compared to these methods. Additionally, the TT successfully analyzed the fractional Fokker–Planck (FFP) equations, and its results were compared with those obtained using the optimal auxiliary function method (OAFM) [49]. The TT was found to produce better results than the OAFM [49]. Moreover, the TT successfully analyzed the fractional nonlinear fourth-order Cahn–Hilliard equations [50]. Due to the promising results of this technique, it was used to study various fractional nonlinear phenomena in different plasma systems. For instance, it was applied to investigate fractional ion-acoustic KdV solitons in non-Maxwellian, unmagnetized, electronegative plasmas (ENPs) by analyzing the planar fractional KdV (FKdV) equation [51]. It was found that the plasma model studied in Ref. [51] supports both positive and negative solitary waves; therefore, the same research team investigated the IA fractional mKdV-solitons for the fractional modified KdV (FmKdV) equation, which is only applicable for studying the acoustic waves at a specific critical values where the nonlinearity coefficient of the quadratic FKdV equation vanishes [52]. On the other hand, the TT was applied to study the electron-acoustic fractional KdV cnoidal waves (CWs) in a homogeneous, unmagnetized two-electron plasma [53]. Additionally, nonlinear IA FKdV-solitons were investigated in a collisionless, unmagnetized, nonthermal plasma [54]. Moreover, the NIM has been used to analyze many fractional evolutionary wave equations, and it was found that the approximations generated using this method are distinguished by their high accuracy [62,63,64]. Building on the research of the TT and the NIM in examining different nonlinear fractional phenomena that arise and propagate in different plasma models, this study will look into the characteristics of fractional DA shock waves that arise and propagate in electron-depleted nonthermal dusty plasmas made up of negatively charged dust particles and nonthermal ions, taking into account the dust’s kinematic viscosity [34]. In addition, in this investigation, the approximations generated using the TT will be compared with those generated using the NIM to assess their precision.

2. Basic Definitions

This section presents definitions and theorems pertinent to the analysis of the equations considered.

(I)

For $m-1<p\le m$, ∀$m\in \mathbb{N}$, the Caputo fractional time derivative (CFTD) of order p to the function $\Phi \equiv \Phi (x,t)$ reads [65,66]

$$D_t^p\Phi \left(x,t\right)={\displaystyle \frac{1}{\Gamma (m-p)}}{\int }_0^t{(t-T)}^{m-p-1}{\Phi }^{\left(m\right)}(x,T)dT,$$

(9)

and for $m=p$, we get $D_t^m\Phi \left(x,t\right)\equiv {\mathsf{\partial }}_t^m\Phi \left(x,t\right)\equiv {\displaystyle \frac{{\mathsf{\partial }}^m\Phi \left(x,t\right)}{\mathsf{\partial }{t}^m}}$.

Note that for $\Phi \left(x,t\right)=t^{\alpha }$, we get [65,66]

$$D_t^p{t}^{\alpha }={\displaystyle \frac{\Gamma (\alpha +1)}{\Gamma (\alpha +1-p)}}{t}^{\alpha -p},\forall m-1<p\le m,\alpha >m-1\&\left(p,\alpha \right)\in \mathbb{R}$$

(10)

For $0<p<1$, the CFTD definition (9) can be reduced to the following form $\left(m=1\right)$ [67]

$$D_t^p\Phi ={\displaystyle \frac{1}{\Gamma (1-p)}}{\int }_0^t{(t-T)}^{-p}{\mathsf{\partial }}_T\Phi (x,T)dT.$$

(11)

Note that for $p\to 1$, the CFTD converges to the standard first-order time derivative [4], i.e., $D_t^1\Phi ={\mathsf{\partial }}_t\Phi$.

(II)

The Laplace transform (LT) for the function $\Phi \equiv \Phi (x,t)$ and of exponential order $\xi$ is given by [65,66];

$$\mathcal{L}[\Phi ]\equiv \Psi (x,s)={\int }_0^{\infty }{e}^{-st}\Phi \left(x,t\right)dt,t\ge 0.$$

(12)

For $m-1<p\le m$, where $m\in \mathbb{N}$, LT to the CFD $D_t^p\Phi (x,t)$ is defined by [65,66]

$$\mathcal{L}\left[D_t^p\Phi \left(x,t\right)\right]=s^p\mathcal{L}\left[\Phi \left(x,t\right)\right]-\sum _{k=0}^{m-1}{\displaystyle \frac{D_t^k\Phi \left(x,0\right)}{s^{k-p+1}}}.$$

(13)

3. Physical Plasma Model and Fractional Burgers-Type Equations

Plasma physics is recognized as one of the most fertile physical disciplines, encompassing a wide range of nonlinear phenomena, including the propagation of solitary waves, shock waves, rogue waves, periodic waves, and many others. All these phenomena are nonlinear and can be explained by reducing the fluid equations for any plasma system into one of the wave equations, which can describe many of the nonlinear structures mentioned earlier, as well as others. For example, we consider an electron-depleted, unmagnetized, collisionless plasma consisting of inertial negatively charged dust grains and inertialess nonextensive ions [34]. Thus, the neutrality condition is expressed as $n_i^{\left(0\right)}=Z_d{n}_d^{\left(0\right)}$, where $Z_d$ denotes the charge number of electrons that reside on the surface of the dust grains, while $n_d^{\left(0\right)}$ and $n_i^{\left(0\right)}$ signify the unperturbed number densities of the dust grains and ions, respectively. This model presents the dimensionless fluid equations that regulate the nonlinear dynamics of low-frequency dust-acoustic waves as follows:

$$\left\{\begin{array}{c}{\mathsf{\partial }}_{\tau }{n}_d+{\mathsf{\partial }}_{\xi }\left(n_d{u}_d\right)=0,\\ {\mathsf{\partial }}_{\tau }{u}_d+u_d{\mathsf{\partial }}_{\xi }{u}_d-{\mathsf{\partial }}_{\xi }\varphi -\eta {\mathsf{\partial }}_{\xi }^2{u}_d=0,\\ {\mathsf{\partial }}_{\xi }^2\varphi -n_d+n_i=0,\end{array}\right.$$

(14)

with the following normalized nonextensive electrons and ion number densities:

$$\begin{array}{cc}\hfill n_i& ={\mu }_i{\left[1-\left(q-1\right)\varphi \right]}^{{\displaystyle \frac{q+1}{2\left(q-1\right)}}}\hfill \\ \hfill & ={\alpha }_0-{\alpha }_1\varphi -{\alpha }_2{\varphi }^2+\dots ,\hfill \end{array}$$

(15)

with

$${\alpha }_0={\mu }_i,{\alpha }_1={\displaystyle \frac{{\mu }_i}{2}}\left(q+1\right)\&{\alpha }_2={\displaystyle \frac{{\mu }_i}{8}}\left(q^2-2q-3\right).$$

Here, $n_d$ and $n_i$ are, respectively, the normalized number density of dust particles and ions, while $u_d$ indicates the normalized velocity of the fluid dust particles, $\varphi$ denotes the normalized electrostatic wave potential, and $\eta$ represents the normalized dust viscosity coefficient. For more details on this model, the reader is referred to Ref. [34]. The independent variables $\xi$ and $\tau$ refer to normalized space and time.

For studying the characteristic behavior of the nonlinear dust-acoustic shock waves that can arise and propagate in this model, the reductive perturbation method is applied. Consequently, the independent variables $\left(\xi ,\tau \right)$ are stretched as follows:

$$x=\epsilon \left(\xi -{\lambda }_{ph}\tau \right)\&t={\epsilon }^2\tau ,$$

(16)

and the dependent perturbed physical quantities $\left(n_d,u_d,\varphi \right)\equiv \left(n\left(\xi ,\tau \right),u\left(\xi ,\tau \right),\varphi \left(\xi ,\tau \right)\right)$ are expanded about their equilibrium values as follows:

$$\left\{\begin{array}{c}{n}_d=1+\epsilon n_d^{\left(1\right)}+{\epsilon }^2{n}_d^{\left(2\right)}+{\epsilon }^3{n}_d^{\left(3\right)}+\dots ,\\ u_d=0+\epsilon u_d^{\left(1\right)}+{\epsilon }^2{u}_d^{\left(2\right)}+{\epsilon }^3{u}_d^{\left(3\right)}+\dots ,\\ \varphi =0+\epsilon {\varphi }^{\left(1\right)}+{\epsilon }^2{\varphi }^{\left(2\right)}+{\epsilon }^3{\varphi }^{\left(3\right)}+\dots ,\end{array}\right.$$

(17)

where ${\lambda }_{ph}$ indicates the normalized DA phase velocity and $\epsilon$ is a real and smallness parameter ($\epsilon <<1$).

