Efficient techniques for nonlinear dynamics: a study of fractional generalized quintic Ginzburg-Landau equation

Abstract

In this research, we explore the solution of the fractional Generalized Quintic Complex Ginzburg-Landau equation (GQCGLE) using the Controlled Picard Transform Method. We solve the equation by incorporating the Laplace transform (LT) and the Caputo-Fabrizio fractional derivative (CF). The fractional GQCGLE is a complex-valued equation that exhibits intricate dynamics and plays a significant role in various physical phenomena, particularly in the realm of nonlinear optics. The suggested method can efficiently address a wide variety of differential equations, encompassing both integer and fractional-order equations. This is achieved by integrating an extra parameter that enhances convergence, making it especially suitable for non-linear differential equations. To confirm the precision and convergence of our approach, we validate it by comparing it with other methods, presenting graphical representations to clarify the effects of different parameters on the solution behaviour, and confirming its precision and convergence. The existence and uniqueness of the solutions are also examined.

Keywords:

• Fractional generalized quintic complex Ginzburg-Landau equation
• controlled Picard's technique
• Laplace transform
• Caputo-Fabrizio fractional derivative

1. Introduction

Fractional calculus (FC) has a long history that dates back over two centuries. There were definitions of fractional derivatives both locally and globally. There are two main categories of nonlocal fractional operators: those include derivatives with singular kernels, such as the conventional Riemann-Liouville and Caputo derivatives, and derivatives with nonsingular kernels, like the more recently applied Caputo-Fabrizio (CF) and Atangana-Baleanu fractional derivatives [1–4]. FC finds applications in diverse fields including physics, engineering, and applied mathematics. Some examples include modelling viscoelastic materials, analyzing electrical circuits, studying diffusion processes, and understanding complex systems [5–8]. There are numerous definitions of fractional derivatives that are used to formulate issues as systems of these equations or fractional partial differential equations (FPDEs), see [9–11]. The fractional GQCGLE is a mathematical model used to describe complex behaviours in physical systems, particularly in the field of nonlinear optics. This equation is an extension of the well-known Ginzburg-Landau equation, which is widely used to study phenomena like pattern formation, wave propagation, and phase transitions in various physical systems. The GQCGLE provides us to evaluate various applicable and practical applications models, and it exhibits rich phenomena and serves as an example of how the structure of spatio-temporal dynamics can degenerate into chaos, studying nonlinear optics, examining reaction-diffusion systems, investigating Rayleigh-Benard convection, exploring Taylor-Couette flow, analyzing Poiseuille flow, and delving into biochemical turbulence, see [12–14]. Other types of Ginzburg equations, such as those discussed in [15–17], can also be considered within this framework. The general form of fractional GQCGLE is typically written as: (1) $\begin{array}{rl}{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)& =(1+\mathsf{i}\varrho ){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}+\xi \mathrm{U}-(1+\mathsf{i}\sigma )\mathrm{U}|\mathrm{U}{|}^{2n}\\ & -(1+\mathsf{i}\vartheta )\mathrm{U}|\mathrm{U}{|}^{4n},0<\alpha ⩽1,n\ge 0,\end{array}$(1) subject to (2) $\mathrm{U}\left(\varkappa ,\mathsf{t}\right)={\mathrm{U}}_0\left(\varkappa ,0\right).$(2) $\mathrm{U}(\varkappa ,\mathsf{t})$ represents a complex-valued field describing the behaviour of the system under consideration. It depends on both the spatial coordinate ϰ and time $\mathsf{t}$. ${\mathsf{D}}_{\mathsf{t}}^{\alpha }$ is a fractional derivative operator with respect to time, where the parameter α governs the degree of the fractional derivative. Additionally, ϱ is a real constant that appears in the linear dispersion term. The value of ϱ influences the dispersion properties of the equation and can affect wave propagation and stability. The parameter ξ represents a linear gain or loss term in the equation, determining the balance between amplification and dissipation in the system. The parameters σ and ϑ affect the strength and sign of the quintic nonlinearity term. Recently, numerous scholars used various techniques to evaluate and obtain certain intriguing results for the Ginzburg-Landau (GL) equation. As a way to solve a nonlinear GL problem, Wang and Huang suggested an efficient difference scheme, and they demonstrated that the numerical solution is restricted and convergent [18]. An analysis of 1D and 2D-dimensional complex GL equations was performed by Wang and Huang employing a split-step quasicompact finite-difference technique [19]. In [20], scientists employed a tailored Jacobi elliptic expansion technique to the nonlinear GL equation. The travelling wave and the Riccati formula were used in a comparison study by Saima Arshed et al. [21] utilized two distinct forms of fractional derivatives. The natural decomposition technique has been used by Shao-Wen Yao et al. to solve the modified fractional quintic GL problem [22]. To find solitary results for the fractional GL problem that use beta derivatives, Ouahid et al. used an extended sub-equation approach and the unifed solution technique [23]. Additionally, several academics are interested to investigate the coupled fractional GL equations see [24,25].

The primary objective of this study is to propose a novel approach for solving the time-dependent fractional GQCGLE using a semi-analytical method. The complexities of this equation necessitate the integration of two powerful techniques: the controlled Picard method (CPM) and the LT with CF fractional operators. To enhance the approximation rate and the progression of the method, a modification to the original CPM is introduced by incorporating an additional parameter, denoted as $\mathsf{h}$. This modification, previously suggested by Mourad et al. [26,27], provides better control over the approximation process. In a related work [28], a similar modification was successfully applied to solve the fractional Navier-Stokes equation (FNSE). By combining these techniques, our approach aims to overcome the challenges posed by the fractional GQCGLE equation and provide a more effective solution. This innovative semi-analytical method opens up new opportunities for studying the dynamics of the fractional GQCGLE model, exploring its properties, and making predictions about its behaviour.

This paper is organized into several sections to provide a clear structure and presentation of the research. In Section 2, we provide an explanation of important concepts and terms related to fractional calculus. This section lays the foundation for understanding the mathematical framework used in the subsequent sections. Section 3 focuses on the implementation of the controlled Picard transform approach. We present the details of this technique, explaining their principles and how they are applied to solve differential equations. Moving on to Section 4, we study the uniqueness and existence of the solutions. In Section 5, we implement the CPM to the fractional GQCGLE. Section 6 is dedicated to the analysis of the results and the discussion of their significance. We examine the implications of the obtained solutions and analyze the behaviour of the fractional GQCGLE based on the findings. Finally, we conclude the paper in the last section by summarizing the key points and implications derived from our research.

2. Mathematical preliminaries

In the past few years, there has been an increasing focus on nonlocal fractional derivatives featuring nonsingular kernels. These derivatives, including the Caputo-Fabrizio and Atangana-Baleanu fractional derivatives, have gained attention due to their unique properties and applications in various fields. The Caputo-Fabrizio derivative, for example, provides an alternative approach to fractional differentiation that offers distinct advantages in certain scenarios [29–31].

In this part, we present a summary of fundamental definitions and concepts of fractional calculus. Additionally, we discuss the theorems related to the Laplace transform [32–34].

