On the Tantawy Technique for Analyzing Fractional Kuramoto–Sivashinsky-Type Equations and Modeling Shock Waves in Plasmas and Fluids—Part (I), Planar Case

Abstract

The Kuramoto–Sivashinsky (KS) equation and its fractional generalizations (FKSs) arise as canonical models for a wide class of nonlinear dissipative–dispersive systems, including thin-film flows, combustion fronts, drift–wave turbulence in plasmas, and chemically reacting media, where shock-like and strongly localized structures play a central role in the dynamics. Despite their apparent simplicity, KS-type models become analytically intractable once higher-order dissipation, geometric effects, and memory (fractional) operators are incorporated, and standard perturbative or transform-based schemes often lead to cumbersome recursive structures, slow convergence, or severe restrictions on the initial data. In this work, a novel direct approximation procedure, referred to as the Tantawy Technique (TT), is developed and implemented to solve and analyze planar fractional KS-type equations and their Burgers-type reductions in a systematic manner. The central difficulty is to construct, for a given physically motivated initial profile, a rapidly convergent series in fractional time that remains stable for a broad range of the fractional order and transport coefficients, while still retaining a clear link to the underlying shock-wave physics. To overcome this, the TT combines (i) a Tanh-based exact shock solution of the planar integer-order KS equation, obtained first as a reference via the standard Tanh method, with (ii) a carefully designed fractional-time ansatz in powers of $t^{\rho }$, where the spatial coefficients are determined recursively from the governing equation in the Caputo sense. This construction yields closed-form expressions for the first few terms in the approximation hierarchy and allows one to monitor convergence through residual and absolute error measures.

1. Introduction

In the last few decades, differential Equations (DEs) have become indispensable in bridging the gap between theoretical predictions, experimental measurements, and space observations. It is now possible to describe a wide range of natural, physical, and technological nonlinear phenomena by analyzing diverse evolution equations, in either differential or integral form, derived from first principles or reduced fluid and kinetic models. Various classes of DEs have also proved successful at forecasting the future behavior of complex systems, underscoring their practical role in prediction, control, and design in applied physics and engineering. Consequently, whether in their integer or fractional formulations, the analysis of different DEs has gained increasing importance for explaining natural, engineering, and optical processes, predicting weather- and climate-related events, and clarifying the anomalous material properties exploited in industrial, medical, and technological applications.

Over the past three decades (roughly), mathematicians and physicists have devoted considerable attention to fractional calculus (FC) because of its close connection to the fundamental concepts of memory, nonlocality, and long-range correlations in many physical and engineering systems. In particular, fractional derivatives have been used to model anomalous (sub-)diffusion, unify different diffusion regimes in a single framework, describe fractional random walks, and capture complex wave propagation in plasmas and fluids. Leibniz and L’Hospital [1] pioneered the introduction of the FC concept, which later gained substantial academic recognition due to its broad applicability. Subsequent studies, however, highlighted that some early formulations suffer from drawbacks such as strong sensitivity to initial states and the non-physical behavior of the derivative of a non-zero constant. To overcome these limitations, Caputo proposed a modified fractional operator that treats initial conditions more physically transparently, and most applied models in FC now employ the Caputo derivative. A large body of work has since used FC-based modeling to explain nonlinear phenomena in physical, engineering, chemical, and biological systems [2,3,4,5,6,7,8,9,10]. For example, Momani and Shawagfeh [11] applied the Adomian decomposition method (ADM) to obtain analytical and numerical approximations of fractional Riccati differential equations, while the homotopy analysis method (HAM) has been used to solve fractional (non)linear diffusion and wave equations [3,4]. The HAM has also been employed to treat systems of fractional nonlinear partial differential Equations (PDEs) [4]. Moreover, Adams-type predictor corrector schemes have been implemented for both linear and nonlinear fractional differential Equations (FDEs) [8], and Laplace residual power series techniques in the Caputo framework have been utilized to study time-fractional generalized Burger–Fisher equations and their associated shock structures [12]. In general, FDEs provide an effective description of physical and technological processes with memory, which explains their growing prominence in fields such as electrodynamics [13], signal processing [14], chaos theory [15], finance [16], biological modeling [17], financial mathematics [18], and optics [19].

Among the nonlinear PDEs that have attracted sustained attention, the Kuramoto–Sivashinsky (KS) equation occupies a prominent position because of its ability to capture rich dynamics with relatively simple structure. The KS equation supports a variety of nonlinear behaviors, including traveling waves, pattern formation, and spatiotemporal chaos, and it describes systems in which long-wavelength instabilities are balanced by short-wavelength dissipation through nonlinear mode coupling. It appears naturally in several physical contexts, such as reaction–diffusion systems, long waves on thin films, unstable drift waves in plasmas, and related dissipative media [20]. Historically, the KS equation was first derived in the context of flame front propagation, plasma instabilities, and phase turbulence in reaction–diffusion systems [21]. It has since been discussed in connection with flame-front instability and drift waves in plasmas [22] and with interfacial long waves between viscous fluids [23]. From a numerical viewpoint, generalized KS (gKS) equations have been solved using finite difference and collocation methods [24]. In planar geometry, the KS equation can admit exact traveling-wave solutions, as shown later. However, when additional physical effects, such as nonplanar geometries or more complex dissipative mechanisms, are taken into account, KS-type equations typically cease to be exactly solvable in closed form. This motivates the search for reliable approximation techniques capable of handling such nonintegrable extensions. One of the main goals of the present work is to construct accurate analytical approximations for the planar KS and its fractional counterpart using a modern, stable direct method. In this regard, the direct ansatz method (DAM) has already proved effective in analyzing different nonlinear evolution equations and in deriving analytical approximations that provide physical insight into the associated nonlinear structures [25,26]. These results have helped clarify several nonlinear phenomena in plasmas, fluids, and nonlinear optics by supplying explicit waveforms and parameter dependencies for the underlying evolution Equations [25,26].

The fractional KS (FKS) equation extends the classical KS model by replacing the standard time derivative with a fractional operator, thereby incorporating memory effects into the dynamics. Fractional derivatives are particularly well suited for describing systems with long-range temporal correlations and history-dependent responses. In many contexts, the FKS formulation has been shown to outperform the classical KS equation in accurately reproducing experimentally observed turbulence, anomalous diffusion, or long-time relaxation features. Several analytical and numerical methods have been proposed for solving FKS-type equations. These include the natural decomposition method combined with non-singular kernel operators [27], the q-homotopy analysis transform technique [28], and quintic B-splines in the Caputo sense for linear time FKS Equations [29]. Exact and approximate traveling-wave solutions of stochastic and deterministic fractional KS equations have also been obtained using the $(G^{\prime }/G)$-expansion method, ADM, sub-equation techniques for conformable derivatives, and homotopy perturbation transform methods based on Atangana-Baleanu-Caputo and Liouville–Caputo derivatives [30,31,32,33]. Additional studies have applied various numerical and semi-analytical schemes to the broader KS family, with applications to physical, chemical, and engineering models [34].

In this study, a new variant of the FKS equation is considered in a planar geometry, with coefficients chosen to represent specific plasma configurations of interest, and attention is focused on constructing fractional shock-wave approximations for this model. The main objectives can be summarized as follows:

(I)

First, derive exact planar shock wave solutions for the integer-order KS equation using the Tanh method (TM) [35,36,37]. These solutions are then used as reference profiles and initial conditions to analyze planar FKS-type equations, including the planar fractional Burgers equation and the planar FKS equation.