By including both stretching (16) and expansion (17) in the system (14) and performing simple calculations, we ultimately obtain the following nonlinear one-dimensional BE: ($\Phi \equiv {\varphi }^{\left(1\right)}$) [34]

$${\mathsf{\partial }}_t\Phi +A\Phi {\mathsf{\partial }}_x\Phi -B{\mathsf{\partial }}_x^2\Phi =0,$$

(18)

with

$$A=-{\displaystyle \frac{{\lambda }_{ph}^3}{2}}\left[{\displaystyle \frac{3}{{\lambda }_{ph}^4}}+2{\alpha }_2\right]\&B={\displaystyle \frac{\eta }{2}}\&{\lambda }_{ph}=\sqrt{{\displaystyle \frac{1}{{\alpha }_1}}}.$$

The exact shock wave solution to Equation (18) reads

$$\Phi \left(x,t\right)={\Phi }_{max}\left[1-tanh\left({\displaystyle \frac{x-u_{0}t}{\delta }}\right)\right],$$

(19)

where ${\Phi }_{max}=u_0/A$ and $\delta =2B/u_0$ represent the peak amplitude and width of the shock waves, whereas $u_0$ denotes the speed of the shock waves.

To see how memory affects the behavior and features of DA shock waves in the current plasma model, we need to convert the integer BE (18) into its fractional form. To achieve this, we can follow the same methodology used by El-Wakil [2,3] in converting the integer KdV equation to its fractional form. This, in turn, leads to the fractional one-dimensional BE:

$$D_t^p\Phi +A\Phi {\mathsf{\partial }}_x\Phi -B{\mathsf{\partial }}_x^2\Phi =0,\forall 0<p\le 1,$$

(20)

where $D_t^p\Phi$ represents the CFD of order p to the function $\Phi$ and $\Phi \equiv \Phi \left(x,t\right)$.

Here, we consider three test examples to solve them using both TT and NIM.

3.1. Analyzing One-Dimensional FBE

Now, we proceed to apply both the TT and the NIM for analyzing the following planar one-dimensional FBE

$$D_t^p\Phi +A\Phi {\mathsf{\partial }}_x\Phi -B{\mathsf{\partial }}_x^2\Phi =0,\forall 0<p\le 1,$$

(21)

with the IC

$$\Phi \left(x,0\right)\equiv {\Phi }_0={\Phi }_{max}\left(1-tanh\left({\displaystyle \frac{x}{\delta }}\right)\right).$$

(22)

3.1.1. The TT for Analyzing One-Dimensional FBE

This technique has previously been utilized to examine various physical and engineering problems [48,49,50,51,52,53,54,55,56,57]. It has proven to be accurate, efficient, and rapid in analyzing the most complicated nonlinear problems. Additionally, it has been demonstrated to be a cost-effective computational approach, in contrast to other methods, including the one discussed in the following portion of this study. Motivated by the encouraging outcomes observed in prior research, this study aims to examine the one-dimensional and two-dimensional Burgers equations and to juxtapose the results of this method with those of the NIM. This technique can be summarized in the following concise points.

Step (1)

Assume that the approximate solution to Equation (21) is expressed in the following infinite convergence series form:

$$\Phi ={\Phi }_0+{\displaystyle \sum _{i=1}^{\infty }}{\Phi }_i{t}^{ip},$$

(23)

where ${\Phi }_0\equiv {\Phi }_0\left(x\right)$ denotes the IC and ${\Phi }_i\equiv {\Phi }_i\left(x\right)$ are unknown functions that will be determined later.

Step (2)

Plugging the assumed solution (23) into Equation (21) and for a finite number of terms of the assumed solution, say $i=3$, we get

$$\begin{array}{cc}\hfill & D_t^p\left({\Phi }_0+{\displaystyle \sum _{i=1}^3}{\Phi }_i{t}^{ip}\right)+A\left({\Phi }_0+{\displaystyle \sum _{i=1}^3}{\Phi }_i{t}^{ip}\right){\mathsf{\partial }}_x\left({\Phi }_0+{\displaystyle \sum _{i=1}^3}{\Phi }_i{t}^{ip}\right)\hfill \\ \hfill & -B{\mathsf{\partial }}_x^2\left({\Phi }_0+{\displaystyle \sum _{i=1}^3}{\Phi }_i{t}^{ip}\right)=0,\hfill \end{array}$$

which leads to

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{i=1}^3}{\Phi }_i\left({\displaystyle \frac{\Gamma (ip+1)}{\Gamma (\left(i-1\right)p+1)}}{t}^{\left(i-1\right)p}\right)\hfill \\ \hfill & +A\left({\Phi }_0+{\displaystyle \sum _{i=1}^3}{\Phi }_i{t}^{ip}\right){\mathsf{\partial }}_x\left({\Phi }_0+{\displaystyle \sum _{i=1}^3}{\Phi }_i{t}^{ip}\right)\hfill \\ \hfill & -B{\mathsf{\partial }}_x^2\left({\Phi }_0+{\displaystyle \sum _{i=1}^3}{\Phi }_i{t}^{ip}\right)=0.\hfill \end{array}$$

(24)

Step (3)

Collecting all terms that have the same power of $t^{ip}$, the following equation is obtained:

$$H_0+H_1{t}^p+H_2{t}^{2p}+\dots =0,$$

(25)

with

$$\begin{array}{cc}\hfill H_0& =Q_0{\Phi }_1+A{\Phi }_0{\Phi }_0^{\prime }-B{\Phi }_0^{″},\hfill \\ \hfill H_1& =Q_1{\Phi }_2+A{\Phi }_1{\Phi }_0^{\prime }+A{\Phi }_0{\Phi }_1^{\prime }-B{\Phi }_1^{″},\hfill \\ \hfill H_2& =Q_2{\Phi }_3+A{\Phi }_2{\Phi }_0^{\prime }+A{\Phi }_1{\Phi }_1^{\prime }+A{\Phi }_0{\Phi }_2^{\prime }-B{\Phi }_2^{″},\hfill \\ \hfill & ⋮,\hfill \end{array}$$

and

$$Q_i={\displaystyle \frac{\Gamma \left(1+\left(i+1\right)p\right)}{\Gamma \left(1+ip\right)}}\forall i=0,1,2,\dots .$$

Step (4)

Since Equation (25) must hold for all values of t, therefore, each coefficient $H_i$ must equal zero. Consequently, we have the subsequent system of equations

$$\begin{array}{cc}\hfill Q_0{\Phi }_1+A{\Phi }_0{\Phi }_0^{\prime }-B{\Phi }_0^{″}& =0,\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill Q_1{\Phi }_2+A{\Phi }_1{\Phi }_0^{\prime }+A{\Phi }_0{\Phi }_1^{\prime }-B{\Phi }_1^{″}& =0,\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill Q_2{\Phi }_3+A{\Phi }_2{\Phi }_0^{\prime }+A{\Phi }_1{\Phi }_1^{\prime }+A{\Phi }_0{\Phi }_2^{\prime }-B{\Phi }_2^{″}& =0.\hfill \end{array}$$

(28)

Step (5)

By solving Equation (26) in ${\Phi }_1$, we ultimately obtain the value of the function ${\Phi }_1$ as a function of ${\Phi }_0$ as follows:

$${\Phi }_1={\displaystyle \frac{-A{\Phi }_0{\Phi }_0^{\prime }+B{\Phi }_0^{″}}{{\Gamma }_1}}.$$

(29)

Here, ${\Gamma }_i\equiv \Gamma \left(ip+1\right)$∀$i=1,2,3,\dots$, is used only for simplicity.