Definition 2.1

[29]

The Riemann-Liouville fractional integral with order α, where $\alpha >0$, applied to a function $\mathrm{\Xi }\left(\mathsf{t}\right)$ is expressed as: $\{\begin{array}{l}{\mathsf{J}}_{\mathsf{t}}^{\alpha }\mathrm{\Xi }\left(\mathsf{t}\right)\\ ={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\displaystyle {\int }_0^{\mathsf{t}}(\mathsf{t}-\zeta {)}^{(\alpha -1)}\mathrm{\Xi }(\zeta )\mathrm{d}\zeta ,}}& \alpha \in {\mathsf{R}}^{+},\mathsf{t}>0,\\ {\mathsf{J}}_{\mathsf{t}}^0\mathrm{\Xi }\left(\mathsf{t}\right)=\mathrm{\Xi }\left(\mathsf{t}\right),& \end{array}$The notation ${\mathsf{R}}^{+}$ is commonly used to represent the set of positive real numbers.

Definition 2.2

[29]

The Caputo fractional derivative for a function $\mathrm{\Xi }\left(\mathsf{t}\right)$ is defined as follows: ${\mathsf{D}}^{\alpha }\mathrm{\Xi }\left(\mathsf{t}\right)=\{\begin{array}{ll}{\mathsf{J}}^{\mathsf{m}-\alpha }\mathrm{\Xi }\left(\mathsf{t}\right)={\displaystyle \frac{{\mathsf{d}}^{\mathsf{m}}}{\mathsf{d}{\mathsf{t}}^{\mathsf{m}}}\mathrm{\Xi }\left(\mathsf{t}\right),}& \mathsf{m}-1<\alpha <\mathsf{m}\\ {\displaystyle \frac{{\mathsf{d}}^{\mathsf{m}}}{\mathsf{d}{\mathsf{t}}^{\mathsf{m}}}\mathrm{\Xi }\left(\mathsf{t}\right),}& \alpha =\mathsf{m}.\end{array}$Some of its properties that contribute to its importance and usefulness in the domain of FC, including: $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\left[{\mathsf{J}}^{\alpha }\mathrm{\Xi }\right(\mathsf{t}\left)\right]=\mathrm{\Xi }\left(\mathsf{t}\right).\\ & {\mathsf{D}}_{\mathsf{t}}^{\alpha }{\mathsf{t}}^{\mathsf{n}}=\frac{\mathrm{\Gamma }(\mathsf{n}+1)}{\mathrm{\Gamma }(\mathsf{n}+1-\alpha )}{\mathsf{t}}^{(\mathsf{n}-\alpha )}.\end{array}$If $0<\alpha ⩽1$, the LT for the Caputo fractional derivative is defined in the following manner: $\mathcal{L}\left\{{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{\Xi }\right(\varkappa ,\mathsf{t}\left)\right\}={\mathsf{s}}^{\alpha }\mathcal{L}\left\{\mathrm{\Xi }\right(\varkappa ,\mathsf{t}\left)\right\}-{\mathsf{s}}^{\alpha -1}\left[\mathrm{\Xi }\right(\varkappa ,0\left)\right].$

Definition 2.3

[34]

The expression for the CF fractional derivative is: $\begin{array}{rl}{}^{\mathrm{CF}}{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{\Xi }(\varkappa ,\mathsf{t})& =\frac{(2-\alpha ){\rm Y}\left(\alpha \right)}{2(\mathsf{n}-\alpha )}{\int }_0^{\mathsf{t}}\exp (-\alpha \frac{(t-\theta )}{\mathsf{n}-\alpha })\\ & \times \frac{{\mathsf{d}}^{\mathsf{n}}}{\mathsf{d}{\mathsf{t}}^{\mathsf{n}}}\mathrm{\Xi }\left(\varkappa ,\theta \right)\mathsf{d}\theta .\end{array}$The function ${\rm Y}\left(\alpha \right)$ is a normalization function, and it equals 1 when α is either 0 or 1. This fractional operator utilizes the exponential rule with a nonsingular kernel.

Definition 2.4

[34]

The CF fractional integral is expressed as: $\begin{array}{rl}{}_0^{\mathrm{CF}}{\mathsf{J}}_{\mathsf{t}}^{\alpha }\mathrm{\Xi }(\varkappa ,\mathsf{t})& =\frac{2(\mathsf{1}-\alpha )}{(2-\alpha ){\rm Y}\left(\alpha \right)}\mathrm{\Xi }(\varkappa ,\mathsf{t})\\ & +\frac{2\alpha }{(2-\alpha ){\rm Y}\left(\alpha \right)}{\int }_0^{\mathsf{t}}\mathrm{\Xi }\left(\varkappa ,\theta \right)\mathsf{d}\theta .\end{array}$Additionally, it is important to observe that fractional integrals on the left and right sided fractional integrals of the equation for ${}_a^{\mathrm{CF}}{\mathsf{D}}^{\alpha }\mathsf{t}$ have been established as separate entities, as described in [30]: $\begin{array}{rl}& {}_a^{\mathrm{CF}}{\mathsf{J}}_{\mathsf{t}}^{\alpha }\mathrm{\Xi }(\varkappa ,\mathsf{t})=\frac{2(\mathsf{1}-\alpha )}{{\rm Y}\left(\alpha \right)}\mathrm{\Xi }(\varkappa ,\mathsf{t})+\frac{\alpha }{{\rm Y}\left(\alpha \right)}{\int }_a^{\mathsf{t}}\mathrm{\Xi }\left(\varkappa ,\theta \right)\mathsf{d}\theta .\\ & {}_b^{\mathrm{CF}}{\mathsf{J}}_{\mathsf{t}}^{\alpha }\mathrm{\Xi }(\varkappa ,\mathsf{t})=\frac{2(\mathsf{1}-\alpha )}{{\rm Y}\left(\alpha \right)}\mathrm{\Xi }(\varkappa ,\mathsf{t})+\frac{\alpha }{{\rm Y}\left(\alpha \right)}{\int }_{\mathsf{t}}^b\mathrm{\Xi }\left(\varkappa ,\theta \right)\mathsf{d}\theta .\end{array}$In the case where $0<\alpha ⩽1$, we establish the LT for the CF fractional derivatives in the following manner: $\mathcal{L}\left\{{}^{\mathrm{CF}}{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{\Xi }\right(\varkappa ,\mathsf{t}\left)\right\}={\rm Y}\left(\alpha \right)\left(\frac{\mathsf{s}\mathcal{L}\left[\mathrm{\Xi }\right(\varkappa ,\mathsf{t}\left)\right]-\mathrm{\Xi }(\varkappa ,0)}{\mathsf{s}+\alpha (1-\mathsf{s})}\right).$

3. Basic idea of controlled picard transform technique

This approach employs the controlled Picard iteration method in combination with the Laplace transform to tackle equations that incorporate fractional derivatives. It serves as a method for addressing non-linear PDEs where the order of differentiation is fractional (non-integer). More precisely, we will explore fundamental principles and strategies for addressing partial differential equations that characterize phenomena with non-integer order rates of change. The Laplace transform, which plays a key role in this technique, serves as a powerful tool for solving intricate equations involving fractional derivatives. Let's now explore these equations' general form: (3) $\begin{array}{rl}& {}^{\mathrm{CF}}{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}(\varkappa ,\mathsf{t})+\mathsf{L}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{R}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]\\ & =\nu (\varkappa ,\mathsf{t}),\mathsf{t}>0,0<\alpha ⩽1,\end{array}$(3) subject to (4) $\mathrm{U}(\varkappa ,0)={\Psi }_0\left(\varkappa \right).$(4) In this context, $\mathsf{L}$ and $\mathsf{R}$ represent both nonlinear and linear differential operators, involving partial derivatives concerning the spatial variable ϰ, and ν stands for an inhomogeneous source expression. Here, we are dealing with the CF definition. Furthermore, there is a complex-valued function denoted as $\mathrm{U}(\varkappa ,\mathsf{t})$ that we aim to determine.