(II)

Second, introduce and implement the Tantawy Technique (TT) [38,39,40,41] as a direct and flexible tool for analyzing the planar FKS equation and deriving higher-order analytical approximations without resorting to complicated transformations or linearization or decomposition. The TT constructs a fractional-time series with spatially dependent coefficients determined recursively from the governing equation, enabling rapid, accurate approximations while keeping the analysis transparent and physically interpretable.

Within this framework, special attention is paid to (i) the role of the chosen initial profile in controlling the quality and convergence of the approximations and (ii) the influence of the fractional parameter $\rho$ on the evolution of planar fractional shock waves. The subsequent sections present the general TT for fractional differential equations, its application to the planar fractional Burgers and FKS equations, and a detailed numerical study of a representative complex plasma model.

2. The Algorithm of the Tantawy Technique (TT) for Analyzing FDEs

This section clearly describes how to carry out the TT for analyzing different types of FDEs [38,39,40,41]. To facilitate a comprehensive and rapid comprehension of this technique, we shall delineate it in the following straightforward points:

Step (1)

Let us consider the subsequent general form to fractional DE:

$$D_t^{\rho }u=P\left[x,t,u,{\partial }_{x}u,{\partial }_x^{2}u,{\partial }_x^{3}u,\dots \right],$$

(1)

where $u\equiv u\left(x,t\right)$ and $D_t^{\rho }u$ indicates the the time fractional Caputo derivative operator (FCDO) of order $\rho$ $\left(0<\rho <1\right)$.

Step (2)

According to the TT, the approximate solution to Equation (1) can be introduced in the subsequent Ansatz form

$$u=f+\sum _{n=1}^{\infty }{g}_n{t}^{n\rho }=f+g_1{t}^{\rho }+g_2{t}^{2\rho }+g_3{t}^{3\rho }+\cdots ,$$

(2)

where the functions $g_n\equiv g_n(x)$ are to be determined later and $f\equiv f(x)$ indicates any initial condition (IC) to Equation (1).

For the first three approximations in the solution series (2), we can rewrite Ansatz (2) in the following manner

$$u=f+\sum _{n=1}^{\infty }{g}_n{t}^{n\rho }.$$

(3)

Step (3)

According to the Ansatz (3), the FCDO $D_t^{\rho }u$, can be defined as follows

$$D_t^{\rho }u=\sum _{n=1}^{\infty }{g}_n{D}_t^{\rho }\left[t^{\rho n}\right]=\sum _{n=1}^{\infty }{g}_n{\displaystyle \frac{\mathrm{\Gamma }\left[n\rho +1\right]}{\mathrm{\Gamma }\left[(n-1)\rho +1\right]}}{t}^{(n-1)\rho },$$

(4)

where $\mathrm{\Gamma }\left(n+1\right)=n!$ represents gamma function.

Step (4)

By substituting Ansatz (3) and the FCDO (4) into Equation (1), we have

$$\begin{array}{cc}\hfill & \sum _{n=1}^{\infty }{g}_n{\displaystyle \frac{\mathrm{\Gamma }\left[n\rho +1\right]}{\mathrm{\Gamma }\left[(n-1)\rho +1\right]}}{t}^{(n-1)\rho }\hfill \\ \hfill & =P\left[x,t,\left(f+\sum _{n=1}^{\infty }{g}_n{t}^{n\rho }\right), {\partial }_x\left(f+\sum _{n=1}^{\infty }{g}_n{t}^{n\rho }\right), {\partial }_x^2\left(f+\sum _{n=1}^{\infty }{g}_n{t}^{n\rho }\right),\dots \right].\hfill \end{array}$$

(5)

Step (5)

Rearranging all terms of Equation (5) and consolidating the coefficients of identical powers of $t^{n\rho }$ implies

$$H_0+H_1{t}^{\rho }+H_2{t}^{2\rho }+H_3{t}^{3\rho }+\cdots =0,$$

(6)

with

$$\begin{array}{cc}\hfill H_0& \equiv H_0\left(f,f_1,f_2,f_3,\dots ,g_1\right)\hfill \\ \hfill H_1& \equiv H_1\left(f,f_1,f_2,f_3,\dots ,g_1,{\dot{g}}_1,{\ddot{g}}_1,\dots ,g_2\right)\hfill \\ \hfill H_2& \equiv H_2\left(f,f_1,f_2,f_3,\dots ,g_1,{\dot{g}}_1,{\ddot{g}}_1,\dots ,g_2,{\dot{g}}_2,{\ddot{g}}_2,\dots ,g_3\right)\hfill \\ \hfill & ⋮.\hfill \end{array}$$

For simplicity, the following notations are considered in all subsequent calculations for any n,

$$\left\{\begin{array}{c}{f}^{(n)}(x)\equiv f^{(n)}\equiv f_n,f(x)\equiv f\equiv f_0,\\ {\dot{g}}_n\equiv g_n^{\prime },{\ddot{g}}_n\equiv g_n^{″},g_n^{‴}\equiv {\stackrel{⃛}{g}}_n,g_n^{⁗}\equiv \stackrel{⃜}{g}.\end{array}\right.$$

(7)

Step (6)

Equating to zero the coefficients $T_0$, $T_1$, and $T_2$, and solving them in $g_1$, $g_2$, and $g_3$, we finally obtain the values of $g_1$, $g_2$, and $g_3$, as functions of f and its derivatives

$$\begin{array}{cc}\hfill g_1& =\left(h_1\left(\rho \right),f,f_1,f_2,f_3,\dots \right)\hfill \\ \hfill g_2& =\left(h_2\left(\rho \right),f,f_1,f_2,f_3,\dots ,g_1,{\dot{g}}_1,{\ddot{g}}_1,\dots \right)\hfill \\ \hfill g_3& =\left(h_3\left(\rho \right),f,f_1,f_2,f_3,\dots ,g_1,{\dot{g}}_1,{\ddot{g}}_1,\dots ,g_2,{\dot{g}}_2,{\ddot{g}}_2,\dots \right)\hfill \\ \hfill & ⋮.\hfill \end{array}$$

(8)

Here, $h_i\left(\rho \right)$ ∀ $i=1,2,3,\dots ,$ are known functions derived from the application of the Caputo derivative operator.

Step (7)

By collecting the obtained values of $g_i\left(x\right)$ ∀ $i=1,2,3$,⋯, into the Ansatz (3), we ultimately obtain the approximation to Equation (1). In the following section, we proceed to apply this novel and straightforward technique to solve and analyze some evolutionary plasma wave equations, such as the planar fractional Burgers’ equation and planar FKS equation. Firstly, we apply TM to derive shock wave solutions for the planar KS equation.

