Approximate solutions of nonlinear fractional Kundu-Eckhaus and coupled fractional massive Thirring equations emerging in quantum field theory using conformable residual power series method

Nonlinear complex FPDEs are generalized models of classical nonlinear PDEs with arbitrary orders involving functions of complex-valued, which have comprehensive and diverse uses in applied mathematics, theoretical physics, and engineering [1–4]. Due to its immense ability to simulate, embody reality, and modeling natural phenomena, it has already attracted the attention of many other scholars who considered it a powerful tool as well as a symbolic language for contemporary mathematics. To name a few, many models that have long-term temporal and spatial memory interactions in microbiology, biochemistry, epidemiology, finance, geomagnetic, astronomy, and economics can be expressed using nonlinear FPDEs with complex functions [5–10]. The categories of fractional operators differ partially from classical in engineering and physical interpretations, and there is no clear and specific physical interpretation as in the integer parameter that indicates the velocity and acceleration of particles. In addition, there are also in the literature many definitions of fractional derivation that have evolved since the origins of fractional calculus, and not like the differentiation and integration of the integer parameter that relies on an explicit and specific definition [11–16]. Anyhow, Liouville-Caputo, Grünwald-Letnikov, Caputo-Fabrizio, and Atangana-Baleanu are some of these common concepts. Recently, a new definition of fractional calculus, called CFO, has been developed based on the essential sense of continuity, derivatives, and integrals using the limits. Since then, numerous theories, characteristics, and rules have been proven in conjunction with the development of the basic theory of the CFO concept [17–23].

Dynamical processes associated with complex partial systems often possess a non-local characteristic that can be distinguished by long-time memory and wide-ranging hereditary potential where the fractional operator excels in dealing with those properties. Therefore, it became possible through fractional calculus to optimize the formation of physical systems and obtain useful and realistic dynamic models. Such models can be found in different areas of science, including gaseous flow, fluid dynamics, solid-state physics, electromagnetic, fractal media, wave propagation, nonlinear acoustics, plasma, traffic flow, optical fibers, nuclear physics and so on [24–31]. The FKEE and coupled FMTEs are complex nonlinear models relevant to a wide variety of applications in physics and have received the attention of many researchers. Nevertheless, the KEE was created by Kundu in 1984 [32] from the nonlinear Schrödinger type equations and was autonomously derived by Eckhaus in 1985 as a linearization processing for nonlinear quantum mechanics equation [33]. While the coupled MTEs was introduced by Thirring in 1958 [34]. The KEE is also applied to depict optical soliton propagation of ultrashort pulses in high birefringence nonlinear fibers. González-Gaxiola [35] has used the LADM for solving KEE. In [36], dark and bright soliton solutions for KEE with nonlinear cubic-quintic sense were introduced by using Hirota method. While the authors in [37] have presented new complex analytic solutions for KEE using Bernoulli sub-equation function method. Manafian and Lakestani [38] have also obtained some new exact soliton, periodic, singular kink-type, and singular cupson wave solutions of KEE utilizing the $\tan \left(\varphi \left(\xi \right)\right)$-expansion method. In addition, $\sinh$-Gordon expansion method has been extended to study new complex hyperbolic and complex trigonometric solutions, including bright, dark, soliton, and singular soliton solutions for the KEE [39].

The investigation for analytical and approximate solutions to complex nonlinear conformable FPDEs plays a significant role in the study of natural phenomena in several areas of engineering, theoretical physics, and mathematics, including nonlinear optical fiber, chemical kinetics, and plasma physics. Obtaining approximate and analytical solutions to those equations in the former fields is very essential in analyzing the solution behaviors of the considered models. More specifically, the motivation of this paper is to obtain approximate and numerical solutions for classes of complex nonlinear FPDEs within nonlinear Schrödinger with CFO derivative by using the RPSM. Anyhow, we consider and discuss the following complex nonlinear fractional models.

• The one-dimensional time-FKEE of the form:
  $$\begin{eqnarray}&&\begin{array}{l}i{T}_{\tau }^{\gamma }u\left(x,\tau \right)+\alpha {u}_{xx}\left(x,\tau \right)+\sigma u\left(x,\tau \right){\left({\left|u\left(x,\tau \right)\right|}^{2}\right)}_{x}\\ \,+\,{\delta }^{2}u\left(x,\tau \right){\left|u\left(x,\tau \right)\right|}^{4}=0,\end{array}\end{eqnarray} \tag{ 1 }$$
  supplemented with the following ICC:
  $$\begin{eqnarray}&&u\left(x,0\right)=\mu {e}^{ix}.\end{eqnarray} \tag{ 2 }$$
  Here, $\gamma \in \left(0,\,1\right],$ $x\in \left[a,b\right],$ $\tau \in \left[a,\infty \right),$ ${i}^{2}=-1,$ and $\mu ,a,a,b\in {\mathbb{R}}$ with $\mu \ne 0.$ In (1) and (2); $u$ is a complex-valued function represents the wave profile of two real independent variables x and $\tau$ that symbolize spatial and temporal terms, respectively. While, $\alpha ,\sigma ,{\delta }^{2}$ are non-zero real constants for group velocity dispersion, nonlinear effect, and quantic nonlinearity of propagation, respectively. $i{T}_{\tau }^{\gamma }u$ is temporal evolution of the nonlinear wave and ${T}_{\tau }^{\gamma }$ is the CTFD of order $\gamma .$ This model illustrates the optical soliton propagation of pulse waves in dispersive lines [40].
• The one-dimensional two-coupled time-FMTEs of the form:

$$\begin{eqnarray}&&\begin{array}{l}i\left({T}_{\tau }^{\gamma }u\left(x,\tau \right)+{u}_{x}\left(x,\tau \right)\right)+v\left(x,\tau \right)+u\left(x,\tau \right){\left|v\left(x,\tau \right)\right|}^{2}=0,\\ i\left({T}_{\tau }^{\gamma }v\left(x,\tau \right)-{v}_{x}\left(x,\tau \right)\right)+u\left(x,\tau \right)+v\left(x,\tau \right){\left|u\left(x,\tau \right)\right|}^{2}=0,\end{array}\end{eqnarray} \tag{ 3 }$$

supplemented with the following ICC:

$$\begin{eqnarray}&&\begin{array}{l}u\left(x,0\right)={\mu }_{0}{e}^{ix},\\ v\left(x,0\right)={\xi }_{0}{e}^{ix}.\end{array}\end{eqnarray} \tag{ 4 }$$

Here, $\gamma \in \left(0,\,1\right],$ $x\in \left[a,b\right],$ $\tau \in \left[a,\infty \right),$ ${i}^{2}=-1,$ and ${\mu }_{0},{\xi }_{0},a,a,b\in {\mathbb{R}}$ with $0\ne {\mu }_{0},{\xi }_{0}\ne 0.$ In (3) and (4); $u$ and $v$ are complex-valued functions represent the wave profiles of two real independent variables x and $\tau$ that symbolize spatial and temporal terms, respectively, and ${T}_{\tau }^{\gamma }$ is the CTFD of order $\gamma .$ This model is a state of dynamic systems that presents chaotic behavior and combines the propagation of nonlinear waves with diffusion effects, but, in fact, it has no exact solution in the literature.

These considered models, as we noted, are nonlinear complex FPDEs that are difficult to obtain exact solutions for particularly complex potentials. Therefore, advanced numerical and approximation techniques are generally applied to obtain solutions. The most common techniques used are homotopy analysis method, reproducing kernel method, homotopy perturbation method, Adomain decomposition method, differential transform method, traveling wave method, expansion method, and first integral method [41–45]. Anyhow, the fractional versions of such models are just simple generalization to arbitrary order derivatives that have many applications in quantum field theory, dispersive media, and nonlinear optics. For instance, some novel exact and solitary traveling wave solutions for FKEE are given in [40] by using a novel expansion method. In [41], Arafa and Hagag have applied the qHATM to get approximate solutions for FKEE and FMTEs. Additionally, an efficient technique based on Demirci and Ozalp transformation method has been employed to find exact complex solutions of Kundu-Eckhaus and MTEs [42]. In [43], F-expansion method has been implemented to find different forms of analytical soliton wave solutions for nonlinear wave FKEE. Kadkhoda and Jafari [44] have used sine-Gordon expansion method to get analytic solutions of space-time FKEE with CFO derivative utilizing symbolic computations. While some new exact travelling wave solutions for space-time FKEE have been obtained by (G'/G2)-expansion method as presented in [45].

Dynamical systems respond to their local environment and can be derived and written down with causal nature in the sense used. Typically, the mathematical construction of temporal differential models depends not only on the current state and influences of its behavior, but also on the past behavior to accomplish the desired goal. On a local view of causality, the time domain most naturally used for expressions of causal behavior. In this direction, specific roles of cause and effect are assigned for the components of dynamical models in which the time-derivative term refers to the effect while the other terms to causes, and so the rate changes of the effect depend on the strength of some causes. Thus, it requires knowing the initial conditions and the on-going behavior of the environment to solve. Anyhow, we should aware that caution must be taken when providing causal interpretations for physical processes. Also, the simultaneous existence of the left and right derivatives makes it more difficult. Generally, the theory developed for causal differential equations covers the theory of different dynamic systems in a single framework [46]. Fractional derivations play a more effective role in dealing with this formalism. Further, a fractional differential equation is said to be causal if it involves fractional derivatives of a single type. In physics, the left fractional derivative according to the time domain represents the previous state of the dynamical process, while the right fractional derivative indicates the future one [47, 48]. In the application of quantum mechanics, the right fractional derivatives can be used to describe the velocity of an antiparticle due to Feynman's and Stückelberg's views about 'antiparticles propagate backward in time'.

The RPSM is an alternative attractive analytical as well as approximate technique in obtaining fractional power series solutions by the means of residual errors for a vast range of applications in physics, applied mathematics, and engineering. It was successfully implemented to investigate exact and approximate solutions of several nonlinear FPDEs, uncertain differential equations, chaos models, and complex systems [23, 49–62]. This systematic method is easy to employ for obtaining infinite series patterns that lead to closed-form solutions in many cases without resort to any truncation, discretization, or linearization. It differs from the classical PSM, which does not need to compare the corresponding coefficients together, nor to find recursive relations, nor to switch to ordinary differential systems with auxiliary equations or even the creation of algebraic systems. In this orientation, by deriving the residual errors many-times, the desirable coefficients can be directly obtained considering the type of fractional operators.

This study consists of five sections. Section 2 is given to provide some fundamental concepts of CFO derivatives and their properties. In section 3, a description of the conformable RPSM and its application to time-FKEE and time-FMTEs are introduced by reviewing the steps necessary to do so. The proposed method relies on finding desirable approximate solutions to physical problems by equating multiple derivatives of residual errors to zero and finding conformable fractional coefficients of infinite power series in a systematic way at a rate of rapid convergence. Approximate and numerical solutions to time-FKEE and time-FMTEs are obtained with conformable sense in section 4. Some conclusions are summarized in section 5.