To make use of the Controlled Picard Transform technique, we rephrase (3(3) $\begin{array}{rl}& {}^{\mathrm{CF}}{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}(\varkappa ,\mathsf{t})+\mathsf{L}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{R}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]\\ & =\nu (\varkappa ,\mathsf{t}),\mathsf{t}>0,0<\alpha ⩽1,\end{array}$(3) ) as follows: (5) ${}^{\mathrm{CF}}{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}(\varkappa ,\mathsf{t})+\mathsf{L}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{R}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]-\nu (\varkappa ,\mathsf{t})=0.$(5) When we employ the Laplace transform on Equation (5(5) ${}^{\mathrm{CF}}{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}(\varkappa ,\mathsf{t})+\mathsf{L}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{R}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]-\nu (\varkappa ,\mathsf{t})=0.$(5) ), we derive the following expression: (6) $\begin{array}{rl}& \mathcal{L}\left\{{}^{\mathrm{CF}}{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\right(\varkappa ,\mathsf{t})+\mathsf{L}[\mathrm{U}(\varkappa ,\mathsf{t})]+\mathsf{R}[\mathrm{U}(\varkappa ,\mathsf{t})]-\nu (\varkappa ,\mathsf{t}\left)\right\}\\ & =0.\end{array}$(6) Using Definition 2.3, we can express Equation (6(6) $\begin{array}{rl}& \mathcal{L}\left\{{}^{\mathrm{CF}}{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\right(\varkappa ,\mathsf{t})+\mathsf{L}[\mathrm{U}(\varkappa ,\mathsf{t})]+\mathsf{R}[\mathrm{U}(\varkappa ,\mathsf{t})]-\nu (\varkappa ,\mathsf{t}\left)\right\}\\ & =0.\end{array}$(6) ) as follows: (7) $\begin{array}{rl}& \left(\frac{\mathsf{s}\mathcal{L}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]-\mathrm{U}(\varkappa ,0)}{\mathsf{s}+\alpha (1-\mathsf{s})}\right)\\ & +\mathcal{L}\left\{\mathsf{L}\right[\mathrm{U}(\varkappa ,\mathsf{t})]+\mathsf{R}[\mathrm{U}(\varkappa ,\mathsf{t})]-\nu (\varkappa ,\mathsf{t}\left)\right\}=0.\end{array}$(7) Now, we will utilize Picard's technique to continue our analysis. (8) $\begin{array}{rl}& \mathcal{L}\left\{{\mathrm{U}}_{\mathrm{m}+1}\right(\varkappa ,\mathsf{t}\left)\right\}\\ & =\frac{\mathrm{U}(\varkappa ,0)}{\mathsf{s}}-\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}\mathcal{L}\left\{\mathsf{L}\right[{\mathrm{U}}_{\mathrm{m}}(\varkappa ,\mathsf{t})]\\ & +\mathsf{R}\left[{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right]-\nu (\varkappa ,\mathsf{t})\}.\end{array}$(8) The following results are obtained by adding and subtracting the terms $\mathcal{L}{\mathsf{D}}_{\mathsf{t}}^{\alpha }{\mathrm{U}}_m(\varkappa ,\mathsf{t})$ from the right side of (8(8) $\begin{array}{rl}& \mathcal{L}\left\{{\mathrm{U}}_{\mathrm{m}+1}\right(\varkappa ,\mathsf{t}\left)\right\}\\ & =\frac{\mathrm{U}(\varkappa ,0)}{\mathsf{s}}-\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}\mathcal{L}\left\{\mathsf{L}\right[{\mathrm{U}}_{\mathrm{m}}(\varkappa ,\mathsf{t})]\\ & +\mathsf{R}\left[{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right]-\nu (\varkappa ,\mathsf{t})\}.\end{array}$(8) ): (9) $\begin{array}{rl}& \mathcal{L}\left\{{\mathrm{U}}_{\mathrm{m}+1}\right(\varkappa ,\mathsf{t}\left)\right\}\\ & =\mathcal{L}\left\{{\mathsf{D}}_{\mathsf{t}}^{\alpha }{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right\}+\frac{\mathrm{U}(\varkappa ,0)}{\mathsf{s}}\\ & -\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}\mathcal{L}\left\{{\mathsf{D}}_{\mathsf{t}}^{\alpha }{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t})+\mathsf{L}[{\mathrm{U}}_{\mathrm{m}}(\varkappa ,\mathsf{t})]\\ & +\mathsf{R}\left[{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right]-\nu (\varkappa ,\mathsf{t})\}.\end{array}$(9) Utilizing the inverse LT on both sides of Equation (9(9) $\begin{array}{rl}& \mathcal{L}\left\{{\mathrm{U}}_{\mathrm{m}+1}\right(\varkappa ,\mathsf{t}\left)\right\}\\ & =\mathcal{L}\left\{{\mathsf{D}}_{\mathsf{t}}^{\alpha }{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right\}+\frac{\mathrm{U}(\varkappa ,0)}{\mathsf{s}}\\ & -\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}\mathcal{L}\left\{{\mathsf{D}}_{\mathsf{t}}^{\alpha }{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t})+\mathsf{L}[{\mathrm{U}}_{\mathrm{m}}(\varkappa ,\mathsf{t})]\\ & +\mathsf{R}\left[{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right]-\nu (\varkappa ,\mathsf{t})\}.\end{array}$(9) ) produces the subsequent outcomes: (10) $\begin{array}{rl}& {\mathrm{U}}_{\mathrm{m}+1}(\varkappa ,\mathsf{t})\\ & ={\mathrm{U}}_{\mathrm{m}}(\varkappa ,\mathsf{t})-{\mathcal{L}}^{-1}\left[\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}\mathcal{L}\right\{{\mathsf{D}}_{\mathsf{t}}^{\alpha }{\mathrm{U}}_{\mathrm{m}}(\varkappa ,\mathsf{t})\\ & +\mathsf{L}\left[{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{R}\left[{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right]-\nu (\varkappa ,\mathsf{t})\left\}\phantom{\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}}\right].\end{array}$(10) Equation (10(10) $\begin{array}{rl}& {\mathrm{U}}_{\mathrm{m}+1}(\varkappa ,\mathsf{t})\\ & ={\mathrm{U}}_{\mathrm{m}}(\varkappa ,\mathsf{t})-{\mathcal{L}}^{-1}\left[\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}\mathcal{L}\right\{{\mathsf{D}}_{\mathsf{t}}^{\alpha }{\mathrm{U}}_{\mathrm{m}}(\varkappa ,\mathsf{t})\\ & +\mathsf{L}\left[{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{R}\left[{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right]-\nu (\varkappa ,\mathsf{t})\left\}\phantom{\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}}\right].\end{array}$(10) ) outlines an iterative procedure that has been derived for estimating the solution to (5(5) ${}^{\mathrm{CF}}{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}(\varkappa ,\mathsf{t})+\mathsf{L}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{R}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]-\nu (\varkappa ,\mathsf{t})=0.