3. Tanh Method (TM) for Solving the Planar Integer KS Equation

External variables can affect the physical structure of various systems, including physical, engineering, and optical systems, hence influencing the nonlinear phenomena that emerge and propagate inside them. These impacts encompass external noise and disturbances, as well as those arising within the system itself. One method to depict these influences is to examine the fourth-order dissipative effect on the Burgers equation: $u_t+\alpha u{u}_x-\beta u_{xx}=0$. This phenomenon resembles the findings of the Japanese research team [42,43] in their investigation of ion-acoustic waves in magnetized plasma. A substantial discrepancy was identified between the experimental and theoretical outcomes. The study team amended the evolutionary wave equation by integrating a fifth-order dispersion term to rectify this deficiency. Subsequently, the research demonstrated robust concordance between the actual and theoretical outcomes when the fifth-order dispersion effect was incorporated into the KdV equation. Inspired by this discovery, we can include a fourth-order dissipative term into the Burgers equation to derive the SK equation and employ the TM to achieve accurate solutions in integer form. These solutions can thereafter be employed to examine the fractional representation of the SK equation to investigate the influence of memory on the behavior and properties of the shock waves delineated by this family.

Here, the TM is applied to analyze the planar non-fractional KS equation to find some exact solutions to this model [35,36,37]:

$$KS\equiv u_t+\alpha u{u}_x-\beta u_{xx}+\gamma u_{xxxx}=0,$$

(9)

where $u\equiv u(x,t)$ and the coefficients $\left(\alpha ,\beta ,\gamma \right)$ are contingent upon the specific physical model being examined, such as certain plasma models. Remember that $\alpha \ne 0$ ensures the presence of nonlinearity, $\beta >0$ guarantees dissipative effect, and $\gamma \ne 0$ is required to retain the fourth-order spatial derivative that characterizes the planar KS dynamics and distinguishes it from the standard planar Burgers equation.

To find an exact shock wave solution to Equation (9), the subsequent Ansatz is introduced according to the TM

$$u=\sum _{j=0}^n{A}_j{z}^j,$$

(10)

with $z=tanh(\eta )$, where $\eta =\left(kx-\mu t+{\xi }_0\right)$ and ${\xi }_0$ is a shift and can be taken as zero.

The value of n can be determined from the balance between the higher-order nonlinearity and the higher-order derivative of model (9): $\left(n+\left(n+1\right)\right)=\left(n+4\right):\Rightarrow n=3$, which leads to

$$u=A+Bz+C{z}^2+D{z}^3,$$

(11)

Note that for $\gamma =0$, the Burgers equation is recovered.

Now, by plugging the Ansatz (11) into Equation (9), and after tedious but straightforward calculations, and for $\beta >0$, we obtain

$$B=-{\displaystyle \frac{45{\beta }^{3/2}}{19\alpha \sqrt{19\gamma }}},D={\displaystyle \frac{15{\beta }^{3/2}}{19\alpha \sqrt{19\gamma }}},k={\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{\beta }{19\gamma }}},\mu ={\displaystyle \frac{A\alpha }{2}}\sqrt{{\displaystyle \frac{\beta }{19\gamma }}},$$

(12)

which lead to the following two shocks solutions

$$u_{Ex}=A-{\displaystyle \frac{45{\beta }^{3/2}}{19\alpha \sqrt{19\gamma }}}tanh\left(\eta \right)+{\displaystyle \frac{15{\beta }^{3/2}}{19\alpha \sqrt{19\gamma }}}{tanh}^3\left(\eta \right),\beta >0,$$

(13)

where $C=0$ and A is an arbitrary constant. Note that the shock wave solution (13) does not recover the shock solution to the following Burgers equation

$$Bs\equiv u_t+\alpha u{u}_x-\beta u_{xx}=0,$$

(14)

because the coefficient of fourth-order dissipative appears in the denominator, and it cannot be set to zero to recover shock wave solution for Burgers Equation (14). The planar Burgers Equation (14) can be solved in the same way by applying the TM, which in this case the balance between the higher-order nonlinearity and the higher-order derivative of model (14) leads to $\left(n+\left(n+1\right)\right)=\left(n+2\right):\Rightarrow n=1$;

$$u=A+Bz,$$

(15)

and by inserting the Ansatz (15) into Equation (14), we ultimately obtain the following planar shock wave solution

$$u={\displaystyle \frac{\mu }{\alpha }}\left[1-tanh\left[{\displaystyle \frac{\mu }{2\beta }}\left(x-\mu t\right)\right]\right].$$

(16)

In the following section, we proceed to apply the TT in order to analyze the planar KS Equation (9) in its fractional form.

4. TT for Analyzing Planar Fractional Burgers Equation (FBE)

Numerous studies have been conducted on various plasma models to investigate the properties of shock waves that arise and propagate in different plasma models. This can be achieved by reducing the fundamental fluid equations of a plasma model to some related evolutionary wave Equations (EWEs) using the reductive perturbation method (RPM), such as the planar Burgers equation with second-order dissipative and KdV-Burgers (KdVB) equation with third-order dispersion and second-order dissipative. For instance, the fluid-governed equations for a collisionless, unmagnetized, electron-depleted complex plasma, consisting of inertial negatively charged dust grains and inertialess nonextensive ions, have been reduced to the following planar Burgers’ equation utilizing the RPM [44].

$$u_t+\alpha u{u}_x-\beta u_{xx}=0,$$

(17)

with the following dimensionless nonlinearity and dissipative coefficients

$$\alpha =-{\displaystyle \frac{V_p^3}{2}}\left[{\displaystyle \frac{{\mu }_i}{4}}\left(q+1\right)\left(q-3\right)+{\displaystyle \frac{3}{V_p^4}}\right]\&\beta ={\displaystyle \frac{\eta }{2}},$$

(18)

and the normalized phase velocity of the dust-acoustic waves reads

$$V_p=\sqrt{{\displaystyle \frac{2}{{\mu }_i\left(q+1\right)}}},$$

where ${\mu }_i$ represents the normalized ion concentration, q refers to the nonextensive parameter, and $\eta$ indicates the normalized dust kinematic viscosity coefficient. For a numerical example, we can use the same physical values as mentioned in Ref. [44]: $\left(q,\eta \right)=\left(0.7,0.1\right)$ with the following exact shock wave solution to Equation (17).

$$u=u_m\left[1-tanh\left[{\displaystyle \frac{\left(x-U_{0}t\right)}{W}}\right]\right],$$

(19)

where $u_m=U_0/\alpha$ and $W=2\beta /U_0$ are, respectively, the amplitude and width of the shock waves.

To examine the influence of the time-fractional parameter (fractionality) on the dynamics of shock waves in the current complex plasma model, it is necessary to convert the integer Burgers Equation (17) to its fractional counterpart. By adhering to the identical methodology outlined in Refs. [45,46,47], we can convert Equation (17) to the following time-fractional Burgers equation:

$$D_t^{\rho }u+\alpha u{u}_x-\beta u_{xx}=0,\forall 0<\rho <1,$$

(20)

Now, the following brief steps are considered for analyzing Equation (20) using the TT [38,39,40,41]:

Step (1)

Rewrite Equation (20) in the following initial value problem (I.V.P.) form

$$\left\{\begin{array}{c}{D}_t^{\rho }u+\alpha u{u}_x-\beta u_{xx}=0,\forall 0<\rho <1,\hfill \\ u(x,0)=f(x)=u_m\left[1-tanh\left[{\displaystyle \frac{x}{W}}\right]\right],\hfill \\ 0\le t\le T \mathrm{and} -X\le x\le X.\hfill \end{array}\right.$$

(21)

Step (2)

According to the TT, the approximate solution to problem (21) until the first-three approximations can be introduced in the subsequent convergence Ansatz form

$$u=f+\sum _{n=1}^3{g}_n{t}^{\rho }=f+g_1{t}^{\rho }+g_2{t}^{2\rho }+g_3{t}^{3\rho },$$

(22)

where the spatial functions $g_n\equiv g_n(x)$ are undetermined values and $f\equiv f(x)$ indicates the IC to Equation (21).