• Then, move to Equation (27), which includes ${\Phi }_2$, and use the obtained value of ${\Phi }_1$ given in Equation (29), and we finally get the value of the function ${\Phi }_2$ as a function of ${\Phi }_0$ as follows:

  $${\Phi }_2={\displaystyle \frac{1}{{\Gamma }_2}}\left[\begin{array}{c}{A}^2{\Phi }_0^2{\Phi }_0^{″}+2A{\Phi }_0\left(A{\left({\Phi }_0^{\prime }\right)}^2-B{\Phi }_0^{\left(3\right)}\right)\\ +B\left(-4A{\Phi }_0^{\prime }{\Phi }_0^{″}+B{\Phi }_0^{\left(4\right)}\right)\end{array}\right].$$

  (30)
• Also, by moving to Equation (28), which includes ${\Phi }_3$, and using the obtained values of ${\Phi }_1$ and ${\Phi }_2$ given in Equations (29) and (30), we finally get the value of the function ${\Phi }_3$ as a function of ${\Phi }_0$ as follows:

  $${\Phi }_3={\displaystyle \frac{1}{{\Gamma }_1^2{\Gamma }_3}}\left[I_0+A{\Phi }_0{I}_1+B{I}_2\right].$$

  (31)

where the coefficients $I_0$, $I_1$, and $I_2$ are given in Appendix A.1.

Step (6)

Now, by inserting the value of the IC ${\Phi }_0$ given in Equation (22) into Equations (29)–(31), the following explicit values of ${\Phi }_1$, ${\Phi }_2$, and ${\Phi }_3$ are obtained:

$$\begin{array}{cc}\hfill {\Phi }_1& ={\displaystyle \frac{{\Phi }_{max}{S}_0{\sec \mathrm{h}}^2\left(\frac{x}{\delta }\right)}{{\delta }^2{\Gamma }_1}},\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill {\Phi }_2& ={\displaystyle \frac{2{\Phi }_{max}{\sec \mathrm{h}}^4\left(\frac{x}{\delta }\right)}{{\delta }^4{\Gamma }_2}}\left(S_1+S_2\right),\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill {\Phi }_3& ={\displaystyle \frac{{\Phi }_{max}{\sec \mathrm{h}}^2\left(\frac{x}{\delta }\right)}{{\delta }^6{\Gamma }_1^2{\Gamma }_3}}\times \hfill \\ & \left\{\begin{array}{c}A\delta S_3{\Phi }_{max}{\Gamma }_2{\sec \mathrm{h}}^2\left({\displaystyle \frac{x}{\delta }}\right)\\ +2{\Gamma }_1^2\left[S_4+S_5+{\displaystyle \frac{16B^3{e}^{\frac{2x}{\delta }}-16{(B-A\delta {\Phi }_{max})}^3}{\left(e^{\frac{2x}{\delta }}+1\right)}}\right]\end{array}\right\},\hfill \end{array}$$

(34)

where the coefficients $S_0$, $S_1$,$\dots ,$$S_5$ are given in Appendix A.2.

Step (7)

Substituting the values obtained of ${\Phi }_1$, ${\Phi }_2$, ${\Phi }_3,$…, into Equation (23), we ultimately obtain the subsequent approximation of the 3rd-order.

$$\begin{array}{cc}\hfill {\Phi }_T& \equiv \Phi ={\Phi }_0+{\Phi }_1{t}^p+{\Phi }_2{t}^{2p}+{\Phi }_3{t}^{3p}+\dots \hfill \\ \hfill & ={\Phi }_{max}\left(1-tanh\left({\displaystyle \frac{x}{\delta }}\right)\right)+{\displaystyle \frac{{\Phi }_{max}{S}_0{\sec \mathrm{h}}^2\left(\frac{x}{\delta }\right)}{{\delta }^2{\Gamma }_1}}{t}^p\hfill \\ \hfill & +{\displaystyle \frac{2{\Phi }_{max}{\sec \mathrm{h}}^4\left(\frac{x}{\delta }\right)}{{\delta }^4{\Gamma }_2}}\left(S_1+S_2\right)t^{2p}+{\displaystyle \frac{{\Phi }_{max}{\sec \mathrm{h}}^2\left(\frac{x}{\delta }\right)}{{\delta }^6{\Gamma }_1^2{\Gamma }_3}}\hfill \\ \hfill & \times \left\{\begin{array}{c}A\delta S_3{\Phi }_{max}{\Gamma }_2{\sec \mathrm{h}}^2\left({\displaystyle \frac{x}{\delta }}\right)\\ +2{\Gamma }_1^2\left[S_4+S_5+{\displaystyle \frac{16B^3{e}^{\frac{2x}{\delta }}-16{(B-A\delta {\Phi }_{max})}^3}{\left(e^{\frac{2x}{\delta }}+1\right)}}\right]\end{array}\right\}{t}^{3p}+\dots .\hfill \end{array}$$

(35)

3.1.2. NIM for Analyzing One-Dimensional FBE

To analyze Equation (21) using the NIM, the following brief steps are introduced:

Step (1)

Rearrange Equation (21) in the following form:

$$D_t^p\Phi =-A\Phi {\mathsf{\partial }}_x\Phi +B{\mathsf{\partial }}_x^2\Phi .$$

(36)

Step (2)

Applying LT to Equation (36), we get

$$\mathcal{L}[D_t^p\Phi ]=\mathcal{L}\left[-A\Phi {\mathsf{\partial }}_x\Phi +B{\mathsf{\partial }}_x^2\Phi \right].$$

(37)

Step (3)

Using the definition of the LT of the CFD

$$\mathcal{L}\left[D_t^p\Phi \right]=s^p\mathcal{L}\left[\Phi \right]-\sum _{k=0}^{m-1}{\displaystyle \frac{D_t^k\Phi \left(x,0\right)}{s^{k-p+1}}}=s^p\mathcal{L}\left[\Phi \right]-{\displaystyle \frac{{\Phi }_0}{s^{1-p}}}.$$

(38)

in Equation (37), we get

$$\begin{array}{cc}\hfill \mathcal{L}[\Phi ]& ={\displaystyle \frac{1}{s^p}}\left\{{\displaystyle \frac{{\Phi }_0}{s^{1-p}}}+\mathcal{L}\left[-A\Phi {\mathsf{\partial }}_x\Phi +B{\mathsf{\partial }}_x^2\Phi \right]\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\Phi }_0}{s}}+{\displaystyle \frac{1}{s^p}}\mathcal{L}\left[-A\Phi {\mathsf{\partial }}_x\Phi +B{\mathsf{\partial }}_x^2\Phi \right].\hfill \end{array}$$

(39)

Step (4)

The application of the inverse LT to Equation (39) produces

$$\Phi ={\mathcal{L}}^{-1}\left[{\displaystyle \frac{\Phi \left(x,0\right)}{s}}+{\displaystyle \frac{1}{s^p}}\mathcal{L}\left[-A\Phi {\mathsf{\partial }}_x\Phi +B{\mathsf{\partial }}_x^2\Phi \right]\right],$$

(40)

Step (5)

The estimated approximation based on the NIM up to the $m^{th}$-order approximations is given by

$$\Phi =\sum _{k=0}^m{\Phi }_k.$$

(41)

Step (6)

Accordingly, both linear and nonlinear terms can be decomposed as follows:

$$R\left(\Phi \right)=R\left[\sum _{m=0}^{\infty }{\Phi }_m\right]=B{\mathsf{\partial }}_x^2\sum _{m=0}^{\infty }{\Phi }_m,$$

and

$$\begin{array}{cc}\hfill N\left(\Phi \right)& =N\left({\Phi }_0\right)+\sum _{m=1}^{\infty }\left\{N\left[\sum _{i=0}^m{\Phi }_i\right]-N\left[\sum _{i=0}^{m-1}{\Phi }_i\right]\right\}\hfill \\ \hfill & =-A\left({\Phi }_0{\mathsf{\partial }}_x{\Phi }_0\right)-A\left(\sum _{k=0}^m{\Phi }_k\right)\left({\mathsf{\partial }}_x\sum _{k=0}^m{\Phi }_k\right)\hfill \\ \hfill & +A\left(\sum _{k=0}^{m-1}{\Phi }_k\right)\left({\mathsf{\partial }}_x\sum _{k=0}^{m-1}{\Phi }_k\right),\forall m=1,2,3,\dots .\hfill \end{array}$$

Step (7)

The zeroth-order approximation can be obtained by using $k=0$ in Equation (40),

$${\Phi }_0={\mathcal{L}}^{-1}\left[{\displaystyle \frac{\Phi (x,0)}{s}}\right]={\Phi }_0.$$

(42)

Step (8)