$(5) ) for both integer and fractional orders. In this stage, we introduce an additional parameter denoted by the symbol $\mathsf{h}$ into Picard's iterative transformation technique. This parameter is introduced to manage and adjust the convergence interval for the approximated solution of (5(5) ${}^{\mathrm{CF}}{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}(\varkappa ,\mathsf{t})+\mathsf{L}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{R}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]-\nu (\varkappa ,\mathsf{t})=0.$(5) ). Let's now analyze the non-linear differential Equation (5(5) ${}^{\mathrm{CF}}{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}(\varkappa ,\mathsf{t})+\mathsf{L}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{R}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]-\nu (\varkappa ,\mathsf{t})=0.$(5) ) with fractional order in the following manner: (11) $\begin{array}{rl}& T(\varkappa ,\mathsf{t},\mathrm{U}(\varkappa ,\mathsf{t}),\alpha )\\ & ={\mathsf{D}}_{\mathsf{t}}^{\alpha }{\mathrm{U}}_{\mathrm{m}}(\varkappa ,\mathsf{t})+\mathsf{L}\left[{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{R}\left[{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right]\\ & -\nu (\varkappa ,\mathsf{t})=0.\end{array}$(11) By multiplying $\mathsf{h}$ on both sides of Equation (11(11) $\begin{array}{rl}& T(\varkappa ,\mathsf{t},\mathrm{U}(\varkappa ,\mathsf{t}),\alpha )\\ & ={\mathsf{D}}_{\mathsf{t}}^{\alpha }{\mathrm{U}}_{\mathrm{m}}(\varkappa ,\mathsf{t})+\mathsf{L}\left[{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{R}\left[{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right]\\ & -\nu (\varkappa ,\mathsf{t})=0.\end{array}$(11) ), the resulting equation is obtained: (12) $\mathsf{h}T(\varkappa ,\mathsf{t},\mathrm{U}(\varkappa ,\mathsf{t}),\alpha )=0.$(12) The results below are derived by incorporating and subtracting the terms ${\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}(\varkappa ,\mathsf{t})$ from the right side of Equation (12(12) $\mathsf{h}T(\varkappa ,\mathsf{t},\mathrm{U}(\varkappa ,\mathsf{t}),\alpha )=0.$(12) ): (13) ${\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}(\varkappa ,\mathsf{t})+\mathsf{h}T(\varkappa ,\mathsf{t},\mathrm{U}(\varkappa ,\mathsf{t}),\alpha )-{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}(\varkappa ,\mathsf{t})=0.$(13) Picard's transform iteration formula (10(10) $\begin{array}{rl}& {\mathrm{U}}_{\mathrm{m}+1}(\varkappa ,\mathsf{t})\\ & ={\mathrm{U}}_{\mathrm{m}}(\varkappa ,\mathsf{t})-{\mathcal{L}}^{-1}\left[\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}\mathcal{L}\right\{{\mathsf{D}}_{\mathsf{t}}^{\alpha }{\mathrm{U}}_{\mathrm{m}}(\varkappa ,\mathsf{t})\\ & +\mathsf{L}\left[{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{R}\left[{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right]-\nu (\varkappa ,\mathsf{t})\left\}\phantom{\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}}\right].\end{array}$(10) ) is applied to Equation (13(13) ${\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}(\varkappa ,\mathsf{t})+\mathsf{h}T(\varkappa ,\mathsf{t},\mathrm{U}(\varkappa ,\mathsf{t}),\alpha )-{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}(\varkappa ,\mathsf{t})=0.$(13) ) to produce: (14) $\begin{array}{rl}& {\mathrm{U}}_{\mathrm{m}+1}(\varkappa ,\mathsf{t},\mathsf{h})\\ & ={\mathrm{U}}_{\mathrm{m}}(\varkappa ,\mathsf{t})-{\mathcal{L}}^{-1}\left[\frac{\mathsf{h}(\mathsf{s}+(1-\alpha \left)\mathsf{s}\right)}{\mathsf{s}}\mathcal{L}\right\{{\mathsf{D}}_{\mathsf{t}}^{\alpha }{\mathrm{U}}_{\mathrm{m}}(\varkappa ,\mathsf{t})\\ & +\mathsf{L}\left[{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{R}\left[{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right]-\nu (\varkappa ,\mathsf{t})\left\}\phantom{\frac{\mathsf{h}(\mathsf{s}+(1-\alpha \left)\mathsf{s}\right)}{\mathsf{s}}}\right].\end{array}$(14) And the initial solution is: (15) ${\Psi }_0\left(x\right)={\mathcal{L}}^{-1}\left[\frac{\mathrm{U}(\varkappa ,0)}{\mathsf{s}}\right].$(15) Following is a formula for the Picard's transform technique's iterative method that includes an auxiliary parameter from (5(5) ${}^{\mathrm{CF}}{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}(\varkappa ,\mathsf{t})+\mathsf{L}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{R}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]-\nu (\varkappa ,\mathsf{t})=0.$(5) ): (16) $\begin{array}{rl}& {\mathrm{U}}_{\mathrm{m}+1}(\varkappa ,\mathsf{t},\mathsf{h})\\ & =(1-\mathsf{h}){\mathrm{U}}_{\mathrm{m}}(\varkappa ,\mathsf{t})+\mathsf{h}{\Psi }_0\left(\varkappa \right)\\ & -\mathsf{h}{\mathcal{L}}^{-1}\left[\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}\mathcal{L}\right\{\mathsf{L}\left[{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right]\\ & +\mathsf{R}\left[{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right]-\nu (\varkappa ,\mathsf{t})\left\}\phantom{\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}}\right].\end{array}$(16) Utilizing Mathematica 11 and beginning with initial approximations ${\Psi }_0\left(\varkappa \right)$ that satisfy the initial conditions, we can compute the estimated solution ${\mathrm{U}}_{m+1}(x,t)$. However, the parameter $\mathsf{h}$ is still undetermined. To determine an appropriate value for$\mathsf{h}$ ensuring the convergence of the approximate solution within the applicable range, we can utilize the notion of the $\mathsf{h}$-curve, as detailed in [27,28]. By pointing out the horizontal region which is equivalent to the eligible interval of $\mathsf{h}$, this curve aids in choosing a good value for $\mathsf{h}$.