Step (3)

Based on Ansatz (22), the FCDO $D_t^{\rho }u$ to the function u, can be defined as follows

$$\begin{array}{cc}{D}_t^{\rho }u& =D_t^{\rho }\left[f+{\displaystyle \sum _{n=1}^3}{g}_n{t}^{\rho }\right]={\displaystyle \sum _{n=1}^3}{g}_n{D}_t^{\rho }\left[t^{\rho n}\right]\hfill \\ & ={\displaystyle \sum _{n=1}^3}{g}_n{\displaystyle \frac{\mathrm{\Gamma }\left[n\rho +1\right]}{\mathrm{\Gamma }\left[(n-1)\rho +1\right]}}{t}^{(n-1)\rho },\hfill \end{array}$$

(23)

where $n=1,2,3,\cdots ,$ and $\mathrm{\Gamma }\left[n+1\right]=n!$ indicates gamma function.

Step (4)

Substituting Ansatz (22) and the FCDO (23) into problem (21) yields

$$\begin{array}{cc}{\displaystyle \sum _{n=1}^3}{g}_n{\displaystyle \frac{\mathrm{\Gamma }\left[n\rho +1\right]}{\mathrm{\Gamma }\left[(n-1)\rho +1\right]}}{t}^{(n-1)\rho }& =-\alpha \left(f+g_1{t}^{\rho }+g_2{t}^{2\rho }+g_3{t}^{3\rho }\right)\hfill \\ & \times {\left(f+g_1{t}^{\rho }+g_2{t}^{2\rho }+g_3{t}^{3\rho }\right)}_x\hfill \\ & +\beta {\left(f+g_1{t}^{\rho }+g_2{t}^{2\rho }+g_3{t}^{3\rho }\right)}_{xx}.\hfill \end{array}$$

(24)

Step (5)

By rearranging all terms of Equation (24) and collecting the coefficients of the same power of $t^{n\rho }$, we obtain

$$H_0+H_1{t}^{\rho }+H_2{t}^{2\rho }+H_3{t}^{3\rho }+\cdots =0,$$

(25)

with

$$\begin{array}{cc}\hfill H_0& ={\mathrm{\Gamma }}_1{g}_1+\alpha f{f}_1-\beta f_2,\hfill \\ \hfill H_1& ={\displaystyle \frac{{\mathrm{\Gamma }}_2{g}_2}{{\mathrm{\Gamma }}_1}}+\alpha g_1{f}_1+\alpha f{\dot{g}}_1-\beta {\ddot{g}}_1,\hfill \\ \hfill H_2& ={\displaystyle \frac{{\mathrm{\Gamma }}_3{g}_3}{{\mathrm{\Gamma }}_2}}+\alpha f_1{g}_2+\alpha g_1{\dot{g}}_1+\alpha f{\dot{g}}_2-\beta {\ddot{g}}_2,\hfill \\ \hfill & ⋮.\hfill \end{array}$$

For simplicity, the following notations are considered: ${\mathrm{\Gamma }}_n\equiv \mathrm{\Gamma }\left[n+\rho \right]$, ∀ $n=1,2,3,\cdots$.

Step (6)

Equating to zero the coefficients $T_0$, $T_1$, and $T_2$, and solving them in $g_1$, $g_2$, and $g_3$, we finally obtain the values of $g_1$, $g_2$, and $g_3$, as follows

$$\begin{array}{cc}{g}_1& ={\displaystyle \frac{1}{{\mathrm{\Gamma }}_1}}\left(-\alpha f{f}_1+\beta f_2\right),\hfill \\ g_2& ={\displaystyle \frac{1}{{\mathrm{\Gamma }}_2}}\left[\begin{array}{c}2{\alpha }^{2}f{f}_1^2+{\alpha }^2{f}^2{f}_2\\ -4\alpha \beta f_1{f}_2-2\alpha \beta f{f}_3+{\beta }^2{f}_4\end{array}\right],\hfill \\ g_3& ={\displaystyle \frac{1}{{\mathrm{\Gamma }}_1^2{\mathrm{\Gamma }}_3}}\left[\begin{array}{c}{\beta }^3{f}_6{\mathrm{\Gamma }}_1^2-3\alpha {\beta }^{2}f{f}_5{\mathrm{\Gamma }}_1^2\\ -9\alpha {\beta }^2{f}_1{f}_4{\mathrm{\Gamma }}_1^2+3{\alpha }^2\beta f^2{f}_4{\mathrm{\Gamma }}_1^2\\ -\alpha {\beta }^2{f}_2{f}_3{\mathrm{\Gamma }}_2-14\alpha {\beta }^2{f}_2{f}_3{\mathrm{\Gamma }}_1^2\\ +{\alpha }^2\beta f{f}_1{f}_3{\mathrm{\Gamma }}_2+16{\alpha }^2\beta f{f}_1{f}_3{\mathrm{\Gamma }}_1^2\\ -{\alpha }^3{f}^3{f}_3{\mathrm{\Gamma }}_1^2+{\alpha }^2\beta f{f}_2^2{\mathrm{\Gamma }}_2\\ +10{\alpha }^2\beta f{f}_2^2{\mathrm{\Gamma }}_1^2+{\alpha }^2\beta f_1^2{f}_2{\mathrm{\Gamma }}_2\\ +16{\alpha }^2\beta f_1^2{f}_2{\mathrm{\Gamma }}_1^2-{\alpha }^3{f}^2{f}_1{f}_2{\mathrm{\Gamma }}_2\\ -7{\alpha }^3{f}^2{f}_1{f}_2{\mathrm{\Gamma }}_1^2-{\alpha }^{3}f{f}_1^3{\mathrm{\Gamma }}_2-4{\alpha }^{3}f{f}_1^3{\mathrm{\Gamma }}_1^2,\end{array}\right]\hfill \\ & ⋮.\hfill \end{array}$$

(26)

Step (7)

Inserting the following IC

$$f(x)=u_m\left[1-tanh\left[{\displaystyle \frac{x}{W}}\right]\right],$$

(27)

into Equation (26), the following explicit values of $g_i\left(x\right)$ ∀ $i=1,2,3$ are obtained as

$$\begin{array}{cc}{g}_1& ={\displaystyle \frac{u_m{S}_0{sech}^2\left(\frac{x}{W}\right)}{W^2{\mathrm{\Gamma }}_1}},\hfill \\ g_2& ={\displaystyle \frac{u_m{sech}^4\left(\frac{x}{W}\right)}{W^4{\mathrm{\Gamma }}_2}}\left[2S_{1}tanh\left({\displaystyle \frac{x}{W}}\right)-S_2\right],\hfill \\ g_3& ={\displaystyle \frac{u_m{sech}^2\left(\frac{x}{W}\right)}{W^6{\mathrm{\Gamma }}_1^2{\mathrm{\Gamma }}_3}}\left\{\begin{array}{c}\alpha W{u}_m{\mathrm{\Gamma }}_2{sech}^2\left({\displaystyle \frac{x}{W}}\right)\left(S_3+S_4\right)\\ +2{\mathrm{\Gamma }}_1^2\left[\begin{array}{c}4{sech}^2\left({\displaystyle \frac{x}{W}}\right)\left(S_5+S_6\right)+S_7\\ -S_8{sech}^4\left({\displaystyle \frac{x}{W}}\right)\left(\alpha W{u}_m-2\beta \right)\end{array}\right]\end{array}\right\},\hfill \end{array}$$

(28)

where the values of the coefficients $S_i$ ∀ $i=0,1,2,\cdots 8$ are given in Appendix A.