Using Equation (42) in Equation (40) implies

$$\Phi ={\Phi }_0+{\Phi }_m,$$

(43)

with

$$\begin{array}{cc}\hfill {\Phi }_m& =B{\mathcal{L}}^{-1}\left[{\displaystyle \frac{1}{s^p}}\mathcal{L}\left[{\mathsf{\partial }}_x^2\sum _{m=0}^{\infty }{\Phi }_m\right]\right]\hfill \\ \hfill & -A{\mathcal{L}}^{-1}\left[{\displaystyle \frac{1}{s^p}}\mathcal{L}\left[\left(\sum _{k=0}^m{\Phi }_k\right)\left({\mathsf{\partial }}_x\sum _{k=0}^m{\Phi }_k\right)\right]\right]\hfill \\ \hfill & +A{\mathcal{L}}^{-1}\left[{\displaystyle \frac{1}{s^p}}\mathcal{L}\left[\left(\sum _{k=0}^{m-1}{\Phi }_k\right)\left({\mathsf{\partial }}_x\sum _{k=0}^{m-1}{\Phi }_k\right)\right]\right].\hfill \end{array}$$

(44)

where $m=1,2,3,\dots .$

Step (9)

From Equation (44), we get the following approximations

$$\begin{array}{cc}\hfill {\Phi }_1& ={\mathcal{L}}^{-1}\left[{\displaystyle \frac{1}{s^p}}\mathcal{L}\left[B{\mathsf{\partial }}_x^2{\Phi }_0\right]\right]-A{\mathcal{L}}^{-1}\left[{\displaystyle \frac{1}{s^p}}\mathcal{L}\left[{\Phi }_0{\mathsf{\partial }}_x{\Phi }_0\right]\right]\hfill \\ & ={\displaystyle \frac{{\Phi }_{max}{P}_0}{{\delta }^2}}{sech}^2\left({\displaystyle \frac{x}{\delta }}\right){\displaystyle \frac{t^p}{{\Gamma }_1}},\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill {\Phi }_2& ={\mathcal{L}}^{-1}\left[{\displaystyle \frac{1}{s^p}}\mathcal{L}\left[B{\mathsf{\partial }}_x^2{\Phi }_1\right]\right]-A{\mathcal{L}}^{-1}\left[{\displaystyle \frac{1}{s^p}}\mathcal{L}\left[\left({\Phi }_0+{\Phi }_1\right){\mathsf{\partial }}_x\left({\Phi }_0+{\Phi }_1\right)\right]\right]\hfill \\ & +A{\mathcal{L}}^{-1}\left[{\displaystyle \frac{1}{s^p}}\mathcal{L}\left[{\Phi }_0{\mathsf{\partial }}_x{\Phi }_0\right]\right]\hfill \\ & ={\displaystyle \frac{{\Phi }_{max}\left(P_1+P_2+P_3\right)}{{\delta }^5}}{sech}^5\left({\displaystyle \frac{x}{\delta }}\right){\displaystyle \frac{t^{2p}}{{\Gamma }_2}},\hfill \end{array}$$

(46)

where the coefficients $P_0$, $P_1$, $P_2$, and $P_3$ are given in Appendix A.3.

Step (10)

By collecting the obtained values of ${\Phi }_1$, ${\Phi }_2$, ${\Phi }_3,$… and adding them to Equation (43), ultimately we obtain the second-order approximation in the following manner:

$$\begin{array}{cc}\hfill {\Phi }_N& \equiv \Phi ={\Phi }_0+{\Phi }_1+{\Phi }_2+\dots \hfill \\ \hfill & ={\Phi }_{max}\left(1-tanh\left({\displaystyle \frac{x}{\delta }}\right)\right)+{\displaystyle \frac{{\Phi }_{max}{P}_0}{{\delta }^2}}{sech}^2\left({\displaystyle \frac{x}{\delta }}\right){\displaystyle \frac{t^p}{{\Gamma }_1}}\hfill \\ \hfill & +{\displaystyle \frac{{\Phi }_{max}\left(P_1+P_2+P_3\right)}{{\delta }^5}}{sech}^5\left({\displaystyle \frac{x}{\delta }}\right){\displaystyle \frac{t^{2p}}{{\Gamma }_2}}+\dots .\hfill \end{array}$$

(47)

To understand how fractionality affects the behavior of the DA shock wave profile in the current plasma model, we will first outline the acceptable physical values for the relevant parameters $\left(q,\eta ,u_0\right)$. Thus, considering $\left(q,\eta ,u_0\right)=\left(0.7,0.1,0.2\right)$ [34], we get $\left(A,B\right)=\left(-0.759257,0.05\right)$. Before proceeding to study the influence of memory on the dynamical behavior of the DA shock waves, the accuracy of the derived approximations (35) and (47) must be verified through several tests, the most important of which is calculating the maximum value of the residual error (MVRR) for these approximations along the whole study domain as follows:

$$\begin{array}{cc}\hfill L_T& =\underset{\Omega }{max}{\left|D_t^p{\Phi }_T+A{\Phi }_T{\mathsf{\partial }}_x{\Phi }_T-B{\mathsf{\partial }}_x^2{\Phi }_T\right|}_{{\Phi }_3=0}\hfill \\ \hfill & =0.0433351\mathrm{at}\left(x,t\right)=\left(0.497197,2\right),\hfill \\ \hfill L_N& =\underset{\Omega }{max}\left|D_t^p{\Phi }_N+A{\Phi }_N{\mathsf{\partial }}_x{\Phi }_N-B{\mathsf{\partial }}_x^2{\Phi }_N\right|\hfill \\ \hfill & =0.217592\mathrm{at}\left(x,t\right)=\left(0.0727606,2\right),\hfill \end{array}$$

where $L_T$ and $L_N$ indicate the MVRR for the approximations (35) and (47), respectively, $\Omega \equiv -2\le x\le 2$ & $0\le t\le 2$. In addition, ${\Phi }_T$ and ${\Phi }_N$ indicate the approximations (35) and (47), respectively.

Additionally, the precision of the approximations (35) and (47) can be verified by comparing these approximations with the exact solution (19) at $p=1$, as illustrated in Figure 1 and Figure 2. Furthermore, the absolute errors for the second- and third-order approximation (35) and the approximation (47) are calculated as shown in

Table 1. The absolute error $R_i$ for the generated approximations (35) and (47) for example (I) is estimated at $\left(t,p\right)=\left(0.1,1\right)$.

x	$\mathit{NIM}\mathit{\left(}{\mathit{2}}^{\mathit{nd}}\mathit{\right)}$	$\mathit{Tantawy}\mathit{\left(}{\mathit{2}}^{\mathit{nd}}\mathit{\right)}$	$\mathit{Tantawy}\mathit{\left(}{\mathit{3}}^{\mathit{rd}}\mathit{\right)}$	$\mathit{Exact}$	${\mathit{R}}_{\mathit{N}}$	${\mathit{R}}_{\mathit{T}\left(2\right)}$	${\mathit{R}}_{\mathit{T}\left(3\right)}$
−2	−0.526668	−0.526668	−0.526668	−0.526668	0.0147656 $\times {10}^{-6}$	0.0147454 $\times {10}^{-6}$	0.0295294 $\times {10}^{-8}$
−1.5	−0.525628	−0.525628	−0.525628	−0.525628	0.108224 $\times {10}^{-6}$	0.107135 $\times {10}^{-6}$	0.210893 $\times {10}^{-8}$
−1	−0.518071	−0.518071	−0.518072	−0.518072	0.752045 $\times {10}^{-6}$	0.697962 $\times {10}^{-6}$	1.1936 $\times {10}^{-8}$
−0.5	−0.46832	−0.468321	−0.468323	−0.468323	3.27381 $\times {10}^{-6}$	1.76408 $\times {10}^{-6}$	1.74589 $\times {10}^{-8}$
0	−0.273952	−0.273952	−0.273946	−0.273946	5.61593 $\times {10}^{-6}$	5.61593 $\times {10}^{-6}$	0.359417 $\times {10}^{-8}$
0.5	−0.0673612	−0.0673597	−0.0673614	−0.0673614	0.216939$\times {10}^{-6}$	1.72667 $\times {10}^{-6}$	1.99569 $\times {10}^{-8}$
1	−0.0102489	−0.0102488	−0.0102495	−0.0102495	0.66801$\times {10}^{-6}$	0.722092 $\times {10}^{-6}$	1.21939 $\times {10}^{-8}$
1.5	−0.00141075	−0.00141075	−0.00141086	−0.00141086	0.11033$\times {10}^{-6}$	0.111419 $\times {10}^{-6}$	0.217471 $\times {10}^{-8}$
2	−0.000191367	−0.000191367	−0.000191382	−0.000191382	0.0153254 $\times {10}^{-6}$	0.0153456 $\times {10}^{-6}$	0.0304844 $\times {10}^{-8}$

:

$$\begin{array}{cc}\hfill R_{T\left(2\right)}& \equiv 2^{nd}-R_T={\left|{\Phi }_T-{\Phi }_{Ex.}\right|}_{{\Phi }_3=0},\hfill \\ \hfill R_{T\left(3\right)}& \equiv 3^{rd}-R_T={\left|{\Phi }_T-{\Phi }_{Ex.}\right|}_{{\Phi }_3\ne 0},\hfill \\ \hfill R_N& \equiv 2^{nd}-R_N=\left|{\Phi }_N-{\Phi }_{Ex.}\right|,\hfill \end{array}$$

where ${\Phi }_{Ex.}$ represents the exact solution (19) for the integer BE (1).