In general, there are many different concepts of fractional computation that have evolved in the past few decades to deal with derivatives and integrations that occur for an arbitrary order. However, Khalil, et al [19] have proposed a new fractional concept, namely CFO, which is presented by means of the 'limits' to cover deficiencies in other existing concepts. Moreover, in this concept, the derivative of any real constant is equal to zero, while the initial conditions for complex nonlinear fractional systems can be handled similarly to those in standard systems. Consequently, many characteristics of the CFOs and their wide applications in the fields of quantum physics, applied physics, and others have been investigated as utilized in [17–23]. Through the rest of the paper, we are standing for the following: ${{\rm{\Omega }}}_{x}:=\left[a,b\right],$ ${{\rm{\Omega }}}_{\tau }=\left[a,\infty \right),$ ${\rm{\Omega }}:{{\rm{\Omega }}}_{x}\times {{\rm{\Omega }}}_{\tau }\to {\mathbb{R}},$ and ${\rm{\Lambda }}={{\rm{\Omega }}}_{x}\times \left[\left.a,a+{\sigma }^{1/\gamma }\right)\right..$

Here, we provide some definitions about CFO derivative and their characteristics. Firstly, if $\gamma \in \left(0,\,1\right]$ and $u$ be differentiable for all $\tau \gt 0,$ then CFO derivative of $u:\left[0,\infty \right)\to {\mathbb{R}}$ of order γ is ${T}_{\tau }^{\gamma }u\left(\tau \right)=\mathop{\mathrm{lim}}\limits_{h\to 0}\tfrac{u\left(\tau +h{\tau }^{1-\alpha }\right)-u\left(\tau \right)}{h}$ provided that $\mathop{\mathrm{lim}}\limits_{\tau \to {0}^{+}}{T}_{\tau }^{\gamma }u\left(\tau \right)$ exists, so that, ${T}_{\tau }^{\gamma }u\left(\tau \right)\,=\mathop{\mathrm{lim}}\limits_{\tau \to {0}^{+}}{T}_{\tau }^{\gamma }u\left(\tau \right)$ and $u\left(\tau \right)$ is γ-differentiable in some $\left(0,a\right)$ with $a\gt 0.$ Secondly, if $\gamma \in \left(0,\,1\right],$ then CFO integral of order γ of $u:{{\rm{\Omega }}}_{\tau }\to {\mathbb{R}}$ is ${{\mathscr{ {\mathcal I} }}}_{\tau }^{\gamma }u\left(\tau \right)=\displaystyle {\int }_{a}^{\tau }\tfrac{u\left(\xi \right)}{{\left(\xi -a\right)}^{1-\gamma }}d\xi$ and ${{\mathscr{ {\mathcal I} }}}_{\tau }^{0}u\left(\tau \right)\,=u\left(\tau \right),$ where $\tau \gt \xi \gt a$ provided that the integral exists.

Definition 1. [21] Let $\gamma \in \left(n-1,n\right]$ with $n\in {\mathbb{N}},$ $x\in {{\rm{\Omega }}}_{x},$ and $\displaystyle \frac{{\partial }^{k}u}{\partial {\tau }^{k}},$ $k=1,2,\ldots ,n-1$ exists on ${\rm{\Omega }}.$ Then, the CTFD of order $\gamma$ for $w:{{\rm{\Omega }}}_{x}\times \left[a,\infty \right)\to {\mathbb{R}}$ is given by

$$\begin{eqnarray}&&{T}_{\tau }^{\gamma }{\mathscr{w}}\left(x,\tau \right)=\mathop{\mathrm{lim}}\limits_{h\to 0}\displaystyle \frac{{\partial }_{\tau }^{n-1}{\mathscr{w}}\left(x,\tau +h{\left(\tau -s\right)}^{n-\alpha }\right)-{\partial }_{\tau }^{n-1}{\mathscr{w}}\left(x,\tau \right)}{h}.\end{eqnarray} \tag{ 5 }$$

Definition 2. [21] Let $\gamma \in \left(n-1,n\right]$ with $n\in {\mathbb{N}}$ and $x\in {{\rm{\Omega }}}_{x}.$ Then, the CFO integral of order $\gamma$ for $w:{\rm{\Omega }}\to {\mathbb{R}}$ is given by

$$\begin{eqnarray}&&\begin{array}{l}{{\mathscr{ {\mathcal I} }}}_{\tau }^{\gamma }{\mathscr{w}}\left(x,\tau \right)=\displaystyle \frac{1}{\left(n-1\right)!}\displaystyle {\int }_{a}^{\tau }\displaystyle \frac{{\left(\tau -\xi \right)}^{n-1}{\mathscr{w}}\left(x,\tau \right)}{{\left(\xi -a\right)}^{n-\gamma }}d\xi ,\\ {{\mathscr{ {\mathcal I} }}}_{\tau }^{0}{\mathscr{w}}\left(x,\tau \right)={\mathscr{w}}\left(x,\tau \right),\end{array}\end{eqnarray} \tag{ 6 }$$

where $\tau \gt \xi \gt a$ provided that the integral exists.

Theorem 1. [21] Let $\gamma \in \left(n-1,n\right]$ with $n\in {\mathbb{N}},$ $x\in {{\rm{\Omega }}}_{x},$ $\tfrac{{\partial }^{k}u}{\partial {\tau }^{k}},$ $k=1,2,\ldots ,\,n-1$ exists for all $\tau \gt a,$ and $w:{\rm{\Omega }}\to {\mathbb{R}}$ be $n$-differentiable. Then, the classical relation between the operators ${T}_{\tau }^{\gamma }$ and ${ {\mathcal I} }_{\tau }^{\gamma }$ is provided by

1. ${T}_{\tau }^{\gamma }\left({{\mathscr{ {\mathcal I} }}}_{\tau }^{\gamma }\left({\mathscr{w}}\left(x,\tau \right)\right)\right)=w\left(x,\tau \right).$

2. ${{\mathscr{ {\mathcal I} }}}_{\tau }^{\gamma }\left({T}_{\tau }^{\gamma }\left({\mathscr{w}}\left(x,\tau \right)\right.\right)={\mathscr{w}}\left(x,\tau \right)-\displaystyle \sum _{k=0}^{n-1}{\partial }_{\tau }^{k}{\mathscr{w}}\left(x,a\right)\tfrac{{\left(\tau -a\right)}^{k}}{k!}.$

Definition 3. [22] For $\gamma \in \left(n-1,n\right]$ with $n\in {\mathbb{N}},$ $x\in {{\rm{\Omega }}}_{x},$ $\gamma \gt 0,$ and $\tau \in {{\rm{\Omega }}}_{\tau },$ the power series of the form

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \sum _{k=0}^{\infty }{{\mathscr{w}}}_{k}\left(x\right){\left(\tau -a\right)}^{k\gamma }={{\mathscr{w}}}_{0}\left(x\right)+{{\mathscr{w}}}_{1}\left(x\right){\left(\tau -a\right)}^{\gamma }\\ \,+\,{{\mathscr{w}}}_{2}\left(x\right){\left(\tau -a\right)}^{2\gamma }\,+\,\ldots ,\end{array}\end{eqnarray} \tag{ 7 }$$

is the MFPS for $\tau =a$ with $\tau \in \left[\left.a,a+{\sigma }^{1/\gamma }\right)\right.$ and $\sigma \gt 0,$ where the sequence ${{\mathscr{w}}}_{k}\left(x\right),$ $k=0,1,2,\ldots$ represents the coefficients of the series and ${\sigma }^{1/\gamma }$ represents the radius of convergence, so that, the MFPS components are disappear when $\tau =a$ with the exception of the initial one ${{\mathscr{w}}}_{0}\left(x\right)$ provided that the MFPS converges when $\tau =a$ and $\left|\tau -a\right|\,\lt {\sigma }^{1/{\mathscr{\gamma }}}$ when $\tau \gt a.$

Theorem 2. [22] Let $\gamma \in \left(n-1,n\right]$ with $n\in {\mathbb{N}},$ $\gamma \gt 0,$ ${\mathscr{w}}:{\rm{\Lambda }}\to {\mathbb{R}}$ has the MFPS at $\tau =a$ of the form

$$\begin{eqnarray}&&{\mathscr{w}}\left(x,\tau \right)=\displaystyle \sum _{k=0}^{\infty }{{\mathscr{w}}}_{k}\left(x\right){\left(\tau -a\right)}^{k\gamma },\end{eqnarray} \tag{ 8 }$$

and ${T}_{\tau }^{\gamma }{\mathscr{w}}$ is continuous on ${\rm{\Lambda }}$ for $k=1,2,\ldots .$ Then, the coefficients ${{\mathscr{w}}}_{k}\left(x\right)$ of FPS (8) can be obtained as

$$\begin{eqnarray}&&{{\mathscr{w}}}_{k}\left(x\right)=\displaystyle \frac{{T}_{\tau }^{k\gamma }{\mathscr{w}}\left(x,a\right)}{\left(k\right)!{\gamma }^{k}},\end{eqnarray} \tag{ 9 }$$

in which ${T}_{\tau }^{k\gamma }$ represents the sequential CFO derivatives of order $k\gamma$ such that ${T}_{\tau }^{k\gamma }({\mathscr{w}}\left(x,\tau \right))=\mathop{\underbrace{{T}_{\tau }^{\gamma }\cdot {T}_{\tau }^{\gamma }\cdot \cdot \cdot {T}_{\tau }^{\gamma }({\mathscr{w}}\left(x,\tau \right))}}\limits_{k-times}.$

Corollary 1. [22] Let $\gamma \in \left(n-1,n\right]$ with $n\in {\mathbb{N}},$ $\gamma \gt 0,$ ${T}_{\tau }^{k\gamma }\left(w\right)$ for $k=0,1,2,\ldots ,\,n+1$ exists for all $\tau \gt a,$ and ${\mathscr{w}}$ has the MFPS (8) about $\tau =a,$ so that, $\left|{T}_{\tau }^{(n+1)\gamma }\left({\mathscr{w}}\left(x,\tau \right)\right)\right|\leqslant {\mathscr{C}}\left(x\right)$ for some $n\in {\mathbb{N}}.$ Then, the error term of MFPS is given as

$$\begin{eqnarray}&&\left|{\varepsilon }_{n}\left(x,\tau \right)\right|\leqslant \displaystyle \frac{{\mathscr{C}}\left(x\right)}{\left(n+1\right)!{\gamma }^{n+1}}{\left(\tau -a\right)}^{\left(n+1\right)\gamma },\end{eqnarray} \tag{ 10 }$$

provided that ${\varepsilon }_{n}\left(x,\tau \right)=\displaystyle {\sum }_{k=n+1}^{\infty }\tfrac{{T}_{\tau }^{k\gamma }{\mathscr{w}}\left(x,a\right)}{\left(k\right)!{\gamma }^{k}}\left(\tau -a\right){}^{k\gamma }.$

The RPSM was first developed by Jordanian mathematician Abu Arqub [49] in 2013 as an alternative analytical and approximation method to deal with a type of differential equations under uncertainty. Subsequently, many different studies have been successfully implemented to deal with realistic and complex physical, medical, biological, economic, and engineering models to represent the generalization of the MFPS associated with fractional derivatives to obtain effective analytical and approximate solutions. Quantum mechanics systems, dynamic systems, chaos systems, bifurcations systems, fluid flow systems, fractal media systems, and oscillating magnetic systems are some of those fractional models [23, 49–62]. Then some improvements were made without resorting to any truncation, discretization, linearization, or even the creation of a set of algebraic systems, taking advantage of the equality of multiple derivatives of the residual functions with zero in finding the required smooth coefficients directly and easily.

Anyhow, in this section, description of the attractive RPSM for obtaining an approximate periodic wave solution of time-FKEE (1) and (2) and time-FMTEs (3) and (4) is introduced in ${\rm{\Omega }}$ by substituting the $n$-term solution into residual errors and allowing the conformable $\left(n-1\right)$-time fractional derivative to be zero, which gives us the ability to estimate truncation coefficients and adding the subsidiary approximations sequentially into the MFPS within rapidly converging rate. To demonstrate the main idea of the proposed method of solving the complex nonlinear FPDEs of CTFD. Let us consider the following general form:

$$\begin{eqnarray}&&i{T}_{\tau }^{\gamma }u\left(x,\tau \right)+{L}_{x}u\left(x,\tau \right)+{N}_{x}u\left(x,\tau \right)=f\left(x,\tau \right),\end{eqnarray} \tag{ 11 }$$

supplemented with the following ICC of periodic wave type:

$$\begin{eqnarray}&&u\left(x,0\right)=\mu {e}^{ix},\end{eqnarray} \tag{ 12 }$$

where $x\in {{\rm{\Omega }}}_{x},$ $\tau \in {{\rm{\Omega }}}_{\tau },$ $\gamma \in \left(0,\,1\right],$ ${i}^{2}=-1,$ and $\mu \in {\mathbb{R}}$ with $\mu \ne 0.$ ${T}_{\tau }^{\gamma }$ is CTFD of order $\gamma$ and $u$ is a complex-valued function to be determined. Further, ${L}_{x},{N}_{x}$ are linear and nonlinear differential operators, respectively, include partial derivatives with respect to spatial term x and $f$ is non-homogeneous source term.