Step (8)

Inserting the obtained values of $g_i\left(x\right)$ ∀ $i=1,2,3$ given in Equation (28) into the Ansatz (22), we ultimately obtain the shock wave approximation to Equation (1) up to the third-order approximation

$$\begin{array}{cc}u& =f+g_1{t}^{\rho }+g_2{t}^{2\rho }+g_3{t}^{3\rho }\hfill \\ & =u_m\left[1-tanh\left[{\displaystyle \frac{x}{W}}\right]\right]+{\displaystyle \frac{u_m{S}_0{sech}^2\left(\frac{x}{W}\right)}{W^2{\mathrm{\Gamma }}_1}}{t}^{\rho }\hfill \\ & +{\displaystyle \frac{u_m{sech}^4\left(\frac{x}{W}\right)}{W^4{\mathrm{\Gamma }}_2}}\left[2S_{1}tanh\left({\displaystyle \frac{x}{W}}\right)-S_2\right]t^{2\rho }\hfill \\ & +{\displaystyle \frac{u_m{sech}^2\left(\frac{x}{W}\right)}{W^6{\mathrm{\Gamma }}_1^2{\mathrm{\Gamma }}_3}}\left\{\begin{array}{c}\alpha W{u}_m{\mathrm{\Gamma }}_2{sech}^2\left({\displaystyle \frac{x}{W}}\right)\left(S_3+S_4\right)\\ +2{\mathrm{\Gamma }}_1^2\left[\begin{array}{c}4{sech}^2\left({\displaystyle \frac{x}{W}}\right)\left(S_5+S_6\right)+S_7\\ -S_8{sech}^4\left({\displaystyle \frac{x}{W}}\right)\left(\alpha W{u}_m-2\beta \right)\end{array}\right]\end{array}\right\}{t}^{3\rho }.\hfill \end{array}$$

(29)

5. TT for Analyzing Planar FKS Equation

To apply the TT for analyzing the planar time FKS equation, first we consider the following form to the FKS problem for $0<\rho \le 1$,

$$\left\{\begin{array}{c}{D}_t^{\rho }u+\alpha u{u}_x-\beta u_{xx}+\gamma u_{xxxx}=0,\hfill \\ u(x,0)=f(x),\hfill \\ 0\le t\le Tand-X\le x\le X,\hfill \end{array}\right.$$

(30)

where $D_t^{\rho }u$ indicates the FCDO of order $\rho$ ∀ $0<\rho <1$. Remember that for $\rho =1$, the standard KS integer-order Equation (9) is recovered.

The TT for analyzing problem (30) can be summarized in following concise steps [38,39,40,41]:

Step (1)

According to the TT, the solution of problem (30) can be introduced in the following convergence series solution

$$u=f(x)+\sum _{n=1}^{\infty }{g}_n(x)t^{\rho n},$$

(31)

where the spatial functions $g_n\equiv g_n(x)$ are to be determined later and $f\equiv f(x)$ indicates any initial solution/condition (IC) to problem (30).

Step (2)

According to Ansatz (31), the FCDO $D_t^{\rho }u$ can be defined as follows

$$\begin{array}{cc}{D}_t^{\rho }u& =D_t^{\rho }\left[f(x)+{\displaystyle \sum _{n=1}^{\infty }}{g}_n(x)t^{\rho n}\right]={\displaystyle \sum _{n=1}^{\infty }}{g}_n{D}_t^{\rho }\left[t^{\rho n}\right]\hfill \\ & ={\displaystyle \sum _{n=1}^{\infty }}{g}_n{\displaystyle \frac{\mathrm{\Gamma }\left[n\rho +1\right]}{\mathrm{\Gamma }\left[(n-1)\rho +1\right]}}{t}^{(n-1)\rho }.\hfill \end{array}$$

(32)

Step (3)

For a specific number of the series solution (say N), following $n^{\mathrm{th}}$-approximation is considered

$$u=f+\sum _{n=1}^N{g}_n{t}^{\rho n}.$$

(33)

Step (4)

For $N=3$, the relation (33) can be rewritten as follows

$$u=f+\sum _{n=1}^3{g}_n{t}^{\rho n}=f+g_1{t}^{\rho }+g_2{t}^{2\rho }+g_3{t}^{3\rho }.$$

(34)

Step (5)

The following residual error definition $\mathcal{R}\equiv \mathcal{R}(x,t)$ is introduced to check the accuracy of the obtained shock wave approximation

$$\begin{array}{cc}\mathcal{R}& =\left|D_t^{\rho }u+\alpha u{u}_x-\beta u_{xx}+\gamma u_{xxxx}\right|\hfill \\ & =\left|\begin{array}{c}{\mathrm{\Gamma }}_1{g}_1+{\displaystyle \frac{{\mathrm{\Gamma }}_2}{{\mathrm{\Gamma }}_1}}{g}_2{t}^{\rho }+{\displaystyle \frac{{\mathrm{\Gamma }}_3}{{\mathrm{\Gamma }}_2}}{g}_3{t}^{2\rho }\\ +\alpha \left\{f+t^{\rho }\left[g_1+t^{\rho }\left(g_2+g_3{t}^{\rho }\right)\right]\right\}\\ \times \left\{f_1+t^{\rho }\left[\dot{g_1}+t^{\rho }\left(\dot{g_2}+\dot{g_3}{t}^{\rho }\right)\right]\right\}\\ -\beta \left\{f_2+t^{\rho }\left[{\ddot{g}}_1+t^{\rho }\left({\ddot{g}}_2+{\ddot{g}}_3{t}^{\rho }\right)\right]\right\}\\ +\gamma \left\{f_4+t^{\rho }\left[{\stackrel{⃜}{g}}_1+t^{\rho }\left({\stackrel{⃜}{g}}_2+{\stackrel{⃜}{g}}_3{t}^{\rho }\right)\right]\right\}\end{array}\right|.\hfill \end{array}$$

(35)

Step (6)

By plugging Equation (32) with solution (34) into problem (30), and collecting the coefficients of the same power of $t^{n\rho }$, we obtain

$$H_0+H_1{t}^{\rho }+H_2{t}^{2\rho }+\cdots =0,$$

(36)

with

$$\begin{array}{c}{H}_0=\alpha f_0{f}_1-\beta f_2+\gamma f_4+g_1{\mathrm{\Gamma }}_1,\\ H_1={\displaystyle \frac{{\mathrm{\Gamma }}_2}{{\mathrm{\Gamma }}_1}}{g}_2+\alpha \left(f_1{g}_1+f_0{\dot{g}}_1\right)-\beta {\ddot{g}}_1+\gamma {\stackrel{⃜}{g}}_1,\\ H_2={\displaystyle \frac{{\mathrm{\Gamma }}_3}{{\mathrm{\Gamma }}_2}}{g}_3+\alpha \left(f_1{g}_2+f_0{\dot{g}}_2+g_1{\dot{g}}_1\right)-\beta {\ddot{g}}_2+\gamma {\stackrel{⃜}{g}}_2.\end{array}$$