4. The existence and uniqueness of the solution

We will demonstrate the existence and uniqueness of the fractional GQCGLE in the context of the CF operator framework in this section. Let's reformat (5(5) ${}^{\mathrm{CF}}{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}(\varkappa ,\mathsf{t})+\mathsf{L}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{R}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]-\nu (\varkappa ,\mathsf{t})=0.$(5) ) as follows: (17) $\begin{array}{rl}{}^{\mathrm{CF}}{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}(\varkappa ,\mathsf{t})& =\mathrm{\Psi }\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{L}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{R}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right],\\ & 0<\alpha ⩽1,\end{array}$(17) where $\mathrm{\Psi }\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]={\mathrm{U}}_{\mathrm{\varkappa \varkappa }},\mathsf{L}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]$ and $\mathsf{R}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]$ represent both nonlinear and linear differential operators.

When we utilize the CF fractional integral, as specified in Definition 2.4, on both sides of (17(17) $\begin{array}{rl}{}^{\mathrm{CF}}{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}(\varkappa ,\mathsf{t})& =\mathrm{\Psi }\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{L}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{R}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right],\\ & 0<\alpha ⩽1,\end{array}$(17) ), we get: (18) $\begin{array}{rl}& \mathrm{U}(\varkappa ,\mathsf{t})-\mathrm{U}(\varkappa ,0)\\ & =\frac{2(\mathsf{1}-\alpha )}{(2-\alpha ){\rm Y}\left(\alpha \right)}\Xi (\varkappa ,\mathsf{t},\mathrm{U})\\ & +\frac{2\alpha }{(2-\alpha ){\rm Y}\left(\alpha \right)}{\int }_0^{\mathsf{t}}\Xi (\varkappa ,\eta ,\mathrm{U})\mathsf{d}\eta .\end{array}$(18) In this context, Ξ stands for the term on the right side of the provided equation. Although it is originally referred to as the kernel $\Xi (\varkappa ,\mathsf{t},\mathrm{U})$, for simplicity, we will refer to it as $\Xi \left(\mathrm{U}\right)$. It is stated that $\mathrm{U}(\varkappa ,\mathsf{t})$ has an upper bound if the kernel $\Xi (\varkappa ,\mathsf{t},\mathrm{U})$ satisfies the Lipschitzian condition. Hence, the equation can be expressed as follows: (19) $\begin{array}{rl}& \parallel \Xi \left(\mathrm{U}\right)-\Xi \left({\mathrm{U}}_1\right)\parallel \\ & =\parallel \mathrm{\Psi }\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{L}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{R}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]\\ & -\left(\mathrm{\Psi }\right[{\mathrm{U}}_1(\varkappa ,\mathsf{t})]+\mathsf{L}[{\mathrm{U}}_1(\varkappa ,\mathsf{t})]+\mathsf{R}[{\mathrm{U}}_1(\varkappa ,\mathsf{t})\left]\right)\parallel \\ & ⩽\xi \parallel \mathrm{U}-{\mathrm{U}}_1\parallel .\end{array}$(19) Let's assume that $\mathrm{U}(\varkappa ,\mathsf{t})$ has a recursive formula: (20) $\begin{array}{rl}{\mathrm{U}}_{\kappa +1}(\varkappa ,\mathsf{t})& =\frac{2(\mathsf{1}-\alpha )}{(2-\alpha ){\rm Y}\left(\alpha \right)}\Xi \left({\mathrm{U}}_{\kappa }\right)\\ & +\frac{2\alpha }{(2-\alpha ){\rm Y}\left(\alpha \right)}{\int }_0^{\mathsf{t}}\Xi \left({\mathrm{U}}_{\kappa }\right)\mathsf{d}\eta .\end{array}$(20) The recursive terms can be described as follows: (21) $\begin{array}{rl}\mathsf{G}(\varkappa ,\mathsf{t})& ={\mathrm{U}}_{\kappa -1}-{\mathrm{U}}_{\kappa -2}\\ & =\frac{2(\mathsf{1}-\alpha )}{(2-\alpha ){\rm Y}\left(\alpha \right)}\left[\Xi \right({\mathrm{U}}_{\kappa -1})-\Xi ({\mathrm{U}}_{\kappa -2}\left)\right]\\ & +\frac{2\alpha }{(2-\alpha ){\rm Y}\left(\alpha \right)}{\int }_0^{\mathsf{t}}\left[\Xi \right({\mathrm{U}}_{\kappa -1})-\Xi ({\mathrm{U}}_{\kappa -2}\left)\right]\mathsf{d}\eta .\end{array}$(21) The approximate solution is provided by: (22) ${\mathrm{U}}_{\kappa }(\varkappa ,\mathsf{t})=\sum _{i=0}^{\kappa }{\mathsf{G}}_i(\varkappa ,\mathsf{t}).$(22) When we examine both sides of (21(21) $\begin{array}{rl}\mathsf{G}(\varkappa ,\mathsf{t})& ={\mathrm{U}}_{\kappa -1}-{\mathrm{U}}_{\kappa -2}\\ & =\frac{2(\mathsf{1}-\alpha )}{(2-\alpha ){\rm Y}\left(\alpha \right)}\left[\Xi \right({\mathrm{U}}_{\kappa -1})-\Xi ({\mathrm{U}}_{\kappa -2}\left)\right]\\ & +\frac{2\alpha }{(2-\alpha ){\rm Y}\left(\alpha \right)}{\int }_0^{\mathsf{t}}\left[\Xi \right({\mathrm{U}}_{\kappa -1})-\Xi ({\mathrm{U}}_{\kappa -2}\left)\right]\mathsf{d}\eta .\end{array}$(21) ), the following becomes evident: (23) $\begin{array}{rl}& \parallel {\mathsf{G}}_{\kappa }(\varkappa ,\mathsf{t})\\ & =\parallel {\mathrm{U}}_{\kappa }-{\mathrm{U}}_{\kappa -1}\parallel \\ & =\parallel \frac{2(\mathsf{1}-\alpha )}{(2-\alpha ){\rm Y}\left(\alpha \right)}\left[\Xi \right({\mathrm{U}}_{\kappa -1})-\Xi ({\mathrm{U}}_{\kappa -2}\left)\right]\\ & +\frac{2\alpha }{(2-\alpha ){\rm Y}\left(\alpha \right)}{\int }_0^{\mathsf{t}}\left[\Xi \right({\mathrm{U}}_{\kappa -1})-\Xi ({\mathrm{U}}_{\kappa -2}\left)\right]\mathsf{d}\eta \parallel \\ & ⩽\frac{2(\mathsf{1}-\alpha )}{(2-\alpha ){\rm Y}\left(\alpha \right)}\parallel \Xi \left({\mathrm{U}}_{\kappa -1}\right)-\Xi \left({\mathrm{U}}_{\kappa -2}\right)\parallel \\ & +\frac{2\alpha }{(2-\alpha ){\rm Y}\left(\alpha \right)}{\int }_0^{\mathsf{t}}\parallel \Xi \left({\mathrm{U}}_{\kappa -1}\right)-\Xi \left({\mathrm{U}}_{\kappa -2}\right)\parallel \mathsf{d}\eta \\ & ⩽\frac{2(\mathsf{1}-\alpha )}{(2-\alpha ){\rm Y}\left(\alpha \right)}\xi \parallel {\mathrm{U}}_{\kappa -1}-{\mathrm{U}}_{\kappa -2}\parallel \\ & +\frac{2\mathrm{\alpha \xi }}{(2-\alpha ){\rm Y}\left(\alpha \right)}{\int }_0^{\mathsf{t}}\parallel {\mathrm{U}}_{\kappa -1}-{\mathrm{U}}_{\kappa -2}\parallel \mathsf{d}\eta \\ & ⩽(\frac{2(\mathsf{1}-\alpha )\xi }{(2-\alpha ){\rm Y}\left(\alpha \right)}+\frac{2\mathrm{\alpha \xi }{\mathsf{t}}_0}{(2-\alpha ){\rm Y}\left(\alpha \right)})\\ & \times \parallel {\mathrm{U}}_{\kappa -1}-{\mathrm{U}}_{\kappa -2}\parallel .\end{array}$(23)