Principally, solving for temporal evolution term $i{T}_{\tau }^{\gamma }u\left(x,\tau \right),$ we have the following:

$$\begin{eqnarray}&&\begin{array}{c}{T}_{\tau }^{\gamma }u\left(x,\tau \right)=i\left(f\left(x,\tau \right)+{L}_{x}u\left(x,\tau \right)+{N}_{x}u\left(x,\tau \right)\right).\end{array}\end{eqnarray} \tag{ 13 }$$

Now, let the solution $u\left(x,\tau \right)$ of (13) expanded in terms of conformable MFPS about $\tau ={\tau }_{0}$ as

$$\begin{eqnarray}&&u\left(x,\tau \right)=\displaystyle \sum _{k=0}^{\infty }{a}_{k}\left(x\right)\displaystyle \frac{{\left(\tau -{\tau }_{0}\right)}^{k\gamma }}{k!{\gamma }^{k}},\end{eqnarray} \tag{ 14 }$$

provided that the $n$-term solution can be expanded by utilizing the ICC (12) as

$$\begin{eqnarray}&&{u}_{n}\left(x,\tau \right)=\mu {e}^{ix}+\displaystyle \sum _{k=1}^{n}{a}_{k}\left(x\right)\displaystyle \frac{{\left(\tau -{\tau }_{0}\right)}^{k\gamma }}{k!{\gamma }^{k}}.\end{eqnarray} \tag{ 15 }$$

Defined the residual error function as ${{\rm{Res}}}_{u}\left(x,\tau \right)\,=\mathop{\mathrm{lim}}\limits_{n\to \infty }{{\rm{Res}}}_{u}^{n}\left(x,\tau \right)$ and the $n$-term residual error function as

$$\begin{eqnarray}&&\begin{array}{c}{{\rm{Res}}}_{u}^{n}\left(x,\tau \right)\,=\,{T}_{\tau }^{\gamma }{u}_{n}\left(x,\tau \right)-i\left(f\left(x,\tau \right)\right.\\ \,\left.+\,{L}_{x}{u}_{n}\left(x,\tau \right)+{N}_{x}{u}_{n}\left(x,\tau \right)\right),\end{array}\end{eqnarray} \tag{ 16 }$$

provided that ${T}_{\tau }^{(n-1)\gamma }{{\rm{Res}}}_{u}^{n}\left(x,{\tau }_{0}\right)=0$ for $n=1,2,3,\ldots .$

To explain the basic steps in obtaining the unknown coefficients ${a}_{k}\left(x\right),$ $k=1,2,\ldots ,n$ of the fractional series formula (15), the main processes of the RPSM is presented in the next algorithm.

Algorithm 1. Assume that $u\left(x,\tau \right)$ is the unique analytical solution of (13) over the interest region ${\rm{\Lambda }},$ where ${\sigma }^{1/\gamma }$ is the radius of convergence and ${T}_{\tau }^{k\gamma }u\left(x,\tau \right)$ with $\gamma \in \left(0,\,1\right]$ and $k=0,1,2,\ldots ,\,n+1$ exists for all $\tau \in {{\rm{\Omega }}}_{\tau }.$ Then, do the following main steps:

Step A. Let the unique analytical solution of (13) has the following MFPS formula at the initial point $\tau ={\tau }_{0}:$

$$\begin{eqnarray}&&u\left(x,\tau \right)=\displaystyle \sum _{k=0}^{\infty }{a}_{k}\left(x\right)\displaystyle \frac{{\left(\tau -{\tau }_{0}\right)}^{k\gamma }}{k!{\gamma }^{k}}.\end{eqnarray} \tag{ 17 }$$

Step B. Based on initial periodic condition, the $n$-term approximate solution is given as

$$\begin{eqnarray}&&{u}_{n}\left(x,\tau \right)=\mu {e}^{ix}+\displaystyle \sum _{k=1}^{n}{a}_{k}\left(x\right)\displaystyle \frac{{\left(\tau -{\tau }_{0}\right)}^{k\gamma }}{k!{\gamma }^{k}}.\end{eqnarray} \tag{ 18 }$$

Step C. The $n$-term residual error function can be defined as

$$\begin{eqnarray}\begin{array}{ccl}{{\rm{Res}}}_{u}^{n}\left(x,\tau \right) & = & {T}_{\tau }^{\gamma }{u}_{n}\left(x,\tau \right)-i\left(f\left(x,\tau \right)\right.\\ & & \left.+\,{L}_{x}{u}_{n}\left(x,\tau \right)+{N}_{x}{u}_{n}\left(x,\tau \right)\right),\end{array}\end{eqnarray} \tag{ 19 }$$

in which the $\infty$-residual error function is ${{\rm{Res}}}_{\infty }\left(x,\tau \right)\,=\mathop{\mathrm{lim}}\limits_{n\to \infty }{{\rm{Res}}}_{u}^{n}\left(x,\tau \right).$

Step D. Substitute (18) into the $n$-term residual error (19) so that

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{Res}}}_{u}^{n}\left(x,\tau \right)={T}_{\tau }^{\gamma }\left(\mu {e}^{ix}+\displaystyle \sum _{n}^{k=1}{a}_{k}\left(x\right)\displaystyle \frac{{\left(\tau -{\tau }_{0}\right)}^{k\gamma }}{k!{\gamma }^{k}}\right)-\,i\left(f\left(x,\tau \right)\right.\\ \,\,\,+\,{L}_{x}\left(\Space{0ex}{3.70ex}{0ex}\mu {e}^{ix}\right.\left.\,+\,\displaystyle \sum _{n}^{k=1}{a}_{k}\left(x\right)\displaystyle \frac{{\left(\tau -{\tau }_{0}\right)}^{k\gamma }}{k!{\gamma }^{k}}\right)\\ \,\,\,+\,{N}_{x}\left(\Space{0ex}{3.70ex}{0ex}\mu {e}^{ix}\right.\left.\,+\,\displaystyle \sum _{n}^{k=1}{a}_{k}\left(x\right)\displaystyle \frac{{\left(\tau -{\tau }_{0}\right)}^{k\gamma }}{k!{\gamma }^{k}}\right).\end{array}\end{eqnarray} \tag{ 20 }$$

Step E. Apply the operator ${T}_{\tau }^{\gamma }$ multiple times on both sides of resulting residual error ${{\rm{Res}}}_{u}^{n}\left(x,\tau \right)$ of (20) for $n=1,2,\ldots ,$ up to arbitrary number and use the fact ${T}_{\tau }^{(n-1)\gamma }{{\rm{Res}}}_{u}^{n}\left(x,{\tau }_{0}\right)=0$ as in the following subroutines:

• For $n=1,$ compute ${{\rm{Res}}}_{u}^{1}\left(x,\tau \right)$ with respect to

$$\begin{eqnarray}&&{u}_{1}\left(x,\tau \right)=\mu {e}^{ix}+{a}_{1}\left(x\right)\displaystyle \frac{{\left(\tau -{\tau }_{0}\right)}^{\gamma }}{\gamma },\end{eqnarray} \tag{ 21 }$$

and then use the fact ${{\rm{Res}}}_{u}^{1}\left(x,{\tau }_{0}\right)=0$ to obtain the first coefficient ${a}_{1}\left(x\right)$ so that the first approximate solution ${u}_{1}\left(x,\tau \right)$ is provided.

• For $n=2,$ operate the conformable fractional derivative ${T}_{\tau }^{\gamma }$ once on both sides of (20), compute ${{\rm{Res}}}_{u}^{2}\left(x,\tau \right)$ with respect to
  $$\begin{eqnarray}&&\begin{array}{l}{u}_{2}\left(x,\tau \right)=\mu {e}^{ix}+i\left(f\left(x,{\tau }_{0}\right)+{L}_{x}{u}_{1}\left(x,{\tau }_{0}\right)\right.\\ \,\,\,\,\left.+\,{N}_{x}{u}_{1}\left(x,{\tau }_{0}\right)\right)\displaystyle \frac{{\left(\tau -{\tau }_{0}\right)}^{\gamma }}{\gamma }+{a}_{2}\left(x\right)\\ \,\,\,\,\times \,\displaystyle \frac{\left(\tau -{\tau }_{0}\right){2}^{\gamma }}{2{\gamma }^{2}},\end{array}\end{eqnarray} \tag{ 22 }$$
  and then use the fact ${T}_{\tau }^{\gamma }{{\rm{Res}}}_{u}^{2}\left(x,{\tau }_{0}\right)=0$ to obtain the second coefficient ${a}_{2}\left(x\right)$ so that the second approximate solution ${u}_{2}\left(x,\tau \right)$ is provided.
• For $n=3,4,\ldots ,\,k$ operate the conformable fractional derivative ${T}_{\tau }^{\gamma }$ twice, thrice, ..., ($k$−1)-times, respectively. Compute ${{\rm{Res}}}_{u}^{n}\left(x,\tau \right)$ with respect to ${u}_{n}\left(x,\tau \right)$ and then use ${T}_{\tau }^{(n-1)\gamma }{{\rm{Res}}}_{u}^{n}\left(x,{\tau }_{0}\right)=0$ to obtain the $n$-term coefficient ${a}_{n}\left(x\right)$ by utilizing ${T}_{\tau }^{\gamma }{\tau }^{\beta }=0$ for all $\beta \gt \gamma$ at $\tau ={\tau }_{0}$ so that the $n$-term approximate solution ${u}_{n}\left(x,\tau \right)$ is provided.

Step F. Collect the resulting approximate solutions together in terms of fractional series expansion solution so that the $n$-term approximate solutions of (13) is obtained.

Remark 1. If the resulting pattern is regular in terms of infinite fractional series, then the closed-form solution $u(x,\tau )$ of (13) can be provided; otherwise, the resulting pattern is just the $n$-term MFPS solution. Furthermore, if more coefficients of conformable MFPS are computed, the error is reduced and thus a more accurate approximate solution is provided.

Theorem 3. Let $u\left(x,\tau \right),{u}_{n}(x,\tau )$ with $n\in {\mathbb{N}}$ are exact and approximate solutions of (13), recpectivly. If there exists a fixed constant $\rho \in \left[0,1\right]$ such that $\parallel {u}_{n+1}(x,\tau )\parallel \leqslant \rho \parallel {u}_{n}\left(x,\tau \right)\parallel$ for all $(x,\tau )\in {\rm{\Lambda }}$ with $\parallel {u}_{0}\left(x\right)\parallel \lt \infty$ for all $x\in {{\rm{\Omega }}}_{x},$ then ${u}_{n}(x,\tau )$ converges to $u(x,\tau )$ whenever $n\to \infty .$

Proof. For each $(x,\tau )\in {{\rm{\Lambda }}}_{x\tau },$ we have $\parallel {u}_{n+1}(x,\tau )\parallel \,\leqslant \rho \parallel {u}_{n}\left(x,\tau \right)\parallel .$ That is, $\parallel {u}_{1}(x,\tau )\parallel \leqslant \rho \parallel {u}_{0}(x,\tau )\parallel =\rho \parallel {u}_{0}\left(x\right)\parallel ,$ $\parallel {u}_{2}(x,\tau )\parallel \leqslant {\rho }^{2}\parallel {u}_{0}\left(x\right)\parallel ,$ and $\parallel {u}_{k}(x,\tau )\parallel \leqslant {\rho }^{k}\parallel {u}_{0}\left(x\right)\parallel .$ Thus

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \sum _{k=n+1}^{\infty }\parallel {u}_{k}(x,\tau )\parallel \leqslant \displaystyle \sum _{k=n+1}^{\infty }{\rho }^{k}\parallel {u}_{0}\left(x\right)\parallel \\ \,=\,\parallel {u}_{0}\left(x\right)\parallel \displaystyle \sum _{k=n+1}^{\infty }{\rho }^{k}.\end{array}\end{eqnarray} \tag{ 23 }$$

Therefore, it can be concluded that

$$\begin{eqnarray}\begin{array}{ccc}\parallel u\left(x,\tau \right)-{u}_{n}(x,\tau )\parallel & = & \parallel \displaystyle \sum _{k=n+1}^{\infty }{u}_{k}(x,\tau )\parallel \\ & \leqslant & \displaystyle \sum _{k=n+1}^{\infty }\parallel {u}_{k}\left(x,\tau \right)\parallel \\ & \leqslant & \parallel {u}_{0}\left(x\right)\parallel \displaystyle \sum _{k=n+1}^{\infty }{\rho }^{k}\\ & = & \displaystyle \frac{{\rho }_{\,}^{n+1}}{1-\rho }\parallel {u}_{0}\left(x\right)\parallel \\ & & \,\to 0,\end{array}\end{eqnarray} \tag{ 24 }$$

along as $n\to \infty .$ ■

Now, some numerical experiments with CFO derivatives are performed to verify the applicability and accuracy of the RPSM. More specifically, two generalized time-FKEE and two-coupled time-MTEs are solved under conformable sense by the proposed approach based on residual error concept. The solution behavior for all models is plotted in 3D-dimensional spaces for different fractional order derivatives. The programs and graphs are implemented by using the wolfram Mathematica package.