One can see from this comparison that the high accuracy of the derived approximations is fulfilled through the complete match between these approximations and the exact solution at $p=1$, as illustrated in Figure 1 and Figure 2. Furthermore, the results of the numerical comparison proved the precision and efficacy of the approaches used, as shown in Table 1. Since both approximations (35) and (47) exhibit the same qualitative behavior for the studied DA shock waves, we can analyze one of them against fractionality p to understand the effect of memory on the behavior of these waves during propagation in the current plasma model. In Figure 3, the influence of fractionality p on the DA shock wave profile is examined. This figure indicates that fractionality p substantially influences the profile of the shock wave (amplitude and width), and the effect becomes more pronounced over long time intervals. In addition, it is observed that decreasing p results in slower wave propagation, increased dispersion, and a change in shock steepness—a consistent trend across the parameter space, consistent with well-established behavior in fractal physical models. This effect is not represented by analyzing the exact shock wave solution to the integer BE (18). Consequently, the derived approximations for fractional cases may be more realistic in describing such phenomena and revealing some behaviors that integer solutions fail to detect. Note that the optimal fractionality values p depend on the experimental results or space observations. For example, when this study is compared with practical results or some observations of spacecraft, the fractionality values p are gradually adjusted within the range $0<p\le 1$. As these values change, the theoretical results match the experimental data or space observations, and based on this, the optimal values of fractionality p are determined. In Figure 3, the choice of fractional values of the parameter p covers the full acceptable range $0<p\le 1$, capturing both strong memory effects (small p) and the classical limit ($p=1$).

3.2. Analyzing Two-Dimensional FBE

For two-dimensional perturbations, in this case, the fluid model equations that govern a physical model can be reduced to a two-dimensional BE [35]. Thus, for the fractional case, we can consider the following planar two-dimensional FBE:

$$D_t^p\Phi +A\Phi \left({\mathsf{\partial }}_x\Phi +{\mathsf{\partial }}_y\Phi \right)-B\left({\mathsf{\partial }}_x^2\Phi +{\mathsf{\partial }}_y^2\Phi \right)=0,$$

(48)

with the IC

$$\Phi \left(x,y,0\right)\equiv {\Phi }_0={\displaystyle \frac{\lambda }{A+Aexp\left(\eta \right)}},$$

(49)

where $\Phi \equiv \Phi \left(x,y,t\right)$, ${\Phi }_0\equiv {\Phi }_0\left(x,y\right)$, and $0<p\le 1$. For simplicity, the following notation is considered: $\eta =\lambda \left(x+y\right)/\left(2B\right).$

The exact shock wave solution for Equation (48) at $p=1$, reads

$$\Phi \left(x,y,t\right)={\displaystyle \frac{\lambda }{A+Aexp\left(\frac{\lambda \left(x+y-\lambda t\right)}{2B}\right)}}.$$

(50)

3.2.1. The TT for Analyzing Two-Dimensional FBE

To analyze Equation (48) using the TT, the following brief steps are introduced:

Step (1)

Assume that the approximate solution to Equation (48) is expressed in the following infinite convergence series form:

$$\Phi ={\Phi }_0+{\displaystyle \sum _{i=1}^{\infty }}{\Phi }_i{t}^{ip},$$

(51)

where ${\Phi }_0\equiv {\Phi }_0\left(x,y\right)$ denotes the IC and ${\Phi }_i\equiv {\Phi }_i\left(x,y\right)$ are unknown functions that will be determined later.

Step (2)

Insert the assumed solution (51) into Equation (48) and for a finite number of terms of the assumed solution, say $i=3$, we get

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{i=1}^3}{\Phi }_i\left({\displaystyle \frac{\Gamma (ip+1)}{\Gamma (\left(i-1\right)p+1)}}{t}^{\left(i-1\right)p}\right)+A\left({\Phi }_0+{\displaystyle \sum _{i=1}^3}{\Phi }_i{t}^{ip}\right)\hfill \\ \hfill & \times \left[{\mathsf{\partial }}_x\left({\Phi }_0+{\displaystyle \sum _{i=1}^3}{\Phi }_i{t}^{ip}\right)+{\mathsf{\partial }}_y\left({\Phi }_0+{\displaystyle \sum _{i=1}^3}{\Phi }_i{t}^{ip}\right)\right]\hfill \\ \hfill & -B\left[{\mathsf{\partial }}_x^2\left({\Phi }_0+{\displaystyle \sum _{i=1}^3}{\Phi }_i{t}^{ip}\right)+{\mathsf{\partial }}_y^2\left({\Phi }_0+{\displaystyle \sum _{i=1}^3}{\Phi }_i{t}^{ip}\right)\right]=0.\hfill \end{array}$$

(52)

Step (3)

Collecting all terms that have the same power of $t^{ip}$, the following equation is obtained:

$$H_0+H_1{t}^p+H_2{t}^{2p}+\dots =0,$$

(53)

with

$$\begin{array}{cc}\hfill H_0& =Q_0{\Phi }_1+A{\Phi }_0{\mathsf{\partial }}_y{\Phi }_0+A{\Phi }_0{\mathsf{\partial }}_x{\Phi }_0-B{\mathsf{\partial }}_x^2{\Phi }_0-B{\mathsf{\partial }}_y^2{\Phi }_0,\hfill \\ \hfill H_1& =Q_1{\Phi }_2+A{\Phi }_1{\mathsf{\partial }}_y{\Phi }_0+A{\Phi }_0{\mathsf{\partial }}_y{\Phi }_1\hfill \\ \hfill & +A{\Phi }_1{\mathsf{\partial }}_x{\Phi }_0+A{\Phi }_0{\mathsf{\partial }}_x{\Phi }_1-B{\mathsf{\partial }}_x^2{\Phi }_1-B{\mathsf{\partial }}_y^2{\Phi }_1\hfill \\ \hfill H_2& =Q_2{\Phi }_3+A{\Phi }_2{\mathsf{\partial }}_y{\Phi }_0+A{\Phi }_1{\mathsf{\partial }}_y{\Phi }_1+A{\Phi }_0{\mathsf{\partial }}_y{\Phi }_2\hfill \\ \hfill & +A{\Phi }_2{\mathsf{\partial }}_x{\Phi }_0+A{\Phi }_1{\mathsf{\partial }}_x{\Phi }_1+A{\Phi }_0{\mathsf{\partial }}_x{\Phi }_2-B{\mathsf{\partial }}_x^2{\Phi }_2-B{\mathsf{\partial }}_y^2{\Phi }_2\hfill \\ \hfill & ⋮,\hfill \end{array}$$

and

$$Q_i={\displaystyle \frac{\Gamma \left(1+\left(i+1\right)p\right)}{\Gamma \left(1+ip\right)}}\forall i=0,1,2,\dots .$$

Step (4)