(37)

Step (7)

Solve system: $\left\{T_0=0,T_1=0,T_2=0\right\}$, the values of the unknown functions $g_1$, $g_2$, and $g_3$ can be obtained as follows

$$\begin{array}{c}{g}_1={\displaystyle \frac{1}{{\mathrm{\Gamma }}_1}}\left[\beta f_2-\alpha f_0{f}_1-\gamma f_4\right],\hfill \\ g_2={\displaystyle \frac{1}{{\mathrm{\Gamma }}_2}}\left[\begin{array}{c}{\mathrm{\Gamma }}_1\left(\beta {\ddot{g}}_1-\gamma {\stackrel{⃜}{g}}_1-\alpha f_0{\dot{g}}_1\right)\\ +{\alpha }^2{f}_0{f}_1^2+\alpha f_1\left(\gamma f_4-\beta f_2\right)\end{array}\right],\hfill \\ g_3={\displaystyle \frac{1}{{\mathrm{\Gamma }}_3}}\left[\alpha {\mathrm{\Gamma }}_1{f}_1\left(-\beta {\ddot{g}}_1+\gamma {\stackrel{⃜}{g}}_1+\alpha f_0{\dot{g}}_1\right)\right.\hfill \\ +{\mathrm{\Gamma }}_2\left(\beta {\ddot{g}}_2-\gamma {\stackrel{⃜}{g}}_2-\alpha f_0{\dot{g}}_2\right)-{\alpha }^3{f}_0{f}_1^3+{\alpha }^2\beta f_2{f}_1^2\hfill \\ \left.-{\alpha }^2\gamma f_4{f}_1^2+{\displaystyle \frac{\alpha {\dot{g}}_1{4}^{\rho }}{\sqrt{\pi }}}\mathrm{\Gamma }\left(\rho +{\displaystyle \frac{1}{2}}\right)\left(\alpha f_0{f}_1-\beta f_2+\gamma f_4\right)\right].\hfill \end{array}$$

(38)

The values of $g_1$, $g_1$, and $g_1$ given in Equation (38) equivalent the following simplified forms

$$\begin{array}{c}{g}_1={\displaystyle \frac{1}{{\mathrm{\Gamma }}_1}}\left[\beta f_2-\alpha f_0{f}_1-\gamma f_4\right],\\ g_2={\displaystyle \frac{-{\mathrm{\Gamma }}_1}{{\mathrm{\Gamma }}_2}}\left[g_1\alpha f_1+\alpha f_0{\dot{g}}_1-\beta {\ddot{g}}_1+\gamma {\stackrel{⃜}{g}}_1\right],\\ g_3={\displaystyle \frac{-{\mathrm{\Gamma }}_2}{{\mathrm{\Gamma }}_3}}\left[g_2\alpha f_1+\left(\alpha f_0{\dot{g}}_2-\beta {\ddot{g}}_2+\gamma {\stackrel{⃜}{g}}_2+\alpha g_1{\dot{g}}_1\right)\right].\end{array}$$

(39)

Step (8)

By solving system (39) in the function f, the explicit form to $g_1$, $g_2$, and $g_3$ can be obtained as the follows

$$\begin{array}{cc}\hfill g_1& ={\displaystyle \frac{1}{{\mathrm{\Gamma }}_1}}\left[\beta f_2-\alpha f_0{f}_1-\gamma f_4\right],\hfill \\ \hfill g_2& ={\displaystyle \frac{1}{{\mathrm{\Gamma }}_2}}\left[\begin{array}{c}{f}_1\left(6\alpha \gamma f_4-4\alpha \beta f_2\right)+\gamma \left(10\alpha f_2{f}_3-2\beta f_6+\gamma f_8\right)\\ +{\beta }^2{f}_4+{\alpha }^2{f}_{2}f{(x)}^2+2\alpha f(x)\left(\alpha f_1^2-\beta f_3+\gamma f_5\right)\end{array}\right],\hfill \end{array}$$

and

$$g_3=-{\displaystyle \frac{\begin{array}{c}{\mathrm{\Gamma }}_1^2\left(\begin{array}{c}{\gamma }^3{f}^{(12)}(x)+4{\alpha }^2{f}_1^2\left(9\gamma f_4-4\beta f_2\right)+30{\alpha }^2\gamma f_2^3\\ +\alpha f_1\left(130\alpha \gamma f_2{f}_3+9{\beta }^2{f}_4+3\gamma \left(5\gamma f_8-8\beta f_6\right)\right)\\ +14\alpha {\beta }^2{f}_2{f}_3-86\alpha \beta \gamma f_3{f}_4-56\alpha \beta \gamma f_2{f}_5+132\alpha {\gamma }^2{f}_4{f}_5\\ +94\alpha {\gamma }^2{f}_3{f}_6+46\alpha {\gamma }^2{f}_2{f}_7+{\beta }^3\left(-f_6\right)+3{\beta }^2\gamma f_8\\ -3\beta {\gamma }^2{f}_{10}+{\alpha }^3{f}_{3}f{(x)}^3+{\alpha }^{2}f{(x)}^2\left(7\alpha f_1{f}_2-3\beta f_4+3\gamma f_6\right)\\ +\alpha f(x)\left(\begin{array}{c}4{\alpha }^2{f}_1^3+2\alpha f_1\left(11\gamma f_5-8\beta f_3\right)\\ +3\gamma \left(10\alpha f_3^2-2\beta f_7+\gamma f_9\right)\\ -10\alpha \beta f_2^2+46\alpha \gamma f_2{f}_4+3{\beta }^2{f}_5\end{array}\right)\end{array}\right)\\ +\alpha {\mathrm{\Gamma }}_2\left(-\beta f_2+\gamma f_4+\alpha f_{1}f(x)\right)\left(\alpha f_1^2-\beta f_3+\gamma f_5+\alpha f_{2}f(x)\right)\end{array}}{{\mathrm{\Gamma }}_1^2{\mathrm{\Gamma }}_3}}.$$

Step (9)

According to the TM, the following IC $f(x)$ is considered

$$f(x)=A+Btanh(kx)+D{tanh}^3\left(kx\right).$$

(40)

Step (10)

Inserting Equations (38) and (40) into Ansatz (34) and for $\beta >0$, we obtain

$$\begin{array}{cc}\hfill g_1& =-{\displaystyle \frac{k{sech}^2(kx)}{{\mathrm{\Gamma }}_1}}\left(AB\alpha +Y_{0}tanh(kx)\right),\hfill \\ \hfill g_2& =-{\displaystyle \frac{2k^2{sech}^2(kx)}{{\mathrm{\Gamma }}_2}}{\displaystyle \sum _{j=1}^8}{Y}_j,\hfill \\ \hfill g_3& =\mathrm{Extreme}\mathrm{value},\hfill \end{array}$$

which lead to

$$\begin{array}{cc}u& =A+Btanh(kx)+D{tanh}^3\left(kx\right)\hfill \\ & -{\displaystyle \frac{k{sech}^2(kx)}{{\mathrm{\Gamma }}_1}}\left(AB\alpha +Y_{0}tanh(kx)\right)t^{\rho }\hfill \\ & -{\displaystyle \frac{2k^2{sech}^2(kx)}{{\mathrm{\Gamma }}_2}}\left({\displaystyle \sum _{j=1}^8}{Y}_j\right)t^{2\rho }+\cdots ,\hfill \end{array}$$

(41)

where the coefficients $Y_j$ are given in the Appendix B.