The RPSM can be powerfully applied to generate effective approximate solutions to handle complex nonlinear FPDEs without discretization, transformation, or restrictive assumptions. The procedure is performed using the proposed approach with rapid convergence and low computational cost compared to other current methods. Furthermore, the concept of the CFO derivative has many strong and gentle features, including simulation of most properties of classic calculus such as quotient, product, Leibniz, and chain rules while other definitions fail to satisfy such rules. It can also deal directly with nondifferentiable terms, unlike other fractional concepts.

4.1. One-dimensioname-FKl nonlinear tiEE

Let us consider the following time-FKEE:

$$\begin{eqnarray}&&i{T}_{\tau }^{\gamma }u+{u}_{xx}+2u{\left({\left|u\right|}^{2}\right)}_{x}+u{\left|u\right|}^{4}=0,\end{eqnarray} \tag{ 25 }$$

supplemented with the following ICC:

$$\begin{eqnarray}&&u\left(x,0\right)=\mu {e}^{ix},\end{eqnarray} \tag{ 26 }$$

where $\gamma \in \left(0,\,1\right],$ $\left(x,\tau \right)\in {\rm{\Omega }}:\,\left[a,b\right]\times \left[0,\infty \right)\to {\mathbb{R}},$ ${i}^{2}\,=-1,$ and $\mu ,a,b\in {\mathbb{R}}$ with $\mu \ne 0.$ $u=u\left(x,\tau \right)$ is an unknown complex-valued function to be determined in $\left(x,\tau \right)$ and ${T}_{\tau }^{\gamma }$ is the CTFD of order $\gamma .$

In view of mathematical physics, the FKEE is a class of nonlinear PDEs within quantum physics, which have been demonstrated in many fields of physics, including optical fiber, plasma physics, water waves propagation, and quantum mechanics. Anyhow, for $\gamma =1$ the exact solution of (25) and (26) can be given by [42] as follows:

$$\begin{eqnarray}&&u\left(x,\tau \right)={e}^{ix}{\left(1+\left({\mu }^{-4}-1\right){e}^{4i\tau }\right)}^{-1/4}.\end{eqnarray} \tag{ 27 }$$

Consequently, the CFO temporal evolution operator $i{T}_{\tau }^{\gamma }u$ of the nonlinear wave can be written as

$$\begin{eqnarray}&&{T}_{\tau }^{\gamma }u=i{u}_{xx}+2i\left(u{u}_{x}\bar{u}+{u}^{2}{\bar{u}}_{x}\right)+i{u}^{3}{\bar{u}}^{2},\end{eqnarray} \tag{ 28 }$$

where $\bar{u}$ is conjugate of $u.$

By separating the real and imaginary parts of the solution of (25) and (26) as $u\left(x,\tau \right)=v\left(x,\tau \right)+iw\left(x,\tau \right)$ so that $u\left(x,0\right)={v}_{0}\left(x\right)+i{w}_{0}\left(x\right)$ whereas $i$ is the imaginary unit, we obtain the following system:

$$\begin{eqnarray}&&\begin{array}{l}{T}_{\tau }^{\gamma }v+{w}_{xx}+2w{({v}^{2}+{w}^{2})}_{x}+w{({v}^{2}+{w}^{2})}^{2}=0,\\ {T}_{\tau }^{\gamma }w-{v}_{xx}-2v{({v}^{2}+{w}^{2})}_{x}-v{({v}^{2}+{w}^{2})}^{2}=0,\end{array}\end{eqnarray} \tag{ 29 }$$

supplemented with the following ICC:

$$\begin{eqnarray}&&\begin{array}{l}{v}_{0}\left(x\right)=\mu \,\cos \left(x\right),\\ {w}_{0}\left(x\right)=\mu \,\sin \left(x\right),\end{array}\end{eqnarray} \tag{ 30 }$$

Suppose the solutions $v(x,\tau )$ and $w(x,\tau )$ of (29) and (30) to be as the following form:

$$\begin{eqnarray}&&\begin{array}{l}v\left(x,\tau \right)={v}_{0}\left(x\right)+\displaystyle \sum _{k=1}^{\infty }{v}_{k}\left(x,\tau \right),\\ w\left(x,\tau \right)={w}_{0}\left(x\right)+\displaystyle \sum _{k=1}^{\infty }{w}_{k}\left(x,\tau \right).\end{array}\end{eqnarray} \tag{ 31 }$$

To perform the proposed RPSM approach, let the truncated residual error can be written as

$$\begin{eqnarray}&&\begin{array}{l}\begin{array}{l}{{\rm{Res}}}_{v}^{n}\left(x,\tau \right)={T}_{\tau }^{\gamma }{v}_{n}+{({w}_{n})}_{xx}+2w{({v}_{n}^{2}+{w}_{n}^{2})}_{x}\\ \,\,\,\,\,+\,w{({v}_{n}^{2}+{w}_{n}^{2})}^{2},\end{array}\\ \begin{array}{l}{{\rm{Res}}}_{w}^{n}\left(x,\tau \right)={T}_{\tau }^{\gamma }{w}_{n}-{\left({v}_{n}\right)}_{xx}-2v{({v}_{n}^{2}+{w}_{n}^{2})}_{x}\\ \,\,\,\,\,-\,v{({v}_{n}^{2}+{w}_{n}^{2})}^{2},\end{array}\end{array}\end{eqnarray} \tag{ 32 }$$

provided that the truncated conformable fractional series solutions are given as

$$\begin{eqnarray}&&\begin{array}{c}{v}_{n}\left(x,\tau \right)=\mu \,\cos \left(x\right)+\displaystyle \sum _{k=1}^{n}{a}_{k}\left(x\right)\displaystyle \frac{{\tau }^{k\gamma }}{{\gamma }^{k}k!},\\ {w}_{n}\left(x,\tau \right)=\mu \,\sin \left(x\right)+\displaystyle \sum _{k=1}^{n}{b}_{k}\left(x\right)\displaystyle \frac{{\tau }^{k\gamma }}{{\gamma }^{k}k!}.\end{array}\end{eqnarray} \tag{ 33 }$$

To complete the algorithm description clearly for finding the coefficients ${a}_{k},{b}_{k}$ for $k=1,2,\ldots ,\,n$ of the conformable MFPS (33), we consider the first approximate solutions of (32) as

$$\begin{eqnarray}&&\begin{array}{l}{v}_{1}\left(x,\tau \right)=\mu \,\cos \left(x\right)+\displaystyle \frac{{a}_{1}\left(x\right)}{\gamma }{\tau }^{\gamma },\\ {w}_{1}\left(x,\tau \right)=\mu \,\sin \left(x\right)+\displaystyle \frac{{b}_{1}\left(x\right)}{\gamma }{\tau }^{\gamma },\end{array}\end{eqnarray} \tag{ 34 }$$

so that ${a}_{1}\left(x\right)$ and ${b}_{1}\left(x\right)$ can be obtained by substituting ${v}_{1}\left(x,\tau \right)$ and ${w}_{1}\left(x,\tau \right)$ into ${{\rm{Res}}}_{v}^{1}\left(x,\tau \right)$ and ${{\rm{Res}}}_{w}^{1}\left(x,\tau \right)$ and then by equating the output to zero when $\tau$ = 0, we get the following:

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{Res}}}_{v}^{1}{\left(x,\tau \right)}_{\left|\tau =0\right.}={a}_{1}\left(x\right)+{\left(-\mu \sin \left(x\right)+\displaystyle \frac{{b}_{1}^{{\prime} ^{\prime} }\left(x\right){\tau }^{\gamma }}{\gamma }\right)}_{\left|\tau =0\right.}\\ \begin{array}{l}+\,\left(2\mu \,\sin \left(x\right)+\displaystyle \frac{2{b}_{1}\left(x\right){\tau }^{\gamma }}{\gamma }\right.\left({\left(\mu \cos \left(x\right)+\displaystyle \frac{{a}_{1}\left(x\right){\tau }^{\gamma }}{\gamma }\right)}^{2}\right.\\ {\left.{\left.+{\left(\mu \sin \left(x\right)+\displaystyle \frac{{b}_{1}\left(x\right){\tau }^{\gamma }}{\gamma }\right)}^{2}\right)}_{x}\right)}_{t=0}+\,\left(\mu \,\sin \left(x\right)+\displaystyle \frac{{b}_{1}\left(x\right){\tau }^{\gamma }}{\gamma }\right.\end{array}\\ \,\times \,{\left({\left(\mu \cos \left(x\right)+\displaystyle \frac{{a}_{1}\left(x\right){\tau }^{\gamma }}{\gamma }\right)}^{2}+{\left(\mu \sin \left(x\right)+\displaystyle \frac{{b}_{1}\left(x\right){\tau }^{\gamma }}{\gamma }\right)}^{2}\right)}_{\left|\tau =0\right.}^{2}\\ =\,0,\end{array}\end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{Res}}}_{w}^{1}{\left(x,\tau \right)}_{\left|\tau =0\right.}={b}_{1}\left(x\right)-{\left(-\mu \cos \left(x\right)+\displaystyle \frac{{a}_{1}^{{\prime\prime} }\left(x\right){\tau }^{\gamma }}{\gamma }\right)}_{\left|\tau =0\right.}\\ \begin{array}{l}\,-\,\left(2\mu \,\cos \left(x\right)+\displaystyle \frac{2{a}_{1}\left(x\right){\tau }^{\gamma }}{\gamma }\right.\left({\left(\mu \cos \left(x\right)+\displaystyle \frac{{a}_{1}\left(x\right){\tau }^{\gamma }}{\gamma }\right)}^{2}\right.\\ {\left.{\left.+{\left(\mu \sin \left(x\right)+\displaystyle \frac{{b}_{1}\left(x\right){\tau }^{\gamma }}{\gamma }\right)}^{2}\right)}_{x}\right)}_{t=0}-\left(\mu \,\cos \left(x\right)+\displaystyle \frac{{a}_{1}\left(x\right){\tau }^{\gamma }}{\gamma }\right.\end{array}\\ \,\times \,{\left({\left(\mu \cos \left(x\right)+\displaystyle \frac{{a}_{1}\left(x\right){\tau }^{\gamma }}{\gamma }\right)}^{2}+{\left(\mu \sin \left(x\right)+\displaystyle \frac{{b}_{1}\left(x\right){\tau }^{\gamma }}{\gamma }\right)}^{2}\right)}_{\left|\tau =0\right.}^{2}\\ \,=\,0.\end{array}\end{eqnarray} \tag{ 36 }$$