Since Equation (53) must hold for all values of t, therefore, each coefficient $H_i$ must equal zero. Therefore, we get the following system of equations:

$$\begin{array}{cc}\hfill & Q_0{\Phi }_1+A{\Phi }_0{\mathsf{\partial }}_y{\Phi }_0+A{\Phi }_0{\mathsf{\partial }}_x{\Phi }_0-B{\mathsf{\partial }}_x^2{\Phi }_0-B{\mathsf{\partial }}_y^2{\Phi }_0=0,\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill & Q_1{\Phi }_2+A{\Phi }_1{\mathsf{\partial }}_y{\Phi }_0+A{\Phi }_0{\mathsf{\partial }}_y{\Phi }_1+A{\Phi }_1{\mathsf{\partial }}_x{\Phi }_0\hfill \\ \hfill & +A{\Phi }_0{\mathsf{\partial }}_x{\Phi }_1-B{\mathsf{\partial }}_x^2{\Phi }_1-B{\mathsf{\partial }}_y^2{\Phi }_1=0,\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill & Q_2{\Phi }_3+A{\Phi }_2{\mathsf{\partial }}_y{\Phi }_0+A{\Phi }_1{\mathsf{\partial }}_y{\Phi }_1+A{\Phi }_0{\mathsf{\partial }}_y{\Phi }_2\hfill \\ \hfill & +A{\Phi }_2{\mathsf{\partial }}_x{\Phi }_0+A{\Phi }_1{\mathsf{\partial }}_x{\Phi }_1+A{\Phi }_0{\mathsf{\partial }}_x{\Phi }_2-B{\mathsf{\partial }}_x^2{\Phi }_2-B{\mathsf{\partial }}_y^2{\Phi }_2=0\hfill \end{array}$$

(56)

Step (5)

By solving Equations (54)–(56) in ${\Phi }_1$, ${\Phi }_2$, ${\Phi }_3$, we ultimately obtain the explicit values of ${\Phi }_1$, ${\Phi }_2$, and ${\Phi }_3$ as follows:

$$\begin{array}{cc}\hfill {\Phi }_1& ={\displaystyle \frac{{\lambda }^3{\sec \mathrm{h}}^2\left(\frac{\eta }{2}\right)}{8AB{\Gamma }_1}},\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill {\Phi }_2& ={\displaystyle \frac{{\lambda }^{5}tanh\left(\frac{\eta }{2}\right){\sec \mathrm{h}}^2\left(\frac{\eta }{2}\right)}{16A{B}^2{\Gamma }_2}},\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill {\Phi }_3& ={\displaystyle \frac{{\lambda }^7{\sec \mathrm{h}}^5\left(\frac{\eta }{2}\right)}{128A{B}^3{\Gamma }_1^2{\Gamma }_3}}\hfill \\ \hfill & \times \left[\begin{array}{c}2{\Gamma }_{2}sinh\left({\displaystyle \frac{\eta }{2}}\right)\\ +{\Gamma }_1^2\left(-4sinh\left({\displaystyle \frac{\eta }{2}}\right)-3cosh\left({\displaystyle \frac{\eta }{2}}\right)+cosh\left({\displaystyle \frac{3\eta }{2}}\right)\right)\end{array}\right].\hfill \end{array}$$

(59)

Step (6)

Substituting the values obtained of ${\Phi }_1$, ${\Phi }_2$, ${\Phi }_3,$…, into Equation (51), ultimately, we obtain the approximation of the third-order in the following manner:

$$\begin{array}{cc}\hfill {\Phi }_T& \equiv \Phi ={\Phi }_0+{\Phi }_1{t}^p+{\Phi }_2{t}^{2p}+{\Phi }_3{t}^{3p}+\dots \hfill \\ \hfill & ={\displaystyle \frac{\lambda }{A+Aexp\left(\eta \right)}}+{\displaystyle \frac{{\lambda }^3{\sec \mathrm{h}}^2\left(\frac{\eta }{2}\right)}{8AB{\Gamma }_1}}{t}^p+{\displaystyle \frac{{\lambda }^{5}tanh\left(\frac{\eta }{2}\right){\sec \mathrm{h}}^2\left(\frac{\eta }{2}\right)}{16A{B}^2{\Gamma }_2}}{t}^{2p}\hfill \\ \hfill & +{\displaystyle \frac{{\lambda }^7{\sec \mathrm{h}}^5\left(\frac{\eta }{2}\right)}{128A{B}^3{\Gamma }_1^2{\Gamma }_3}}\times \left[\begin{array}{c}2{\Gamma }_{2}sinh\left({\displaystyle \frac{\eta }{2}}\right)\\ +{\Gamma }_1^2\left(-4sinh\left({\displaystyle \frac{\eta }{2}}\right)-3cosh\left({\displaystyle \frac{\eta }{2}}\right)+cosh\left({\displaystyle \frac{3\eta }{2}}\right)\right)\end{array}\right]t^{3p}+\dots .\hfill \end{array}$$

(60)

3.2.2. NIM for Analyzing 2D-FBE

To analyze Equation (48) using the NIM, the following brief steps are introduced:

Step (1)

Rearrange Equation (48) in the following form:

$$D_t^p\Phi =-A\Phi \left({\mathsf{\partial }}_x\Phi +{\mathsf{\partial }}_y\Phi \right)+B\left({\mathsf{\partial }}_x^2\Phi +{\mathsf{\partial }}_y^2\Phi \right).$$

(61)

Step (2)

Applying the LT to both sides of Equation (61), we obtain the following result:

$$\mathcal{L}[D_t^p\Phi ]=-\mathcal{L}\left[A\Phi \left({\mathsf{\partial }}_x\Phi +{\mathsf{\partial }}_y\Phi \right)\right]+B\mathcal{L}\left[\left({\mathsf{\partial }}_x^2\Phi +{\mathsf{\partial }}_y^2\Phi \right)\right],$$

(62)

Step (3)

Using the definition of the LT to the CFD $L[D_t^p\Phi ]$ as given in Equation (13)

$$\begin{array}{cc}\hfill \mathcal{L}\left[D_t^p\Phi \right]& =s^p\mathcal{L}\left[\Phi \right]-\sum _{k=0}^{m-1}{\displaystyle \frac{D_t^k\Phi \left(x,0\right)}{s^{k-p+1}}},0<p\le m,\hfill \\ \hfill & =s^p\mathcal{L}\left[\Phi \right]-{\displaystyle \frac{{\Phi }_0}{s^{1-p}}}.\hfill \end{array}$$

(63)

in Equation (62) yields

$$\mathcal{L}[\Phi ]={\displaystyle \frac{{\Phi }_0}{s}}-{\displaystyle \frac{A}{s^p}}\mathcal{L}\left[\Phi \left({\mathsf{\partial }}_x\Phi +{\mathsf{\partial }}_y\Phi \right)\right]+{\displaystyle \frac{B}{s^p}}\mathcal{L}\left[\left({\mathsf{\partial }}_x^2\Phi +{\mathsf{\partial }}_y^2\Phi \right)\right].$$

(64)

Step (4)

Applying the inverse LT to Equation (64) yields

$$\Phi ={\mathcal{L}}^{-1}\left[{\displaystyle \frac{{\Phi }_0}{s}}-{\displaystyle \frac{A}{s^p}}\mathcal{L}\left[\Phi \left({\mathsf{\partial }}_x\Phi +{\mathsf{\partial }}_y\Phi \right)\right]+{\displaystyle \frac{B}{s^p}}\mathcal{L}\left[\left({\mathsf{\partial }}_x^2\Phi +{\mathsf{\partial }}_y^2\Phi \right)\right]\right].$$

(65)

Step (5)

The estimated approximation based on the NIM up to the $m^{th}$-order approximations is given by

$$\Phi =\sum _{k=0}^m{\Phi }_k.$$

(66)

Step (6)

Accordingly, both linear and nonlinear terms can be decomposed as follows:

$$R\left(\Phi \right)=R\left({\Phi }_0\right)+R\left[\sum _{m=1}^{\infty }{\Phi }_m\right]=B\left({\mathsf{\partial }}_x^2\sum _{m=0}^{\infty }{\Phi }_m+{\mathsf{\partial }}_y^2\sum _{m=0}^{\infty }{\Phi }_m\right),$$