Theorem 4.1

Equation (17(17) $\begin{array}{rl}{}^{\mathrm{CF}}{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}(\varkappa ,\mathsf{t})& =\mathrm{\Psi }\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{L}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]+\mathsf{R}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right],\\ & 0<\alpha ⩽1,\end{array}$(17) ) is considered to have a solution when the following inequality holds: (24) $\frac{2(\mathsf{1}-\alpha )\xi }{(2-\alpha ){\rm Y}\left(\alpha \right)}+\frac{2\mathrm{\alpha \xi }{\mathsf{t}}_0}{(2-\alpha ){\rm Y}\left(\alpha \right)}<1.$(24)

Proof.

Considering that the solution scheme relies on a recursive relation (23(23) $\begin{array}{rl}& \parallel {\mathsf{G}}_{\kappa }(\varkappa ,\mathsf{t})\\ & =\parallel {\mathrm{U}}_{\kappa }-{\mathrm{U}}_{\kappa -1}\parallel \\ & =\parallel \frac{2(\mathsf{1}-\alpha )}{(2-\alpha ){\rm Y}\left(\alpha \right)}\left[\Xi \right({\mathrm{U}}_{\kappa -1})-\Xi ({\mathrm{U}}_{\kappa -2}\left)\right]\\ & +\frac{2\alpha }{(2-\alpha ){\rm Y}\left(\alpha \right)}{\int }_0^{\mathsf{t}}\left[\Xi \right({\mathrm{U}}_{\kappa -1})-\Xi ({\mathrm{U}}_{\kappa -2}\left)\right]\mathsf{d}\eta \parallel \\ & ⩽\frac{2(\mathsf{1}-\alpha )}{(2-\alpha ){\rm Y}\left(\alpha \right)}\parallel \Xi \left({\mathrm{U}}_{\kappa -1}\right)-\Xi \left({\mathrm{U}}_{\kappa -2}\right)\parallel \\ & +\frac{2\alpha }{(2-\alpha ){\rm Y}\left(\alpha \right)}{\int }_0^{\mathsf{t}}\parallel \Xi \left({\mathrm{U}}_{\kappa -1}\right)-\Xi \left({\mathrm{U}}_{\kappa -2}\right)\parallel \mathsf{d}\eta \\ & ⩽\frac{2(\mathsf{1}-\alpha )}{(2-\alpha ){\rm Y}\left(\alpha \right)}\xi \parallel {\mathrm{U}}_{\kappa -1}-{\mathrm{U}}_{\kappa -2}\parallel \\ & +\frac{2\mathrm{\alpha \xi }}{(2-\alpha ){\rm Y}\left(\alpha \right)}{\int }_0^{\mathsf{t}}\parallel {\mathrm{U}}_{\kappa -1}-{\mathrm{U}}_{\kappa -2}\parallel \mathsf{d}\eta \\ & ⩽(\frac{2(\mathsf{1}-\alpha )\xi }{(2-\alpha ){\rm Y}\left(\alpha \right)}+\frac{2\mathrm{\alpha \xi }{\mathsf{t}}_0}{(2-\alpha ){\rm Y}\left(\alpha \right)})\\ & \times \parallel {\mathrm{U}}_{\kappa -1}-{\mathrm{U}}_{\kappa -2}\parallel .\end{array}$(23) ), the existence of ${\mathsf{G}}_{\kappa }(\varkappa ,\mathsf{t})$ is established, and it is considered as the solution to the proposed equation.