Hence, the values of ${a}_{1}\left(x\right)$ and ${b}_{1}\left(x\right)$ can be given respectively as

$$\begin{eqnarray}&&\begin{array}{l}{a}_{1}\left(x\right)=-\mu \left({\mu }^{4}-1\right)\sin \left(x\right),\\ {b}_{1}\left(x\right)=\mu \left({\mu }^{4}-1\right)\cos \left(x\right).\end{array}\end{eqnarray} \tag{ 37 }$$

That is, the functions ${v}_{1}\left(x,\tau \right)$ and ${w}_{1}\left(x,\tau \right)$ should be

$$\begin{eqnarray}&&\begin{array}{l}{v}_{1}\left(x,\tau \right)=\mu \,\cos \left(x\right)-\mu \left({\mu }^{4}-1\right)\sin \left(x\right)\displaystyle \frac{{\tau }^{\gamma }}{\gamma },\\ {w}_{1}\left(x,\tau \right)=\mu \,\sin \left(x\right)+\mu \left({\mu }^{4}-1\right)\cos \left(x\right)\displaystyle \frac{{\tau }^{\gamma }}{\gamma },\end{array}\end{eqnarray} \tag{ 38 }$$

which leads to the first approximate solution of (25) and (26) to be as follows:

$$\begin{eqnarray}\begin{array}{rcl}{u}_{1}\left(x,\tau \right) & = & {u}_{0}\left(x,\tau \right)+\mu \left({\mu }^{4}-1\right)\left(i\,\cos \left(x\right)-\,\sin \left(x\right)\right)\displaystyle \frac{{\tau }^{\gamma }}{\gamma }\\ & = & {e}^{ix}\left(\mu +i\mu \left({\mu }^{4}-1\right)\displaystyle \frac{{\tau }^{\gamma }}{\gamma }\right).\end{array}\end{eqnarray} \tag{ 39 }$$

To proceeds more in the solution constructions, let

$$\begin{eqnarray}&&\begin{array}{l}{v}_{2}\left(x,\tau \right)=\mu \,\cos \left(x\right)-\left({\mu }^{5}-\mu \right)\sin \left(x\right)\displaystyle \frac{{\tau }^{\gamma }}{\gamma }+{a}_{2}\left(x\right)\displaystyle \frac{{\tau }^{2\gamma }}{2{\gamma }^{2}},\\ {w}_{2}\left(x,\tau \right)=\mu \,\sin \left(x\right)+\left({\mu }^{5}-\mu \right)\cos \left(x\right)\displaystyle \frac{{\tau }^{\gamma }}{\gamma }+{b}_{2}\left(x\right)\displaystyle \frac{{\tau }^{2\gamma }}{2{\gamma }^{2}}.\end{array}\end{eqnarray} \tag{ 40 }$$

Thus, the values of ${a}_{2}\left(x\right)$ and ${b}_{2}\left(x\right)$ can be computed by substituting ${v}_{2}\left(x,\tau \right)$ and ${w}_{2}\left(x,\tau \right)$ into ${T}_{\tau }^{\gamma }{{\rm{Res}}}_{v}^{2}\left(x,\tau \right)$ and ${T}_{\tau }^{\gamma }{{\rm{Res}}}_{w}^{2}\left(x,\tau \right)$ and equating the output to zero when $\tau$ = 0 so that

$$\begin{eqnarray}&&\begin{array}{l}{a}_{2}\left(x\right)=-\mu \left({\mu }^{4}-1\right)\left(1-{\mu }^{4}\right)\cos \left(x\right),\\ {b}_{2}\left(x\right)=-\mu \left({\mu }^{4}-1\right)\left(1-{\mu }^{4}\right)\sin \left(x\right).\end{array}\end{eqnarray} \tag{ 41 }$$

Subsequently, the second approximate solution of (25) and (26) can be given by

$$\begin{eqnarray}&&\begin{array}{l}{u}_{2}\left(x,\tau \right)={u}_{1}\left(x,\tau \right)-\mu \left({\mu }^{4}-1\right)\left(1-{\mu }^{4}\right)\left(\cos \left(x\right)+i\,\sin \left(x\right)\right)\displaystyle \frac{{\tau }^{2\gamma }}{2{\gamma }^{2}}\\ \,\,=\,{e}^{ix}\left(\mu +i\mu \left({\mu }^{4}-1\right)\displaystyle \frac{{\tau }^{\gamma }}{\gamma }-\mu \left({\mu }^{4}-1\right)\left(1-{\mu }^{4}\right)\displaystyle \frac{{\tau }^{2\gamma }}{2{\gamma }^{2}}\right).\end{array}\end{eqnarray} \tag{ 42 }$$

By following the former approach, the values ${a}_{3}\left(x\right)$ and ${b}_{3}\left(x\right)$ can be obtained by substituting

$$\begin{eqnarray}&&\begin{array}{l}{v}_{3}\left(x,\tau \right)={v}_{2}\left(x,\tau \right)+{a}_{3}\left(x\right)\displaystyle \frac{{\tau }^{3\gamma }}{3!{\gamma }^{3}},\\ {w}_{3}\left(x,\tau \right)={w}_{2}\left(x,\tau \right)+{b}_{3}\left(x\right)\displaystyle \frac{{\tau }^{3\gamma }}{3!{\gamma }^{3}},\end{array}\end{eqnarray} \tag{ 43 }$$

into ${T}_{\tau }^{2\gamma }{{\rm{Res}}}_{v}^{3}\left(x,\tau \right)$ and ${T}_{\tau }^{2\gamma }{{\rm{Res}}}_{w}^{3}\left(x,\tau \right)$ and equating the output to zero when $\tau$ = 0. Therefore, by collecting the selected solution components, the approximate solution $u\left(x,\tau \right)$ of (25) and (26) can be given as a series form as

$$\begin{eqnarray}&&\begin{array}{l}u\left(x,\tau \right)=\mu {e}^{ix}+i\mu \left({\mu }^{4}-1\right){e}^{ix}\displaystyle \frac{{\tau }^{\gamma }}{\gamma }\\ \,\,\,\,-\,\mu \left({\mu }^{4}-1\right)\left(1-{\mu }^{4}\right){e}^{ix}\displaystyle \frac{{\tau }^{2\gamma }}{2{\gamma }^{2}}\,+\,\ldots ,\end{array}\end{eqnarray} \tag{ 44 }$$

which is fully consistent with the approximate solution obtained by LADM [35] and qHATM [41].

For numerical outcomes, table 1 describes the numeric approximate wave solutions of (25) and (26) for different fractional values of $\gamma$ such that $\gamma =0.5$ and $\gamma =0.2$ at $x=0.5$ when $\mu =\sqrt[16]{2}$ with temporal step size $0.16.$ It is worth noting here that the approximate solutions are compatible with each other in terms of fractional values, as well as, the closer of fractional values to the classical case $\gamma =1,$ the more accurate solutions will be. To illustrate the ability and applicably of the suggested algorithm, the exact and approximate solutions of (29) and (30) are portrayed in figure 1 on $-6\leqslant x\leqslant 6$ and $0\leqslant \tau \leqslant 2,$ which shows how the approximate solution matches the exact solution. Further, the 3D-space graphs of real and imaginary parts of periodic approximate solutions are, respectively, plotted in figures 2 and 3 on $-20\leqslant x\leqslant 20$ and $0\leqslant \tau \leqslant 5$ with different values of fractional case $\gamma$ such that $\gamma \in \left\{0.95\right.,$ $0.75,$ $0.5,$ $\left.0.25\right\}.$

Table 1.  Numerical results of the solutions of (25) and (26) for different $\gamma .$

${\tau }_{i}$	${\rm{Re}}[u\left(x,\tau \right)]$		${\rm{Im}}[u\left(x,\tau \right)]$
	$\gamma =0.5$	$\gamma =0.2$	$\gamma =0.5$	$\gamma =0.2$
$0.16$	$0.830466$	$0.633115$	$0.794829$	$0.677313$
$0.32$	$0.789165$	$0.683905$	$0.725114$	$0.751474$
$0.48$	$0.755362$	$0.721066$	$0.675861$	$0.796060$
$0.64$	$0.725504$	$0.751101$	$0.636570$	$0.827813$
$0.80$	$0.698208$	$0.776540$	$0.603389$	$0.852299$
$0.96$	$0.672762$	$0.798686$	$0.574413$	$0.872087$

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The absolute errors of exact and periodic approximate solutions in the classical case of (29) and (30) are portrayed in figure 4 on $-100\leqslant x\leqslant 100$ and $0\leqslant \tau \leqslant 0.01,$ which portrays excellent approximate solutions. On the other hand, the solution behaviors for exact and periodic wave approximate solutions with different fractional parameters $\gamma$ are presented in the 2D-spaces of real and imaginary parts for (29) and (30) in figure 5 on $-50\leqslant x\leqslant 50$ and $\tau =1.$

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4.2. One-dimensional nonlinear coupled time-FMTEs

Let us consider the following two-coupled time-FMTEs:

$$\begin{eqnarray}&&\begin{array}{l}i\left({T}_{\tau }^{\gamma }u+{u}_{x}\right)+v+u{\left|v\right|}^{2}=0,\\ i\left({T}_{\tau }^{\gamma }v-{v}_{x}\right)+u+v{\left|u\right|}^{2}=0,\end{array}\end{eqnarray} \tag{ 45 }$$

supplemented with the following ICC:

$$\begin{eqnarray}&&\begin{array}{l}u\left(x,0\right)={\mu }_{0}{e}^{ix},\\ v\left(x,0\right)={\xi }_{0}{e}^{ix},\end{array}\end{eqnarray} \tag{ 46 }$$

where $\gamma \in \left(0,\,1\right],$ $\left(x,\tau \right)\in {\rm{\Omega }}:\,\left[a,b\right]\times \left[0,\infty \right)\to {\mathbb{R}},$ ${i}^{2}\,=-1,$ and $\mu ,a,b\in {\mathbb{R}}$ with $0\ne {\mu }_{0},{\xi }_{0}\ne 0.$ $u=u\left(x,\tau \right)$ and $v=v\left(x,\tau \right)$ are unknowns complex-valued functions to be determined in $\left(x,\tau \right)$ and ${T}_{\tau }^{\gamma }$ is the CTFD of order $\gamma .$

In view of mathematical physics, the coupled MTE is a class of nonlinear PDEs within quantum physics, which have been demonstrated in many fields of physics, including ultrashort pulse propagation, weakly dispersive water waves, and quadratic electro-optic effect.