For simplicity, in approximation (41), we can consider “$u^{\left(1\right)}$” for the first-order approximation ($g_1\ne 0$ & $g_{2,3}=0$), “$u^{\left(2\right)}$” for the second-order approximation ($g_{1,2}\ne 0$ & $g_3=0$), and “$u^{\left(3\right)}$” for the third-order approximation ($g_{1,2,3}\ne 0$).

Numerical example: According to the above discussed plasma model and by considering higher-order dissipative, the following time FKS problem is considered

$$D_t^{\rho }u+\alpha u{u}_x-\beta u_{xx}+\gamma u_{xxxx}=0,$$

(42)

where the coefficients $\left(\alpha ,\beta \right)$ are defined in Equation (18) and the coefficient of fourth-order dissipative $\gamma$ takes the range $0<\gamma \le 1$. According to the physical values to the related parameters that have been discussed in Ref. [44], we consider the following values: $\left(q,\eta \right)=\left(0.7,0.1\right)$ which leads to $\left(\alpha ,\beta \right)=\left(-0.759257,0.05\right)$. Furthermore, the following IC is considered

$$u(x,0)\equiv f(x)=A+Btanh\left(kx\right)+D{tanh}^3\left(kx\right),\beta >0,$$

(43)

with

$$B=-{\displaystyle \frac{45{\beta }^{3/2}}{19\alpha \sqrt{19\gamma }}},D={\displaystyle \frac{15{\beta }^{3/2}}{19\alpha \sqrt{19\gamma }}}\&k={\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{\beta }{19\gamma }}}.$$

The numerical and analytical results obtained for the planar fractional Burgers and FKS equations highlight several key physical and methodological features of the TT. From the methodological side, the comparison between the exact planar KS shock solution in the integer case (13) and the TT-based approximations (41) at $\rho =1$ shows excellent agreement, as evidenced by the graphical comparison as illustrated in Figure 1. This agreement confirms that the fractional-time approximation (41) recovers the known integer-order dynamics in the appropriate limit and that the truncation at relatively low order in $t^{\rho }$ is sufficient to capture the essential structure of the shock wave profile with high accuracy.

Additionally, the absolute errors according to the following relations and for various order approximations (e.g., first-order approximation $\left(g_{2,3}=0\right)$, second-order approximation $\left(g_3=0\right)$, and third-order approximation $\left(g_{2,3}\ne 0\right)$) are quantitatively computed at $A=0.01$ and $A=0.1,$ as seen in

Table 1. The absolute error for the third-order approximation (41) is estimated at $\tau =1$ and $A=0$.

x	Exact	${\left.{\mathit{u}}^{(1)}(41)\right|}_{\mathit{\rho }=1}$	${\left.{\mathit{u}}^{\left(2\right)}(41)\right|}_{\mathit{\rho }=1}$	${\left.{\mathit{u}}^{\left(3\right)}(41)\right|}_{\mathit{\rho }=1}$	${10}^{-13}\times {\mathit{L}}_1$	${10}^{-17}\times {\mathit{L}}_2$	${10}^{-17}\times {\mathit{L}}_3$
−75	−0.0421345	−0.0421345	−0.0421345	−0.0421345	5.29396	5.2945	5.2945
−65	−0.0419503	−0.0419503	−0.0419503	−0.0419503	0.661744	0.660785	0.660785
−55	−0.0414736	−0.0414736	−0.0414736	−0.0414736	9.26442	9.26794	9.26794
−45	−0.040296	−0.040296	−0.040296	−0.040296	2.64698	2.64019	2.64019
−35	−0.0375916	−0.0375916	−0.0375916	−0.0375916	5.29396	5.28814	5.28814
−25	−0.0320238	−0.0320238	−0.0320238	−0.0320238	29.1168	29.1161	29.1161
−15	−0.022201	−0.022201	−0.022201	−0.022201	10.5879	10.5821	10.5821
−5	−0.00803834	−0.00803834	−0.00803834	−0.00803834	7.94093	7.94029	7.94029
5	0.00803834	0.00803834	0.00803834	0.00803834	7.94093	7.94029	7.94029
15	0.022201	0.022201	0.022201	0.022201	10.5879	10.5821	10.5821
25	0.0320238	0.0320238	0.0320238	0.0320238	29.1168	29.1161	29.1161
35	0.0375916	0.0375916	0.0375916	0.0375916	5.29396	5.28814	5.28814
45	0.040296	0.040296	0.040296	0.040296	2.64698	2.64019	2.64019
55	0.0414736	0.0414736	0.0414736	0.0414736	9.26442	9.26794	9.26794
65	0.0419503	0.0419503	0.0419503	0.0419503	0.661744	0.660785	0.660785
75	0.0421345	0.0421345	0.0421345	0.0421345	5.29396	5.2945	5.2945

and

Table 2. The absolute error for the third-order approximation (41) is estimated at $\tau =1$ and $A=0.1$.

x	Exact	${\left.{\mathit{u}}^{\left(1\right)}(41)\right|}_{\mathit{\rho }=1}$	${\left.{\mathit{u}}^{\left(2\right)}(41)\right|}_{\mathit{\rho }=1}$	${\left.{\mathit{u}}^{\left(3\right)}(41)\right|}_{\mathit{\rho }=1}$	${10}^{-9}\times {\mathit{L}}_1$	${10}^{-13}\times {\mathit{L}}_2$	${10}^{-17}\times {\mathit{L}}_3$
−75	0.0578656	0.0578656	0.0578656	0.0578656	0.0491621	0.15099	0.
−65	0.0580499	0.0580499	0.0580499	0.0580499	0.126071	0.370121	0.693889
−55	0.0585271	0.0585271	0.0585271	0.0585271	0.30604	0.830655	0.693889
−45	0.0597057	0.0597057	0.0597057	0.0597057	0.679598	1.60143	2.77556
−35	0.0624122	0.0624122	0.0624122	0.0624122	1.3081	2.3348	2.08167
−25	0.0679834	0.0679834	0.0679834	0.0679834	2.00448	1.7427	5.55112
−15	0.0778106	0.0778106	0.0778106	0.0778106	2.10526	1.54057	11.1022
−5	0.0919767	0.0919767	0.0919767	0.0919767	0.945327	5.5848	6.41848
5	0.108053	0.108053	0.108053	0.108053	0.946443	5.5835	6.59195
15	0.122213	0.122213	0.122213	0.122213	2.10557	1.53839	10.7553
25	0.132031	0.132031	0.132031	0.132031	2.00413	1.74374	4.85723
35	0.137595	0.137595	0.137595	0.137595	1.30763	2.33459	0.
45	0.140298	0.140298	0.140298	0.140298	0.679278	1.60094	2.08167
55	0.141474	0.141474	0.141474	0.141474	0.305874	0.830308	2.08167
65	0.141951	0.141951	0.141951	0.141951	0.125997	0.370051	0.693889
75	0.142135	0.142135	0.142135	0.142135	0.0491319	0.150921	0.693889