(67)

and

$$\begin{array}{cc}\hfill N\left(\Phi \right)& =N\left({\Phi }_0\right)+\sum _{m=1}^{\infty }\left\{N\left[\sum _{i=0}^m{\Phi }_i\right]-N\left[\sum _{i=0}^{m-1}{\Phi }_i\right]\right\}\hfill \\ \hfill & =-A\left({\Phi }_0{\mathsf{\partial }}_x{\Phi }_0+{\Phi }_0{\mathsf{\partial }}_y{\Phi }_0\right)\hfill \\ \hfill & -A\left(\sum _{k=0}^m{\Phi }_k\right)\left({\mathsf{\partial }}_x\sum _{k=0}^m{\Phi }_k+{\mathsf{\partial }}_y\sum _{k=0}^m{\Phi }_k\right)\hfill \\ \hfill & +A\left(\sum _{k=0}^{m-1}{\Phi }_k\right)\left({\mathsf{\partial }}_x\sum _{k=0}^{m-1}{\Phi }_k+{\mathsf{\partial }}_y\sum _{k=0}^{m-1}{\Phi }_k\right),\forall m=1,2,3,\dots .\hfill \end{array}$$

(68)

Step (7)

The zeroth-order approximation can be determined using $k=0$ in Equation (65),

$${\Phi }_0={\mathcal{L}}^{-1}{\left[{\displaystyle \frac{1}{s^p}}\left(\sum _{k=0}^{m-1}{\displaystyle \frac{D_t^k\Phi \left(x,y,0\right)}{s^{k-p+1}}}\right)\right]}_{k=0}={\mathcal{L}}^{-1}\left[{\displaystyle \frac{\Phi (x,y,0)}{s}}\right]={\Phi }_0.$$

(69)

Step (8)

Using Equations (66)–(68) in Equation (65), we obtain

$$\Phi ={\Phi }_0+{\Phi }_m,$$

(70)

with

$$\begin{array}{cc}\hfill {\Phi }_m& =B{\mathcal{L}}^{-1}\left[{\displaystyle \frac{1}{s^p}}\mathcal{L}\left[{\mathsf{\partial }}_x^2\sum _{m=0}^{\infty }{\Phi }_m+{\mathsf{\partial }}_y^2\sum _{m=0}^{\infty }{\Phi }_m\right]\right]\hfill \\ \hfill & -A{\mathcal{L}}^{-1}\left[{\displaystyle \frac{1}{s^p}}\mathcal{L}\left[\left(\sum _{k=0}^m{\Phi }_k\right)\left({\mathsf{\partial }}_x\sum _{k=0}^m{\Phi }_k+{\mathsf{\partial }}_y\sum _{k=0}^m{\Phi }_k\right)\right]\right]\hfill \\ \hfill & +A{\mathcal{L}}^{-1}\left[{\displaystyle \frac{1}{s^p}}\mathcal{L}\left[\left(\sum _{k=0}^{m-1}{\Phi }_k\right)\left({\mathsf{\partial }}_x\sum _{k=0}^{m-1}{\Phi }_k+{\mathsf{\partial }}_y\sum _{k=0}^{m-1}{\Phi }_k\right)\right]\right].\hfill \end{array}$$

(71)

Step (9)

From Equation (71), we get the following approximations:

$$\begin{array}{cc}\hfill {\Phi }_1& =B{\mathcal{L}}^{-1}\left[{\displaystyle \frac{1}{s^p}}\mathcal{L}\left[{\mathsf{\partial }}_x^2{\Phi }_0+{\mathsf{\partial }}_y^2{\Phi }_0\right]\right]\hfill \\ \hfill & -A{\mathcal{L}}^{-1}\left[{\displaystyle \frac{1}{s^p}}\mathcal{L}\left[{\Phi }_0{\mathsf{\partial }}_x{\Phi }_0+{\Phi }_0{\mathsf{\partial }}_y{\Phi }_0\right]\right]\hfill \\ \hfill & ={\displaystyle \frac{{\lambda }^3{sech}^2\left(\eta \right)}{8AB}}{\displaystyle \frac{t^p}{{\Gamma }_1}},\hfill \end{array}$$

(72)

$$\begin{array}{cc}\hfill {\Phi }_2& ={\mathcal{L}}^{-1}\left[{\displaystyle \frac{B}{s^p}}\mathcal{L}\left[{\mathsf{\partial }}_x^2{\Phi }_1+{\mathsf{\partial }}_y^2{\Phi }_1\right]\right]\hfill \\ \hfill & -A{\mathcal{L}}^{-1}\left[{\displaystyle \frac{1}{s^p}}\mathcal{L}\left[\left(\sum _{k=0}^1{\Phi }_k\right)\left({\mathsf{\partial }}_x\sum _{k=0}^1{\Phi }_k+{\mathsf{\partial }}_y\sum _{k=0}^1{\Phi }_k\right)\right]\right]\hfill \\ \hfill & +A{\mathcal{L}}^{-1}\left[{\displaystyle \frac{1}{s^p}}\mathcal{L}\left[{\Phi }_0{\mathsf{\partial }}_x{\Phi }_0+{\Phi }_0{\mathsf{\partial }}_y{\Phi }_0\right]\right]\hfill \\ \hfill & ={\displaystyle \frac{t^{2p}{\lambda }^5{J}_{0}exp\left(\eta \right)\left(-1+exp\left(\eta \right)\right)}{4A{B}^3{\left(1+e^{\eta }\right)}^5}},\hfill \end{array}$$

(73)

and

$$\begin{array}{cc}\hfill {\Phi }_3& ={\mathcal{L}}^{-1}\left[{\displaystyle \frac{B}{s^p}}\mathcal{L}\left[{\mathsf{\partial }}_x^2{\Phi }_2+{\mathsf{\partial }}_y^2{\Phi }_2\right]\right]\hfill \\ \hfill & -A{\mathcal{L}}^{-1}\left[{\displaystyle \frac{1}{s^p}}\mathcal{L}\left[\left(\sum _{k=0}^2{\Phi }_k\right)\left({\mathsf{\partial }}_x\sum _{k=0}^2{\Phi }_k+{\mathsf{\partial }}_y\sum _{k=0}^2{\Phi }_k\right)\right]\right]\hfill \\ \hfill & +A{\mathcal{L}}^{-1}\left[{\displaystyle \frac{1}{s^p}}\mathcal{L}\left[\left(\sum _{k=0}^1{\Phi }_k\right)\left({\mathsf{\partial }}_x\sum _{k=0}^1{\Phi }_k+{\mathsf{\partial }}_y\sum _{k=0}^1{\Phi }_k\right)\right]\right]\hfill \\ \hfill & ={\displaystyle \frac{t^{3p}{\lambda }^7{e}^{\eta }}{8A{B}^7{\left(1+e^{\eta }\right)}^{11}}}\left\{J_1+J_2+{\displaystyle \frac{1}{{\Gamma }_3}}\left[J_5+{\displaystyle \frac{{\lambda }^4{t}^{2p}{e}^{2\eta }}{{\Gamma }_1^2}}\left(J_6+J_7+J_8\right)\right]\right\},\hfill \end{array}$$

(74)

where the coefficients $J_0,J_1,\dots ,$ and $J_8$ are given in the Appendix A.

Step (10)

By collecting the obtained values of ${\Phi }_1$, ${\Phi }_2$, ${\Phi }_3,$… and adding them to Equation (70), ultimately we obtain the approximation of the third order in the following manner:

$$\begin{array}{cc}\hfill {\Phi }_N& \equiv \Phi ={\Phi }_1+{\Phi }_2+{\Phi }_3+\dots \hfill \\ \hfill & ={\displaystyle \frac{1}{1+exp\left(\frac{x+y}{2B}\right)}}+{\displaystyle \frac{{\lambda }^3{sech}^2\left(\eta \right)}{8AB}}{\displaystyle \frac{t^p}{{\Gamma }_1}}\hfill \\ \hfill & +{\displaystyle \frac{t^{2p}{\lambda }^5{J}_{0}exp\left(\eta \right)\left(-1+exp\left(\eta \right)\right)}{4A{B}^3{\left(1+e^{\eta }\right)}^5}}\hfill \\ \hfill & +{\displaystyle \frac{t^{3p}{\lambda }^7{e}^{\eta }}{8A{B}^7{\left(1+e^{\eta }\right)}^{11}}}\left\{J_1+J_2+{\displaystyle \frac{1}{{\Gamma }_3}}\left[J_5+{\displaystyle \frac{{\lambda }^4{t}^{2p}{e}^{2\eta }}{{\Gamma }_1^2}}\left(J_6+J_7+J_8\right)\right]\right\}+\dots .\hfill \end{array}$$

(75)