For equivalent simplicity, the coupled time-FMTEs (45) and (46) can be converted into the following system:

$$\begin{eqnarray}&&\begin{array}{l}{T}_{\tau }^{\gamma }u+{u}_{x}-iv-iu{\left|v\right|}^{2}=0,\\ {T}_{\tau }^{\gamma }v-{v}_{x}-iu-iv{\left|u\right|}^{2}=0.\end{array}\end{eqnarray} \tag{ 47 }$$

To perform the RPSM, let the solutions of (47) and (46) can be represented as follows

$$\begin{eqnarray}&&\begin{array}{l}u\left(x,\tau \right)={u}_{0}\left(x,\tau \right)+\displaystyle \sum _{k=1}^{\infty }{u}_{k}\left(x,\tau \right),\\ v\left(x,\tau \right)={v}_{0}\left(x,\tau \right)+\displaystyle \sum _{k=1}^{\infty }{v}_{k}\left(x,\tau \right),\end{array}\end{eqnarray} \tag{ 48 }$$

where ${u}_{0}\left(x,\tau \right)={\mu }_{0}{e}^{ix}$ and ${v}_{0}\left(x,\tau \right)={\xi }_{0}{e}^{ix}$ provided with the following $n$-term truncated residual functions:

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{Res}}}_{u}^{n}\left(x,\tau \right)={T}_{\tau }^{\gamma }{u}_{n}+{({u}_{n})}_{x}-i{v}_{n}-i{u}_{n}{\left|{v}_{n}\right|}^{2},\\ {{\rm{Res}}}_{v}^{n}\left(x,\tau \right)={T}_{\tau }^{\gamma }{u}_{n}-{({v}_{n})}_{x}-i{u}_{n}-i{v}_{n}{\left|{u}_{n}\right|}^{2}.\end{array}\end{eqnarray} \tag{ 49 }$$

Consequently, substitute the approximate solutions

$$\begin{eqnarray}&&\begin{array}{l}{u}_{n}\left(x,\tau \right)={\mu }_{0}{e}^{ix}+\displaystyle \sum _{k=1}^{n}{a}_{k}\left(x\right)\displaystyle \frac{{\tau }^{k\gamma }}{{\gamma }^{k}k!},\\ {v}_{n}\left(x,\tau \right)={\xi }_{0}{e}^{ix}+\displaystyle \sum _{k=1}^{n}{b}_{k}\left(x\right)\displaystyle \frac{{\tau }^{k\gamma }}{{\gamma }^{k}k!},\end{array}\end{eqnarray} \tag{ 50 }$$

for $n=1,2,\ldots$ up to arbitrary $m$ into the $n$-term truncated residual functions (49) to get the required components for expansion (48) as in the following procedure: for first approximations ${u}_{1}\left(x,\tau \right)={\mu }_{0}{e}^{ix}+{a}_{1}\left(x\right)\displaystyle \frac{{\tau }^{\gamma }}{\gamma }$ and ${v}_{1}\left(x,\tau \right)\,={\xi }_{0}{e}^{ix}+{b}_{1}\left(x\right)\displaystyle \frac{{\tau }^{\gamma }}{\gamma },$ we have

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{Res}}}_{u}^{1}\left(x,\tau \right)={a}_{1}\left(x\right)+{\left({\mu }_{0}{e}^{ix}+\displaystyle \frac{{a}_{1}\left(x\right){\tau }^{\gamma }}{\gamma }\right)}_{x},\\ \,\,\,\,\,-\,i\left({\xi }_{0}{e}^{ix}+\displaystyle \frac{{b}_{1}\left(x\right){\tau }^{\gamma }}{\gamma }\right)\\ \,\,\,\,\,-\,i\left({\mu }_{0}{e}^{ix}+\displaystyle \frac{{a}_{1}\left(x\right){\tau }^{\gamma }}{\alpha }\right)\\ \,\,\,\,\,\times \,{\left|{\xi }_{0}{e}^{ix}+\displaystyle \frac{{b}_{1}\left(x\right){\tau }^{\gamma }}{\gamma }\right|}^{2},\end{array}\end{eqnarray} \tag{ 51 }$$

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{Res}}}_{v}^{1}\left(x,\tau \right)={b}_{1}\left(x\right)-{\left({\xi }_{0}{e}^{ix}+\displaystyle \frac{{b}_{1}\left(x\right){\tau }^{\gamma }}{\gamma }\right)}_{x}\\ \,\,\,\,\,-\,i\left({\mu }_{0}{e}^{ix}+\displaystyle \frac{{a}_{1}\left(x\right){\tau }^{\gamma }}{\gamma }\right)\\ \,\,\,\,\,-\,i\left({\xi }_{0}{e}^{ix}+\displaystyle \frac{{b}_{1}\left(x\right){\tau }^{\gamma }}{\gamma }\right)\\ \,\,\,\,\,\times \,{\left|{\mu }_{0}{e}^{ix}+\displaystyle \frac{{a}_{1}\left(x\right){\tau }^{\gamma }}{\gamma }\right|}^{2}.\end{array}\end{eqnarray} \tag{ 52 }$$

By using ${{\rm{Res}}}_{u}^{1}{\left(x,\tau \right)}_{\left|\tau =0\right.}=0$ and ${{\rm{Res}}}_{v}^{1}{\left(x,\tau \right)}_{\left|\tau =0\right.}=0$ the values of those unknown coefficients ${a}_{1}\left(x\right)$ and ${b}_{1}\left(x\right)$ are given by

$$\begin{eqnarray}&&\begin{array}{l}{a}_{1}\left(x\right)={\rm{i}}\left({\xi }_{0}-{\mu }_{0}+{\mu }_{0}^{3}\right){{\rm{e}}}^{{\rm{i}}x},\\ {b}_{1}\left(x\right)={\rm{i}}\left({\mu }_{0}+{\xi }_{0}+{\xi }_{0}^{3}\right){{\rm{e}}}^{{\rm{i}}x}.\end{array}\end{eqnarray} \tag{ 53 }$$

Thus, the first approximate solutions of (47) and (46) can be provided by

$$\begin{eqnarray}&&\begin{array}{l}{u}_{1}\left(x,\tau \right)={\mu }_{0}{e}^{ix}+{\rm{i}}({\xi }_{0}-{\mu }_{0}+{\mu }_{0}^{3}){{\rm{e}}}^{{\rm{i}}x}\displaystyle \frac{{\tau }^{\gamma }}{\gamma },\\ {v}_{1}\left(x,\tau \right)={\xi }_{0}{e}^{ix}+{\rm{i}}\left({\mu }_{0}+{\xi }_{0}+{\xi }_{0}^{3}\right){{\rm{e}}}^{{\rm{i}}x}\displaystyle \frac{{\tau }^{\gamma }}{\gamma }.\end{array}\end{eqnarray} \tag{ 54 }$$

For second approximations ${u}_{2}\left(x,\tau \right)={u}_{1}\left(x,\tau \right)+{a}_{2}\left(x\right)\displaystyle \frac{{\tau }^{2\gamma }}{2{\gamma }^{2}}$ and ${v}_{2}\left(x,\tau \right)={v}_{1}\left(x,\tau \right)+{b}_{2}\left(x\right)\displaystyle \frac{{\tau }^{2\gamma }}{2{\gamma }^{2}},$ we have

$$\begin{eqnarray}\begin{array}{ccl}{{\rm{Res}}}_{u}^{2}\left(x,\tau \right) & = & \left({\rm{i}}({\xi }_{0}-{\mu }_{0}+{\mu }_{0}^{3}){{\rm{e}}}^{ix}+\displaystyle \frac{{a}_{2}\left(x\right){\tau }^{\gamma }}{\gamma }\right)\\ & & +\,\left(i{\mu }_{0}{e}^{ix}-\left({\xi }_{0}-{\mu }_{0}+{\mu }_{0}^{3}\right){{\rm{e}}}^{ix}\displaystyle \frac{{\tau }^{\gamma }}{\gamma }+\displaystyle \frac{{a}_{2}^{{\prime} }\left(x\right){\tau }^{2\gamma }}{2{\gamma }^{2}}\right)\\ & & -\,\left(i{\xi }_{0}{e}^{ix}-\left({\mu }_{0}+{\xi }_{0}+{\xi }_{0}^{3}\right){{\rm{e}}}^{ix}\displaystyle \frac{{\tau }^{\gamma }}{\gamma }+\displaystyle \frac{i{b}_{2}\left(x\right){\tau }^{2\gamma }}{2{\gamma }^{2}}\right)\\ & & -\,\left(i{\mu }_{0}{e}^{ix}-\left({\xi }_{0}-{\mu }_{0}+{\mu }_{0}^{3}\right){{\rm{e}}}^{ix}\displaystyle \frac{{\tau }^{\gamma }}{\gamma }+\displaystyle \frac{i{a}_{2}\left(x\right){\tau }^{2\gamma }}{2{\gamma }^{2}}\right)\\ & & \times \,{\left|{\xi }_{0}{e}^{ix}+{\rm{i}}\left({\mu }_{0}+{\xi }_{0}+{\xi }_{0}^{3}\right){{\rm{e}}}^{ix}\displaystyle \frac{{\tau }^{\gamma }}{\gamma }+\displaystyle \frac{{b}_{2}\left(x\right){\tau }^{2\gamma }}{2{\gamma }^{2}}\right|}^{2},\end{array}\end{eqnarray} \tag{ 55 }$$

$$\begin{eqnarray}\begin{array}{ccl}{{\rm{Res}}}_{v}^{2}\left(x,\tau \right) & = & \left({\rm{i}}({\mu }_{0}+{\xi }_{0}+{\xi }_{0}^{3}){{\rm{e}}}^{{\rm{i}}x}+\displaystyle \frac{{b}_{2}\left(x\right){\tau }^{\gamma }}{\gamma }\right)\\ & & -\,\left(i{\xi }_{0}{e}^{ix}-\left({\mu }_{0}+{\xi }_{0}+{\xi }_{0}^{3}\right){{\rm{e}}}^{{\rm{i}}x}\displaystyle \frac{{\tau }^{\gamma }}{\gamma }+\displaystyle \frac{{b}_{2}^{{\prime} }\left(x\right){\tau }^{2\gamma }}{2{\gamma }^{2}}\right)\\ & & -\,\left(i{\mu }_{0}{e}^{ix}-\left({\xi }_{0}-{\mu }_{0}+{\mu }_{0}^{3}\right){{\rm{e}}}^{{\rm{i}}x}\displaystyle \frac{{\tau }^{\gamma }}{\gamma }+\displaystyle \frac{i{a}_{2}\left(x\right){\tau }^{2\gamma }}{2{\gamma }^{2}}\right)\\ & & -\,\left(i{\xi }_{0}{e}^{ix}-\left({\mu }_{0}+{\xi }_{0}+{\xi }_{0}^{3}\right){{\rm{e}}}^{{\rm{i}}x}\displaystyle \frac{{\tau }^{\gamma }}{\gamma }+\displaystyle \frac{i{b}_{2}\left(x\right){\tau }^{2\gamma }}{2{\gamma }^{2}}\right)\\ & & \times \,{\left|{\mu }_{0}{e}^{ix}+i\left({\xi }_{0}-{\mu }_{0}+{\mu }_{0}^{3}\right){{\rm{e}}}^{{\rm{i}}x}\displaystyle \frac{{\tau }^{\gamma }}{\gamma }+\displaystyle \frac{{a}_{2}\left(x\right){\tau }^{2\gamma }}{2{\gamma }^{2}}\right|}^{2}.\end{array}\end{eqnarray} \tag{ 56 }$$

By applying ${T}_{\tau }^{\gamma }$ on both sides of (56) and utilizing ${T}_{\tau }^{\gamma }{{\rm{Res}}}_{v}^{2}\left(x,0\right)=0$ and ${T}_{\tau }^{\gamma }{{\rm{Res}}}_{w}^{2}\left(x,0\right)=0,$ the values of the unknown coefficients ${a}_{2}\left(x\right)$ and ${b}_{2}\left(x\right)$ are given by

$$\begin{eqnarray}&&\begin{array}{l}\begin{array}{l}{a}_{2}\left(x\right)=-\left({\xi }_{0}^{3}+2{\mu }_{0}+\left(1+2{\rm{i}}\right){\xi }_{0}{\mu }_{0}^{2}-\left(2+2{\rm{i}}\right){\mu }_{0}^{3}\right.\\ \left.\,\,\,\,+\,\left(1+2{\rm{i}}\right){\mu }_{0}^{5}\right){{\rm{e}}}^{{\rm{i}}x},\end{array}\\ \begin{array}{l}{b}_{2}\left(x\right)=-\left({\mu }_{0}^{3}+2{\xi }_{0}+\left(1+2{\rm{i}}\right){\mu }_{0}{\xi }_{0}^{2}+\left(2-2{\rm{i}}\right){\xi }_{0}^{3}\right.\\ \left.\,\,\,\,+\,\left(1+2{\rm{i}}\right){\xi }_{0}^{5}\right){{\rm{e}}}^{{\rm{i}}x}.\end{array}\end{array}\end{eqnarray} \tag{ 57 }$$