To establish the uniqueness of the solution, let's consider that (23(23) $\begin{array}{rl}& \parallel {\mathsf{G}}_{\kappa }(\varkappa ,\mathsf{t})\\ & =\parallel {\mathrm{U}}_{\kappa }-{\mathrm{U}}_{\kappa -1}\parallel \\ & =\parallel \frac{2(\mathsf{1}-\alpha )}{(2-\alpha ){\rm Y}\left(\alpha \right)}\left[\Xi \right({\mathrm{U}}_{\kappa -1})-\Xi ({\mathrm{U}}_{\kappa -2}\left)\right]\\ & +\frac{2\alpha }{(2-\alpha ){\rm Y}\left(\alpha \right)}{\int }_0^{\mathsf{t}}\left[\Xi \right({\mathrm{U}}_{\kappa -1})-\Xi ({\mathrm{U}}_{\kappa -2}\left)\right]\mathsf{d}\eta \parallel \\ & ⩽\frac{2(\mathsf{1}-\alpha )}{(2-\alpha ){\rm Y}\left(\alpha \right)}\parallel \Xi \left({\mathrm{U}}_{\kappa -1}\right)-\Xi \left({\mathrm{U}}_{\kappa -2}\right)\parallel \\ & +\frac{2\alpha }{(2-\alpha ){\rm Y}\left(\alpha \right)}{\int }_0^{\mathsf{t}}\parallel \Xi \left({\mathrm{U}}_{\kappa -1}\right)-\Xi \left({\mathrm{U}}_{\kappa -2}\right)\parallel \mathsf{d}\eta \\ & ⩽\frac{2(\mathsf{1}-\alpha )}{(2-\alpha ){\rm Y}\left(\alpha \right)}\xi \parallel {\mathrm{U}}_{\kappa -1}-{\mathrm{U}}_{\kappa -2}\parallel \\ & +\frac{2\mathrm{\alpha \xi }}{(2-\alpha ){\rm Y}\left(\alpha \right)}{\int }_0^{\mathsf{t}}\parallel {\mathrm{U}}_{\kappa -1}-{\mathrm{U}}_{\kappa -2}\parallel \mathsf{d}\eta \\ & ⩽(\frac{2(\mathsf{1}-\alpha )\xi }{(2-\alpha ){\rm Y}\left(\alpha \right)}+\frac{2\mathrm{\alpha \xi }{\mathsf{t}}_0}{(2-\alpha ){\rm Y}\left(\alpha \right)})\\ & \times \parallel {\mathrm{U}}_{\kappa -1}-{\mathrm{U}}_{\kappa -2}\parallel .\end{array}$(23) ) possesses two solutions, namely $\mathrm{U}\left(\varkappa ,\mathsf{t}\right)$ and $\Theta \left(\varkappa ,\mathsf{t}\right)$. Consequently, we can examine the following: (25) $\begin{array}{rl}& \mathrm{U}\left(\varkappa ,\mathsf{t}\right)-\Theta \left(\varkappa ,\mathsf{t}\right)\\ & =\frac{2(\mathsf{1}-\alpha )}{(2-\alpha ){\rm Y}\left(\alpha \right)}\left[\Xi \right(\mathrm{U})-\Xi (\Theta \left)\right]\\ & +\frac{2\alpha }{(2-\alpha ){\rm Y}\left(\alpha \right)}{\int }_0^{\mathsf{t}}\left[\Xi \right(\mathrm{U})-\Xi (\Theta \left)\right]\mathsf{d}\eta .\end{array}$(25) By taking the norm: (26) $\begin{array}{rl}& \parallel \mathrm{U}\left(\varkappa ,\mathsf{t}\right)-\Theta \left(\varkappa ,\mathsf{t}\right)\parallel \\ & =\parallel \frac{2(\mathsf{1}-\alpha )}{(2-\alpha ){\rm Y}\left(\alpha \right)}\left[\Xi \right(\mathrm{U})-\Xi (\Theta \left)\right]\\ & +\frac{2\alpha }{(2-\alpha ){\rm Y}\left(\alpha \right)}{\int }_0^{\mathsf{t}}\left[\Xi \right(\mathrm{U})-\Xi (\Theta \left)\right]\mathsf{d}\eta \parallel \\ & ⩽\frac{2(\mathsf{1}-\alpha )}{(2-\alpha ){\rm Y}\left(\alpha \right)}\parallel \Xi \left(\mathrm{U}\right)-\Xi \left(\Theta \right)\parallel \\ & +\frac{2\alpha }{(2-\alpha ){\rm Y}\left(\alpha \right)}{\int }_0^{\mathsf{t}}\parallel \Xi \left(\mathrm{U}\right)-\Xi \left(\Theta \right)\parallel \mathsf{d}\eta \\ & ⩽\frac{2(\mathsf{1}-\alpha )\xi }{(2-\alpha ){\rm Y}\left(\alpha \right)}\parallel \mathrm{U}(\varkappa ,\mathsf{t})-\Theta (\varkappa ,\mathsf{t})\parallel \\ & +\frac{2\mathrm{\alpha \xi }}{(2-\alpha ){\rm Y}\left(\alpha \right)}{\int }_0^{\mathsf{t}}\parallel \mathrm{U}(\varkappa ,\eta )-\Theta (\varkappa ,\eta )\parallel \mathsf{d}\eta \\ & ⩽\frac{2(\mathsf{1}-\alpha )\xi }{(2-\alpha ){\rm Y}\left(\alpha \right)}\parallel \mathrm{U}(\varkappa ,\mathsf{t})-\Theta (\varkappa ,\mathsf{t})\parallel \\ & +\frac{2\mathrm{\alpha \xi }\mathsf{t}}{(2-\alpha ){\rm Y}\left(\alpha \right)}\parallel \mathrm{U}(\varkappa ,\mathsf{t})-\Theta (\varkappa ,\mathsf{t})\parallel .\end{array}$(26) So, we can conclude: (27) $\begin{array}{rl}& \parallel \mathrm{U}(\varkappa ,\mathsf{t})-\Theta (\varkappa ,\mathsf{t})\parallel \\ & \times (1-\frac{2(\mathsf{1}-\alpha )\xi }{(2-\alpha )}+\frac{2\mathrm{\alpha \xi }\mathsf{t}}{(2-\alpha ){\rm Y}\left(\alpha \right)})⩽0.\end{array}$(27) Therefore, we can deduce that: (28) $\parallel \mathrm{U}(\varkappa ,\mathsf{t})-\Theta (\varkappa ,\mathsf{t})\parallel =0.$(28) Consequently, we can state that $\mathrm{U}(\varkappa ,\mathsf{t})=\Theta (\varkappa ,\mathsf{t})$.

In the following, we will prove the convergence of the solution under Caputo-Fabrizio fractional derivative For the sequence of the series solution presented in Equation (22(22) ${\mathrm{U}}_{\kappa }(\varkappa ,\mathsf{t})=\sum _{i=0}^{\kappa }{\mathsf{G}}_i(\varkappa ,\mathsf{t}).$(22) ) (29) ${\mathrm{U}}_{\mathit{n}}(\chi ,\mathsf{t})=\sum _{i=0}^{\mathit{n}}{\mathsf{G}}_i(\chi ,\mathsf{t}).$(29) Utilizing the exact solution $\mathsf{w}(\varkappa ,\mathfrak{t})$, we may demonstrate convergence in the same way as this:

The explored formula, expressed as (30) $\begin{array}{rl}& {}^{\mathrm{CF}}{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}(\chi ,\mathsf{t})\\ & =\mathcal{G}(\mathrm{\Psi }\left[\mathrm{U}\right(\chi ,\mathsf{t}\left)\right]+\mathsf{L}\left[\mathrm{U}\right(\chi ,\mathsf{t}\left)\right]+\mathsf{R}\left[\mathrm{U}\right(\chi ,\mathsf{t}\left)\right]),\\ & 0<\alpha ⩽1,\end{array}$(30)