, respectively;

$$\begin{array}{cc}\hfill L_1& ={\left|u_{Ex}(13)-u^{\left(1\right)}(41)\right|}_{g_{2,3}=0},\hfill \\ \hfill L_2& ={\left|u_{Ex}(13)-u^{\left(2\right)}(41)\right|}_{g_3=0},\hfill \\ \hfill L_3& ={\left|u_{Ex}(13)-u^{\left(3\right)}(41)\right|}_{g_{2,3}\ne 0}.\hfill \end{array}$$

Furthermore, the residual error analysis, performed for different values of the fractional parameter and different values of A ($A=0.01$ and $A=0.1$, as shown below, respectively), indicates that the approximations remain stable and convergent throughout the whole spatiotemporal domain considered, with errors decreasing as the approximation order increases:

$$\begin{array}{cc}\underset{x,t}{max}{\left|\mathcal{R}(x,t)\right|}_{\rho =0.1}& =5.14305\times {10}^{-16},\hfill \\ \underset{x,t}{max}{\left|\mathcal{R}(x,t)\right|}_{\rho =0.5}& =3.53717\times {10}^{-14},\hfill \\ \underset{x,t}{max}{\left|\mathcal{R}(x,t)\right|}_{\rho =1}& =2.17727\times {10}^{-12}.\hfill \end{array}$$

(44)

and

$$\begin{array}{cc}\underset{x,t}{max}{\left|\mathcal{R}(x,t)\right|}_{\rho =0.1}& =4.69423\times {10}^{-12},\hfill \\ \underset{x,t}{max}{\left|\mathcal{R}(x,t)\right|}_{\rho =0.5}& =2.58487\times {10}^{-10},\hfill \\ \underset{x,t}{max}{\left|\mathcal{R}(x,t)\right|}_{\rho =1}& =2.7258\times {10}^{-8}.\hfill \end{array}$$

(45)

The obtained results demonstrate high accuracy and stability of the generated approximations across the entire study domain, which confirm the strength of the employed methodology. Furthermore, the analysis reveals that the form of the initial solution/condition plays a significant role in the accuracy of the derived approximations for the fractional problem. Similarly, the values of the relevant physical parameters influence the accuracy of the derived approximations. However, the most essential characteristic of derived approximations is their high accuracy and stability. Additionally, it is also observed that the derived approximations exhibit high accuracy, even at the first-order level. As the order of the approximation increases to third order, the accuracy improves further, as shown in Table 2. However, in Table 1, at small value for $A=0.01$, the accuracy remained almost constant from the beginning of the second order, with the same absolute error of both the second- and third-order approximations, and with a few differences with the absolute error of the first-order approximation. These results demonstrate the convergence of all derived approximations at any level of order, as well as their high accuracy and remarkable stability across the entire study domain.

We also studied the effect of the fractionality on the shock wave profile, as shown in Figure 2 and Figure 3 for $A=0.01$ and $A=0.1$, respectively. It is worth noting that fractionality $\rho$ plays a significant role in the dynamical behavior and propagation of the planar shock waves. It is observed that the effect of the fractionality appears after a long time period, due to the influence of the physical parameters relevant to the model under study. This means that, with changes in the values of these parameters, the effect of fractionality may appear over short time intervals. Physically, the present TT-based analysis makes the influence of fractionality on planar shock waves in KS-type models for complex plasmas more transparent. In particular, the fractional-order $\rho$ emerges as an effective measure of temporal memory: when $\rho$ decreases below unity, the system increasingly “remembers” its past evolution, and the associated shock profile undergoes noticeable changes in amplitude, width, and relaxation characteristics. These Figure 2 and Figure 3 indicate that for smaller $\rho$, the shock fronts develop more slowly, with the onset of steepening or smoothing postponed according to the balance between nonlinear advection and higher-order dissipation. This retardation in the dynamics is a direct manifestation of the nonlocal nature of the Caputo fractional derivative, which distributes the impact of earlier times across the subsequent evolution, and it is particularly relevant for the description of plasmas and fluids in which transport is governed by anomalous diffusion or long-time correlation effects.

Furthermore, in Figure 4, we examined the time effect on the dynamics of shock wave propagation at fixed $\rho =0.7$. It is observed that the profile is not sensitive to the effect at short time intervals, but as time increases, the effect of increasing time on the shock wave profile becomes apparent. Figure 2, Figure 3 and Figure 4 indicate that, for the parameter set considered, the influence of fractionality becomes more pronounced at longer times. This suggests that in laboratory or astrophysical plasmas with different characteristic scales, fractional behavior might emerge either earlier or later in the evolution, depending on the competition between nonlinear steepening, viscous dissipation, and higher-order dissipative processes. Such insights are valuable for interpreting experimental and space observations in regimes where classical integer-order models fail to reproduce the observed shock profiles or relaxation rates.

6. Conclusions

To sum up, in this work, a novel direct approximation framework, the Tantawy Technique (TT), has been introduced and applied to analyze the planar fractional Kuramoto–Sivashinsky (FKS) equation and its Burgers-type reduction. The main objective was to construct reliable fractional shock-wave approximations that retain a clear physical interpretation while accurately capturing memory effects encoded by the Caputo time derivative. The strategy combines an exact planar shock solution of the integer-order KS equation, obtained via the Tanh method, with a fractional-time Ansatz in powers of $t^{\rho }$, whose spatial coefficients are determined recursively from the governing equation. This construction yields explicit low-order approximations that can be systematically improved, and it avoids the algebraic complexity and restrictive assumptions commonly encountered in other analytical methods for fractional PDEs. Numerical examples of a realistic electron-depleted complex plasma with nonextensive ions demonstrate the efficacy of the TT. For $\rho =1$, the obtained approximations reproduce the known planar KS shock with excellent accuracy, as confirmed by direct comparison and by the minor absolute errors. For $0<\rho <1$, the method provides stable, convergent fractional shock profiles, with residual errors remaining small throughout the computational domain. The analysis reveals that the choice of initial profile and the values of the physical parameters strongly influence the rate of convergence and the time scale at which fractional effects become apparent. In particular, for small initial amplitudes, the fractional series converges quickly even when only a few terms in the series of approximations are retained, while larger amplitudes and stronger nonlinearities make the higher-order terms significantly more influential in the overall approximation.

From a physical perspective, this investigation demonstrates how introducing fractional time derivatives changes the propagation characteristics of the planar shock waves in the framework of the fractional KS-type models, and how their profiles evolve in time. The fractional-order parameter $\rho$ acts as a control knob for temporal memory: by varying the fractionality $\rho$, one effectively adjusts how sharply the shock front steepens, how wide it becomes, and how quickly it relaxes. This provides a flexible modeling tool for plasmas and fluids in which classical KS dynamics must be extended to account for anomalous transport or long-time correlations. The TT thus emerges as a robust and straightforward approach for generating analytical approximations to fractional evolution equations in mathematical physics and engineering, with a level of accuracy and stability that makes it suitable for confronting laboratory data and space observations without resorting to intensive numerical computation.

Future work: A forward-looking extension of the present work is to apply the Tantawy Technique [38,39,40,41] to a broader class of fractional nonlinear wave models, with particular emphasis on soliton and shock-wave dynamics in realistic plasma environments [48]. In the next work, as Part–II of the current investigation, we will expand on this idea by including the geometrical effect (wave curvature (cylindrical and spherical)), which is one of the key ideas that no researcher has addressed, by combining the effects of both memory and curvature and exploring in detail how the fractional order reshapes their amplitude, width, and stability properties in comparison with the integer-order case.