Thus, the second approximate solutions of (47) and (46) can be provided by

$$\begin{eqnarray}&&\begin{array}{c}{u}_{2}\left(x,\tau \right)={e}^{ix}\left({\mu }_{0}+{\rm{i}}\displaystyle \frac{{C}_{1}^{{\mu }_{0}{\xi }_{0}}{\tau }^{\gamma }}{\gamma }-\displaystyle \frac{{C}_{3}^{{\mu }_{0}{\xi }_{0}}{\tau }^{2\gamma }}{2{\gamma }^{2}}\right),\\ {v}_{2}\left(x,\tau \right)={e}^{ix}\left({\xi }_{0}+{\rm{i}}\displaystyle \frac{{C}_{2}^{{\mu }_{0}{\xi }_{0}}{\tau }^{\gamma }}{\gamma }-\displaystyle \frac{{C}_{4}^{{\mu }_{0}{\xi }_{0}}{\tau }^{2\gamma }}{2{\gamma }^{2}}\right),\end{array}\end{eqnarray} \tag{ 58 }$$

where the parameters constants ${C}_{i},$ $i=1,2,3,4$ are given as

$$\begin{eqnarray}&&\begin{array}{l}{C}_{1}^{{\mu }_{0}{\xi }_{0}}={\xi }_{0}-{\mu }_{0}+{\mu }_{0}^{3},\\ {C}_{2}^{{\mu }_{0}{\xi }_{0}}={\mu }_{0}+{\xi }_{0}+{\xi }_{0}^{3},\\ \begin{array}{l}{C}_{3}^{{\mu }_{0}{\xi }_{0}}={\xi }_{0}^{3}+2{\mu }_{0}+\left(1+2{\rm{i}}\right){\xi }_{0}{\mu }_{0}^{2}-\left(2+2{\rm{i}}\right){\mu }_{0}^{3}\\ \,\,\,+\,\left(1+2{\rm{i}}\right){\mu }_{0}^{5},\end{array}\\ \begin{array}{l}{C}_{4}^{{\mu }_{0}{\xi }_{0}}={\mu }_{0}^{3}+2{\xi }_{0}+\left(1+2{\rm{i}}\right){\mu }_{0}{\xi }_{0}^{2}+\left(2-2{\rm{i}}\right){\xi }_{0}^{3}\\ \,\,\,+\,\left(1+2{\rm{i}}\right){\xi }_{0}^{5}.\end{array}\end{array}\end{eqnarray} \tag{ 59 }$$

For third unknown coefficients ${a}_{3}\left(x\right)$ and ${b}_{3}\left(x\right),$ substitute the approximate solutions

$$\begin{eqnarray}&&\begin{array}{l}{u}_{3}\left(x,\tau \right)={u}_{2}\left(x,\tau \right)+{a}_{3}\left(x\right)\displaystyle \frac{{\tau }^{3\gamma }}{3!{\gamma }^{3}},\\ {v}_{3}\left(x,\tau \right)={v}_{2}\left(x,\tau \right)+{b}_{3}\left(x\right)\displaystyle \frac{{\tau }^{3\gamma }}{3!{\gamma }^{3}},\end{array}\end{eqnarray} \tag{ 60 }$$

into ${T}_{\tau }^{2\gamma }{{\rm{Res}}}_{v}^{3}\left(x,\tau \right)$ and ${T}_{\tau }^{2\gamma }{{\rm{Res}}}_{w}^{3}\left(x,\tau \right)$ and equating the output to zero when $\tau =0,$ so that

$$\begin{eqnarray}&&\begin{array}{l}{a}_{3}\left(x\right)=-\,i\left(\left(1+2i\right){\xi }_{0}^{5}-\left(1-2i\right){\mu }_{0}{\xi }_{0}^{2}+\left(\left(1-2{\rm{i}}\right)+3{\mu }_{0}^{2}\right)\right.{\xi }_{0}^{3}\\ \,\,\,\,\,+\,{\xi }_{0}(2+(3-2{\rm{i}}){\mu }_{0}^{2}-(1-2{\rm{i}}){\mu }_{0}^{4})\\ \,\,\,\,\,\left.+\,{\mu }_{0}\left(-2+(7+2{\rm{i}}){\mu }_{0}^{2}-(3+4{\rm{i}}){\mu }_{0}^{4}+(1+2{\rm{i}}){\mu }_{0}^{6}\right)\right){{\rm{e}}}^{{\rm{i}}x},\\ {b}_{3}\left(x\right)=-\,i\left((7-2{\rm{i}}){\xi }_{0}^{3}+3{\xi }_{0}^{5}+(1+2{\rm{i}}){\xi }_{0}^{7}+2{\mu }_{0}\right.\\ \,\,\,\,\,-\,(1-2{\rm{i}}){\mu }_{0}{\xi }_{0}^{4}+(2-(1-2{\rm{i}}){\mu }_{0}^{2}){\xi }_{0}\\ \,\,\,\,\,\left.\,+\,(1+2{\rm{i}})({\mu }_{0}^{2}-1){\mu }_{0}^{3}+((-3+2{\rm{i}})+3{\mu }_{0}^{2}){\mu }_{0}{\xi }_{0}^{2}\right){{\rm{e}}}^{{\rm{i}}x}.\end{array}\end{eqnarray} \tag{ 61 }$$

Subsequently, the third approximate solutions of (47) and (46) can be provided by

$$\begin{eqnarray}&&\begin{array}{l}{u}_{3}\left(x,\tau \right)={e}^{ix}\left({\mu }_{0}+{\rm{i}}\displaystyle \frac{{C}_{1}^{{\mu }_{0}{\xi }_{0}}{\tau }^{\gamma }}{\gamma }-\displaystyle \frac{{C}_{3}^{{\mu }_{0}{\xi }_{0}}{\tau }^{2\gamma }}{2{\gamma }^{2}}-i\displaystyle \frac{{C}_{5}^{{\mu }_{0}{\xi }_{0}}{\tau }^{3\gamma }}{3!{\gamma }^{3}}\right),\\ {v}_{3}\left(x,\tau \right)={e}^{ix}\left({\xi }_{0}+{\rm{i}}\displaystyle \frac{{C}_{2}^{{\mu }_{0}{\xi }_{0}}{\tau }^{\gamma }}{\gamma }-\displaystyle \frac{{C}_{4}^{{\mu }_{0}{\xi }_{0}}{\tau }^{2\gamma }}{2{\gamma }^{2}}-i\displaystyle \frac{{C}_{6}^{{\mu }_{0}{\xi }_{0}}{\tau }^{3\gamma }}{3!{\gamma }^{3}}\right),\end{array}\end{eqnarray} \tag{ 62 }$$

where the parameters constants ${C}_{i},$ $i=5,6$ are given as

$$\begin{eqnarray}&&\begin{array}{l}{C}_{5}^{{\mu }_{0}{\xi }_{0}}=\left(1+2i\right){\xi }_{0}^{5}-\left(1-2i\right){\mu }_{0}{\xi }_{0}^{2}+\left(\left(1-2{\rm{i}}\right)+3{\mu }_{0}^{2}\right)\,{\xi }_{0}^{3}\\ \,+\,{\xi }_{0}\left(2+\left(3-2{\rm{i}}\right){\mu }_{0}^{2}-\left(1-2{\rm{i}}\right){\mu }_{0}^{4}\right)\\ \,+\,{\mu }_{0}\left(-2+\left(7+2{\rm{i}}\right){\mu }_{0}^{2}-\left(3+4{\rm{i}}\right){\mu }_{0}^{4}+\left(1+2{\rm{i}}\right){\mu }_{0}^{6}\right),\end{array}\end{eqnarray} \tag{ 63 }$$

$$\begin{eqnarray}\begin{array}{ccl}{C}_{6}^{{\mu }_{0}{\xi }_{0}} & = & (7-2{\rm{i}}){\xi }_{0}^{3}+3{\xi }_{0}^{5}+(1+2{\rm{i}}){\xi }_{0}^{7}+2{\mu }_{0}-\,(1-2{\rm{i}}){\mu }_{0}{\xi }_{0}^{4}\\ & & +\,\left(2-\left(1-2{\rm{i}}\right){\mu }_{0}^{2}\right){\xi }_{0}+\left(1+2{\rm{i}}\right)\left({\mu }_{0}^{2}-1\right){\mu }_{0}^{3}\\ & & +\,\left(\left(-3+2{\rm{i}}\right)+3{\mu }_{0}^{2}\right){\mu }_{0}{\xi }_{0}^{2}\,.\end{array}\end{eqnarray} \tag{ 64 }$$

By continuing with a similar procedure, and utilizing the facts ${T}_{\tau }^{(n-1)\gamma }{{\rm{Res}}}_{u,v}^{n}\left(x,0\right)=0$ for $n\geqslant 4,$ the approximate solutions $u\left(x,\tau \right)$ and $v\left(x,\tau \right)$ of (47) and (46) can be presented as a series form as follows:

$$\begin{eqnarray}&&\begin{array}{l}u\left(x,\tau \right)={e}^{ix}\left({\mu }_{0}+{\rm{i}}\displaystyle \frac{{C}_{1}^{{\mu }_{0}{\xi }_{0}}{\tau }^{\gamma }}{\gamma }+{i}^{2}\displaystyle \frac{{C}_{3}^{{\mu }_{0}{\xi }_{0}}{\tau }^{2\gamma }}{2{\gamma }^{2}}+{i}^{3}\displaystyle \frac{{C}_{5}^{{\mu }_{0}{\xi }_{0}}{\tau }^{3\gamma }}{3!{\gamma }^{3}}+{i}^{4}\displaystyle \frac{{C}_{7}^{{\mu }_{0}{\xi }_{0}}{\tau }^{4\gamma }}{4!{\gamma }^{4}}+\ldots \right),\\ v\left(x,\tau \right)={e}^{ix}\left({\xi }_{0}+{\rm{i}}\displaystyle \frac{{C}_{2}^{{\mu }_{0}{\xi }_{0}}{\tau }^{\gamma }}{\gamma }+{i}^{2}\displaystyle \frac{{C}_{4}^{{\mu }_{0}{\xi }_{0}}{\tau }^{2\gamma }}{2{\gamma }^{2}}+{i}^{3}\displaystyle \frac{{C}_{6}^{{\mu }_{0}{\xi }_{0}}{\tau }^{3\gamma }}{3!{\gamma }^{3}}+{i}^{4}\displaystyle \frac{{C}_{8}^{{\mu }_{0}{\xi }_{0}}{\tau }^{4\gamma }}{4!{\gamma }^{4}}+\ldots \right).\end{array}\end{eqnarray} \tag{ 65 }$$

In the classical case ($\gamma \to 1$), the approximate solutions of (47) and (46) can be written as

$$\begin{eqnarray}&&\begin{array}{l}u\left(x,\tau \right)={e}^{ix}\left({\mu }_{0}+i\left({\xi }_{0}-{\mu }_{0}+{\mu }_{0}^{3}\right)\tau +\displaystyle \frac{{i}^{2}}{2}\left({\xi }_{0}^{3}\right.\right.\\ \,\,\,\,\,\,\,+\,2{\mu }_{0}+\left(1+2{\rm{i}}\right){\xi }_{0}{\mu }_{0}^{2}-\left(2+2{\rm{i}}\right){\mu }_{0}^{3}\\ \left.\left.\,\,\,\,\,\,\,+\,\left(1+2{\rm{i}}\right){\mu }_{0}^{5}\right){\tau }^{2}+\ldots \Space{0ex}{3.0ex}{0ex}\right),\\ v\left(x,\tau \right)={e}^{ix}\left({\xi }_{0}+{\rm{i}}({\mu }_{0}+{\xi }_{0}+{\xi }_{0}^{3})\tau +\displaystyle \frac{{i}^{2}}{2}\left({\mu }_{0}^{3}\right.\right.\\ \,\,\,\,\,\,\,\,+\,2{\xi }_{0}+\left(1+2{\rm{i}}\right){\mu }_{0}{\xi }_{0}^{2}+(2-2{\rm{i}}){\xi }_{0}^{3}\\ \left.\left.\,\,\,\,\,\,\,\,+\,(1+2{\rm{i}}){\xi }_{0}^{5}\right){\tau }^{2}+\ldots \Space{0ex}{3.0ex}{0ex}\right),\end{array}\end{eqnarray} \tag{ 66 }$$

which have similar behavior with those obtained by qHATM [41].