Now, we initially compare the derived approximations (60) and (75) with the exact solution (50) at $p=1$ to validate the precision of these approximations as illustrated in Figure 4 and Figure 5, respectively. In addition, the MVRR for the derived approximations (60) and (75) is estimated throughout the study domain $\Omega \equiv -7\le x\le 7$ &$-7\le y\le 7$ & $0\le t\le 4$ as follows:

$$\begin{array}{cc}\hfill L_T& =\underset{\Omega }{max}\left|D_t^p{\Phi }_T+A{\Phi }_T\left({\mathsf{\partial }}_x{\Phi }_T+{\mathsf{\partial }}_y{\Phi }_T\right)-B\left({\mathsf{\partial }}_x^2{\Phi }_T+{\mathsf{\partial }}_y^2{\Phi }_T\right)\right|\hfill \\ \hfill & =0.0187566\mathrm{at}\left(x,y,t\right)=\left(-1.71424,2.12589,4\right),\hfill \\ \hfill L_N& =\underset{\Omega }{max}\left|D_t^p{\Phi }_N+A{\Phi }_N\left({\mathsf{\partial }}_x{\Phi }_N+{\mathsf{\partial }}_y{\Phi }_N\right)-B\left({\mathsf{\partial }}_x^2{\Phi }_N+{\mathsf{\partial }}_y^2{\Phi }_N\right)\right|\hfill \\ \hfill & =0.091665\mathrm{at}\left(x,y,t\right)=\left(-2.95398,2.97151,4\right),\hfill \end{array}$$

where $L_T$ and $L_N$ indicate the MVRR for the approximations (60) and (75), respectively. In addition, ${\Phi }_T$ and ${\Phi }_N$ indicate the approximations (60) and (75), respectively.

Additionally, a numerical comparison is conducted between the approximations (60) and (75) with the exact solution (50) at $p=1$ by calculating the absolute error of these approximations, as presented in

Table 2. The absolute error R for the generated approximations (35) and (47) for example (II) is estimated at $\left(y,t,p\right)=\left(1,1,1\right)$.

x	NIM	Tantawy	Exact	${\mathit{R}}_{\mathit{N}}$	${\mathit{R}}_{\mathit{T}}$
−8	−0.263219	−0.263219	−0.263219	0.153462 $\times {10}^{-7}$	0.151786 $\times {10}^{-7}$
−6	−0.26197	−0.26197	−0.26197	1.11237 $\times {10}^{-7}$	1.02325$\times {10}^{-7}$
−4	−0.253099	−0.253099	−0.253099	7.25512 $\times {10}^{-7}$	3.31422 $\times {10}^{-7}$
−2	−0.202441	−0.202439	−0.202441	1.04658 $\times {10}^{-7}$	21.268 $\times {10}^{-7}$
0	−0.081666	−0.0816677	−0.0816655	5.10837 $\times {10}^{-7}$	21.9776 $\times {10}^{-7}$
2	−0.0151005	−0.0150997	−0.0151001	4.01212 $\times {10}^{-7}$	3.24312 $\times {10}^{-7}$
4	−0.00215006	−0.00215004	−0.00215015	0.887141 $\times {10}^{-7}$	1.09854 $\times {10}^{-7}$
6	−0.000293043	−0.000293043	−0.000293059	0.160118 $\times {10}^{-7}$	0.164237 $\times {10}^{-7}$
8	−0.0000396972	−0.0000396972	−0.0000396994	0.0224462 $\times {10}^{-7}$	0.0225222 $\times {10}^{-7}$

:

$$\begin{array}{cc}\hfill R_T& =\left|{\Phi }_T-{\Phi }_{Ex.}\right|,\hfill \\ \hfill R_N& =\left|{\Phi }_N-{\Phi }_{Ex.}\right|.\hfill \end{array}$$

where ${\Phi }_{Ex.}$ represents the exact solution (50) for the two-dimensional integer BE (5).

To study the impact of fractionality p on the profile of the 2D shock waves described by the approximations (60) and (75), these approximations are graphically plotted against fractionality p, as shown in Figure 6 and Figure 7, respectively. The results obtained indicate that these approximations are sensitive to any change in the fractional-order parameter p. Consequently, fractional approximations of FDEs offer a more realistic explanation for numerous nonlinear phenomena, given their ability to reveal some ambiguous behaviors that integer solutions fail to detect.

The results obtained confirm that both the TT and NIM are effective methods for solving strong nonlinear FDEs. Therefore, we expect the TT to analyze any FDE and produce higher-order approximations with low cost and fast computations.

4. Conclusions

To sum up, we examined the effects of non-locality and memory on the propagation of dust-acoustic (DA) shock waves that originate and propagate in an electron-depleted complex plasma consisting of inertial, negatively charged dust grains and non-Maxwellian ions following a nonextensive distribution. The fluid equations governing the propagation of nonlinear structures in this plasma model have been reduced to a planar integer/non-fractional Burgers-type equation using a reduced perturbation technique. To explore the influence of memory on the behavior of fractional DA shock waves, the planar integer/non-fractional Burgers-type equations have been transformed into their fractional counterparts using a suitable method. In the Caputo framework, both the Tantawy technique (TT) and the new iteration method (NIM) have been applied to analyze and solve the proposed fractional evolutionary wave equations (EWEs), i.e., the nonlinear one-dimensional fractional Burgers equation (FBE) and two-dimensional fractional Burgers equation (FBE). At this point, we can summarize the results obtained in the following points:

(I)

Using the TT, the one-dimensional FBE has been analyzed, and some analytical approximations up to the third order have been derived. Given the fast computation of this technique and its ability to overcome many complexities, as well as the small size of the derived approximations, this approach can quickly generate high-order approximations with low computational cost. However, the one-dimensional FBE was also analyzed using the NIM, and an approximation up to the second order was produced. It is worth noting that higher-order approximations are possible, but they require significantly higher computational costs, unlike the TT. Additionally, the obtained approximations using the NIM are very large and may not be possible to insert into the text.

(II)

The memory effect, represented by the fractional-order parameter p, has been studied on the dynamical behavior of the DA shock wave profile. Additionally, an analytical comparison between the derived approximations and the exact solution for the non-fractional case ($p=1$) confirmed the high precision of all the derived approximations. The residual error across the whole study domain and the absolute error of the approximations have also been computed. Both methods were found to demonstrate high accuracy and stability throughout the study, even over long periods.

(III)

Additionally, the two-dimensional FBE has been analyzed using both proposed approaches, and some third-order approximations have been generated. Furthermore, the effect of fractionality p on the DA shock wave profile has been examined, revealing that the shock wave profile is sensitive to changes in the fractional-order parameter, exposing behaviors that the integer form of the two-dimensional BE cannot detect. To verify the accuracy and stability of the generated approximations, a graphical comparison has been performed between all derived approximations and the exact solution to the two-dimensional BE, i.e., for $p=1$. The comparison showed complete agreement among all generated approximations and the exact solution. Furthermore, the residual and absolute errors have also been calculated for each approximation, demonstrating that all approximations exhibit high accuracy and confirming the effectiveness of both approaches in analyzing more complex fractional EWEs.

Successful application of the TT and the NIM to the fractional Burgers-type equations proves their flexibility and ability to control problems over a broad spectrum of scientific fields.

(IV)

Advantages of the TT

• Does not need linearization, discretization, or perturbation.
• Does not require writing the linear or nonlinear terms in a complicated manner, such as other methods.
• Fast and low-cost computations.
• Works easily for more complicated nonlinear FDEs.
• Easy to implement computations.
• Provides accurate and stable solutions.
• Higher-order approximations are easy to generate.
• Does not require applying any transform to facilitate the calculation process, unlike other methods.
• All generated approximations are characterized by being short expressions compared to other methods.

Future work: Examining how well these techniques (the TT and NIM) transfer to real-world environments can lead to a deeper understanding of nonlinear phenomena, such as those seen in fluid dynamics and diverse plasma models. For example, current approaches can analyze different fractional nonlinear EWEs used to model nonlinear waves in plasma, such as nonlinear Schrödinger-type equations [68], fractional KdV-type equations, and fractional Kawahara-type equations [69]. Moreover, studying fractional forced Burgers-type equations poses significant challenges due to the presence of an external perturbation force and the fact that the coefficients are not constant [70]. The methods used, especially the TT, are not limited to analyzing the fractional family of Burgers equations but also enable the simulation of a broader range of physical and engineering problems with high accuracy and realism.