For numerical representations, table 2 describes the numerical comparisons of approximate solutions of (47) and (46) between the proposed algorithm and those used in [41] within the classical case $\gamma \to 1$ on $0.5\leqslant x\leqslant 5$ for $\tau =0.01$ and ${\mu }_{0}={\xi }_{0}=1$ with step size $0.5.$ While for the fractional case $\gamma =0.8,$ the comparison is presented in table 3 over $0.5\leqslant x\leqslant 5$ for $\tau =0.01$ and ${\mu }_{0}={\xi }_{0}=1$ with step size $0.5,$ in which the columns of absolute error are calculated by $\left|{u}_{2}\left(x,\tau \right)-{u}_{qHATM}\left(x,\tau \right)\right|$ and $\left|{v}_{2}\left(x,\tau \right)-{v}_{qHATM}\left(x,\tau \right)\right|.$ From these tables, it can be observed that the approximate solutions are in perfect harmony together with maximum error ${10}^{-6}$ in classical case $\gamma =1$ and ${10}^{-3}$ in fractional case $\gamma =0.95.$

Table 2.  Numerical comparison of solutions for (45) and (46) in the classical case.

$\gamma =1$	${x}_{i}$	${u}_{2}(x,\tau )$	${u}_{qHATM}$	$\left|{u}_{2}-{u}_{qHATM}\right|$	${v}_{2}(x,\tau )$	${v}_{q-HATM}$	$\left|{v}_{2}-{v}_{qHATM}\right|$
Real part	$0.5$	$0.8771022$	$0.8771018$	$4.79426\times {10}^{-7}$	$0.8761417$	$0.8761408$	$8.46678\times {10}^{-7}$
	$1.0$	$0.5394608$	$0.5394600$	$8.41471\times {10}^{-7}$	$0.5377768$	$0.5377746$	$2.22056\times {10}^{-6}$
	$1.5$	$0.0697406$	$0.0697396$	$9.97495\times {10}^{-7}$	$0.0677455$	$0.0677417$	$3.78528\times {10}^{-6}$
	$2.0$	$-0.417055$	$-0.417056$	$9.09297\times {10}^{-7}$	$-0.418872$	$-0.418878$	$5.15774\times {10}^{-6}$
	$2.5$	$-0.801740$	$-0.801741$	$5.98472\times {10}^{-7}$	$-0.802936$	$-0.802942$	$6.00190\times {10}^{-6}$
	$3.0$	$-0.990132$	$-0.936104$	$1.41120\times {10}^{-7}$	$-0.990412$	$-0.990418$	$6.11111\times {10}^{-6}$
	$3.5$	$-0.936105$	$-0.936104$	$3.50783\times {10}^{-7}$	$-0.935401$	$-0.935407$	$5.45859\times {10}^{-6}$
	$4.0$	$-0.652887$	$-0.652886$	$7.56802\times {10}^{-7}$	$-0.651372$	$-0.651376$	$4.20413\times {10}^{-6}$
	$4.5$	$-0.209819$	$-0.209818$	$9.77530\times {10}^{-7}$	$-0.207863$	$-0.207866$	$2.65486\times {10}^{-6}$
	$5.0$	$0.2846197$	$0.2846207$	$9.58924\times {10}^{-7}$	$0.2865370$	$0.2865358$	$1.19009\times {10}^{-6}$
Imaginary part	$0.5$	$0.4803015$	$0.4803024$	$8.77583\times {10}^{-7}$	$0.4820557$	$0.4820580$	$2.31586\times {10}^{-6}$
	$1.0$	$0.8420095$	$0.8420100$	$5.40302\times {10}^{-7}$	$0.8430884$	$0.8430915$	$3.06472\times {10}^{-6}$
	$1.5$	$0.9975642$	$0.9975642$	$7.07372\times {10}^{-8}$	$0.9977036$	$0.9080485$	$3.06322\times {10}^{-6}$
	$2.0$	$0.9088803$	$0.9088799$	$4.16147\times {10}^{-7}$	$0.9080462$	$0.9080485$	$2.31175\times {10}^{-6}$
	$2.5$	$0.5976709$	$0.5976701$	$8.01144\times {10}^{-7}$	$0.5960674$	$0.5960684$	$9.94273\times {10}^{-7}$
	$3.0$	$0.1401308$	$0.1401298$	$9.89992\times {10}^{-7}$	$0.1381505$	$0.1381499$	$5.66632\times {10}^{-7}$
	$3.5$	$-0.351718$	$-0.351719$	$9.36457\times {10}^{-7}$	$-0.353590$	$-0.353592$	$1.98881\times {10}^{-6}$
	$4.0$	$-0.757454$	$-0.757455$	$6.53644\times {10}^{-7}$	$-0.758760$	$-0.758763$	$2.92405\times {10}^{-6}$
	$4.5$	$-0.977739$	$-0.977739$	$2.10796\times {10}^{-7}$	$-0.978159$	$-0.978162$	$3.14339\times {10}^{-6}$
	$5.0$	$-0.958639$	$-0.958639$	$2.83662\times {10}^{-7}$	$-0.958070$	$-0.958073$	$2.59311\times {10}^{-6}$

Table 3.  Numerical comparison of solutions for (45) and (46) in the fractional case.

$\gamma =0.95$	${x}_{i}$	${u}_{2}(x,\tau )$	${u}_{qHATM}$	$\left|{u}_{2}-{u}_{qHATM}\right|$	${v}_{2}(x,\tau )$	${v}_{q-HATM}$	$\left|{v}_{2}-{v}_{qHATM}\right|$
Real part	$0.5$	$0.8768679$	$0.8768886$	$2.07134\times {10}^{-5}$	$0.8768678$	$0.8768886$	$6.34344\times {10}^{-5}$
	$1.0$	$0.5390512$	$0.5390875$	$3.63149\times {10}^{-5}$	$0.5365464$	$0.5366561$	$1.09639\times {10}^{-4}$
	$1.5$	$0.0692560$	$0.0692990$	$4.30252\times {10}^{-5}$	$0.0662894$	$0.0664168$	$1.27396\times {10}^{-4}$
	$2.0$	$-0.417495$	$-0.417456$	$3.92015\times {10}^{-5}$	$-0.420198$	$-0.420085$	$1.12359\times {10}^{-4}$
	$2.5$	$-0.802029$	$-0.802004$	$2.57798\times {10}^{-5}$	$-0.803806$	$-0.803737$	$6.82076\times {10}^{-5}$
	$3.0$	$-0.990199$	$-0.990193$	$6.04640\times {10}^{-6}$	$-0.990614$	$-0.990608$	$5.75290\times {10}^{-6}$
	$3.5$	$-0.935933$	$-0.935948$	$1.51674\times {10}^{-5}$	$-0.934885$	$-0.934945$	$5.97143\times {10}^{-5}$
	$4.0$	$-0.652518$	$-0.652551$	$3.26677\times {10}^{-5}$	$-0.650264$	$-0.650377$	$1.12165\times {10}^{-4}$
	$4.5$	$-0.209344$	$-0.209386$	$4.21698\times {10}^{-5}$	$-0.206436$	$-0.206575$	$1.38758\times {10}^{-4}$
	$5.0$	$0.2850849$	$0.2850436$	$4.13473\times {10}^{-5}$	$0.2879353$	$0.2878023$	$1.32983\times {10}^{-4}$
Imaginary part	$0.5$	$0.4807269$	$0.4806890$	$3.78310\times {10}^{-5}$	$0.4833345$	$0.4832202$	$1.14246\times {10}^{-4}$
	$1.0$	$0.8422704$	$0.8422470$	$2.32693\times {10}^{-5}$	$0.8438733$	$0.8438070$	$6.67070\times {10}^{-5}$
	$1.5$	$0.9975967$	$0.9975937$	$3.01046\times {10}^{-6}$	$0.9978026$	$0.9977998$	$2.83622\times {10}^{-5}$
	$2.0$	$0.9086766$	$0.9086945$	$1.79855\times {10}^{-5}$	$0.9074350$	$0.9074968$	$6.1729\times {10}^{-5}$
	$2.5$	$0.5972807$	$0.5973153$	$3.45779\times {10}^{-5}$	$0.5948957$	$0.5950068$	$1.11181\times {10}^{-4}$
	$3.0$	$0.1396497$	$0.1396924$	$4.27045\times {10}^{-5}$	$0.1367051$	$0.1368385$	$1.33412\times {10}^{-4}$
	$3.5$	$-0.352172$	$-0.352132$	$4.03755\times {10}^{-5}$	$-0.354956$	$-0.354833$	$1.22979\times {10}^{-4}$
	$4.0$	$-0.757770$	$-0.757742$	$2.81612\times {10}^{-5}$	$-0.759711$	$-0.759628$	$8.24362\times {10}^{-5}$
	$4.5$	$-0.977840$	$-0.977831$	$9.05205\times {10}^{-6}$	$-0.978462$	$-0.978441$	$2.17106\times {10}^{-5}$
	$5.0$	$-0.958499$	$-0.958512$	$1.22734\times {10}^{-5}$	$-0.957652$	$-0.957696$	$4.43306\times {10}^{-5}$

Moreover, some 3D-dimensional space graphs for approximate solutions of (45) and (46) are presented in figures 6–10 as follows: the approximate solutions of real and imaginary parts of ${u}_{3}\left(x,\tau \right)$ and ${v}_{3}\left(x,\tau \right)$ are portrayed, respectively, in figures 6–9 for ${\mu }_{0}={\xi }_{0}=1$ over $-6\leqslant x\,\leqslant 6$ and $0\leqslant \tau \leqslant 6.$ Here, the fractional values are varied among $1,$ $0.75,$ $0.5,$ and $0.25.$ The geometrical behavior of the approximate solutions ${u}_{3}\left(x,\tau \right)$ and ${v}_{3}\left(x,\tau \right)$ are plotted in figure 10 for the parameter value ${\mu }_{0}={\xi }_{0}=0.25$ on $-20\leqslant x\leqslant 20$ and $0\leqslant \tau \leqslant 4$ within the classical case $\gamma =1$ and fractional case $\gamma =0.5.$

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Figure 11 shows the fractional level curves of ${\rm{Re}}[u\left(x,\tau \right)],$ ${\rm{Im}}\left[u\left(x,\tau \right)\right],$ ${\rm{Re}}[v\left(x,\tau \right)]$ and ${\rm{Im}}[v\left(x,\tau \right)]$ on $-20\leqslant x\leqslant 20$ and $\tau =0.1$ with parameter value ${\mu }_{0}={\xi }_{0}\,=1$ and with $\gamma =1$ (blue line), $\gamma =0.8$ (red line), $\gamma =0.6$ (yellow line) and $\gamma =0.4$ (green line). From graphs, it can be observed that the solution behavior of the approximation for different fractional values is quite consistent during the specified spatial and temporal intervals.

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In this paper, approximate solutions to one-dimensional complex nonlinear time-FKEE and coupled time-FMTEs with conformable derivatives have been successfully obtained by applying the RPSM. The performance of our approach is based on the optimization of approximate solutions by multiple substituting them in the conformable time-fractional residual error concept. The convergence theorem for the selected models has been discussed.

The main innovation of the proposed method lies in the following advantages: It can be applied directly without resorting to discretization to the variables or any limitations on parameters to obtain an accurate waveform solution with smooth coefficients of the concerned system; it is not influenced by the round-off error, and there is no need for a large computation memory and efforts to achieve the desired results; and it is of global nature in the sense of approximate solutions of the physical potential and also, in some situations, exact solutions can be obtained.

Two physical applications have been tested to get an accurate approximate solution and to illustrate the applicability and reliability of the proposed method for long-terms time and space. From the tabular results and graphical representations at the classical and fractional cases, it can be conclueded that the suggested method is systematic to obtain appropriate approximations compatible with each other and with those obtained by other methods of various fractional values. Consequently, such an approach is a powerful tool and can be easily implemented to many nonlinear FPDEs apears in physics and engineering sciences.

The authors gratefully acknowledges Ajman University for support with Grant ID 2020-COVID 19-08: GL: 5211529.

The authors declare that they have no conflict of interest.