Theorem 4.2

Suppose $\mathcal{G}$ is an operator that maps from the Hilbert space $\mathsf{H}\to \mathsf{H}$. Let $\mathrm{U}$ be the exact solution of Equation (30(30) $\begin{array}{rl}& {}^{\mathrm{CF}}{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}(\chi ,\mathsf{t})\\ & =\mathcal{G}(\mathrm{\Psi }\left[\mathrm{U}\right(\chi ,\mathsf{t}\left)\right]+\mathsf{L}\left[\mathrm{U}\right(\chi ,\mathsf{t}\left)\right]+\mathsf{R}\left[\mathrm{U}\right(\chi ,\mathsf{t}\left)\right]),\\ & 0<\alpha ⩽1,\end{array}$(30) ). If the approximated solution given by Equation (29(29) ${\mathrm{U}}_{\mathit{n}}(\chi ,\mathsf{t})=\sum _{i=0}^{\mathit{n}}{\mathsf{G}}_i(\chi ,\mathsf{t}).$(29) ) satisfies the condition $\parallel {\mathrm{U}}_{\mathit{n}+1}(\chi ,\mathsf{t})\parallel \le \delta \parallel {\mathrm{U}}_{\mathit{n}+1}(\chi ,\mathsf{t})\parallel$, for all $\mathit{n}\in N\cup \left\{0\right\}$, where δ is a constant such that $0<\delta \le 1$, hence, $\delta =\frac{2(\mathsf{1}-\alpha )\xi }{(2-\alpha ){\rm Y}\left(\alpha \right)}+\frac{2\mathrm{\alpha \xi }{\mathsf{t}}_0}{(2-\alpha ){\rm Y}\left(\alpha \right)}$ from Equation (23(23) $\begin{array}{rl}& \parallel {\mathsf{G}}_{\kappa }(\varkappa ,\mathsf{t})\\ & =\parallel {\mathrm{U}}_{\kappa }-{\mathrm{U}}_{\kappa -1}\parallel \\ & =\parallel \frac{2(\mathsf{1}-\alpha )}{(2-\alpha ){\rm Y}\left(\alpha \right)}\left[\Xi \right({\mathrm{U}}_{\kappa -1})-\Xi ({\mathrm{U}}_{\kappa -2}\left)\right]\\ & +\frac{2\alpha }{(2-\alpha ){\rm Y}\left(\alpha \right)}{\int }_0^{\mathsf{t}}\left[\Xi \right({\mathrm{U}}_{\kappa -1})-\Xi ({\mathrm{U}}_{\kappa -2}\left)\right]\mathsf{d}\eta \parallel \\ & ⩽\frac{2(\mathsf{1}-\alpha )}{(2-\alpha ){\rm Y}\left(\alpha \right)}\parallel \Xi \left({\mathrm{U}}_{\kappa -1}\right)-\Xi \left({\mathrm{U}}_{\kappa -2}\right)\parallel \\ & +\frac{2\alpha }{(2-\alpha ){\rm Y}\left(\alpha \right)}{\int }_0^{\mathsf{t}}\parallel \Xi \left({\mathrm{U}}_{\kappa -1}\right)-\Xi \left({\mathrm{U}}_{\kappa -2}\right)\parallel \mathsf{d}\eta \\ & ⩽\frac{2(\mathsf{1}-\alpha )}{(2-\alpha ){\rm Y}\left(\alpha \right)}\xi \parallel {\mathrm{U}}_{\kappa -1}-{\mathrm{U}}_{\kappa -2}\parallel \\ & +\frac{2\mathrm{\alpha \xi }}{(2-\alpha ){\rm Y}\left(\alpha \right)}{\int }_0^{\mathsf{t}}\parallel {\mathrm{U}}_{\kappa -1}-{\mathrm{U}}_{\kappa -2}\parallel \mathsf{d}\eta \\ & ⩽(\frac{2(\mathsf{1}-\alpha )\xi }{(2-\alpha ){\rm Y}\left(\alpha \right)}+\frac{2\mathrm{\alpha \xi }{\mathsf{t}}_0}{(2-\alpha ){\rm Y}\left(\alpha \right)})\\ & \times \parallel {\mathrm{U}}_{\kappa -1}-{\mathrm{U}}_{\kappa -2}\parallel .\end{array}$(23) ), then the approximated solution converges to the exact solution $\mathrm{U}$.

Proof.

We aim to prove that ${\mathrm{U}}_{\mathit{n}}{|}_{\mathit{n}=0}^{\mathrm{\infty }}$ is a convergent Cauchy sequence, $\begin{array}{rl}\parallel {\mathrm{U}}_{\mathit{n}+1}-{\mathrm{U}}_{\mathit{n}}\parallel & =\parallel {\mathrm{U}}_{\mathit{n}+1}\parallel \le \delta \parallel {\mathrm{U}}_{\mathit{n}}\parallel \le {\delta }^2\parallel {\mathrm{U}}_{\mathit{n}-1}\parallel \\ & \le \cdots \le {\delta }^{\mathit{n}}\parallel {\mathrm{U}}_1\parallel \le {\delta }^{\mathit{n}+1}\parallel {\mathrm{U}}_0\parallel .\end{array}$For $\mathit{n},\mathit{m}\in \mathsf{N},\mathit{n}>\mathit{m}$, we obtain $\begin{array}{rl}& \parallel {\mathrm{U}}_{\mathit{n}}-{\mathrm{U}}_{\mathit{m}}\parallel \\ & =\parallel ({\mathrm{U}}_{\mathit{n}}-{\mathrm{U}}_{\mathit{n}-1})+({\mathrm{U}}_{\mathit{n}-1}-{\mathrm{U}}_{\mathit{n}-2})+\cdots \\ & +({\mathrm{U}}_{\mathit{m}+1}-{\mathrm{U}}_{\mathit{m}})\parallel \\ & \le \parallel ({\mathrm{U}}_{\mathit{n}}-{\mathrm{U}}_{\mathit{n}-1})\parallel +\parallel ({\mathrm{U}}_{\mathit{n}-1}-{\mathrm{U}}_{\mathit{n}-2})\parallel +\cdots \\ & +\parallel ({\mathrm{U}}_{\mathit{m}+1}-{\mathrm{U}}_{\mathit{m}})\parallel \\ & \le {\delta }^{\mathit{n}}\parallel {\mathrm{U}}_0\left(x\right)\parallel +{\delta }^{\mathit{n}-1}\parallel {\mathrm{U}}_0\left(x\right)\parallel +\cdots \\ & +{\delta }^{\mathit{m}+1}\parallel {\mathrm{U}}_0\left(x\right)\parallel \\ & \le ({\delta }^{\mathit{n}}+{\delta }^{\mathit{n}-1}+\cdots +{\delta }^{\mathit{m}+1})\parallel {\mathrm{U}}_0\left(x\right)\parallel \\ & \le {\delta }^{\mathit{m}+1}\frac{1-{\delta }^{\mathit{n}-\mathit{m}}}{1-\delta }\parallel {\mathrm{U}}_0\left(x\right)\parallel \to 0\mathrm{as}\mathit{n},\mathit{m}\to \mathrm{\infty }.\end{array}$Hence ${\mathrm{U}}_{\mathit{n}}{|}_{\mathit{n}=0}^{\mathrm{\infty }}$ is a convergent Cauchy sequence in $\mathsf{H}$.

5. Numerical solution for the fractional GQCGLE

This section will provide a concise illustration of the procedures associated with applying the controlled Picard-transform approach to solve the fractional GQCGLE (1(1) $\begin{array}{rl}{\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)& =(1+\mathsf{i}\varrho ){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}+\xi \mathrm{U}-(1+\mathsf{i}\sigma )\mathrm{U}|\mathrm{U}{|}^{2n}\\ & -(1+\mathsf{i}\vartheta )\mathrm{U}|\mathrm{U}{|}^{4n},0<\alpha ⩽1,n\ge 0,\end{array}$(1) ), as outlined in [22].

To illustrate the efficacy and dependability of the suggested scheme, let's look at two examples.

5.1. Example 1.

Now, let's examine the the fractional GQCGLE given below [22]: (31) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})\mathrm{U}|\mathrm{U}{|}^2\\ & +(1-4\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4=0,0<\alpha ⩽1,\end{array}$(31) assuming the starting guess: (32) $\mathrm{U}(\varkappa ,0)=e^{\mathsf{i}\varkappa }.$(32) Upon simplification, (31(31) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})\mathrm{U}|\mathrm{U}{|}^2\\ & +(1-4\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4=0,0<\alpha ⩽1,\end{array}$(31) ) can be rewritten as: (33) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})(\mathrm{U}{)}^2\stackrel{-}{\mathrm{U}}\\ & +(1+4\mathsf{i})\left(\mathrm{U}{)}^3\right(\stackrel{-}{\mathrm{U}}{)}^2,\end{array}$(33) where $|\mathrm{U}{|}^2=\mathrm{U}\stackrel{-}{\mathrm{U}}$, where $\stackrel{-}{\mathrm{U}}$ is conjugate of $\mathrm{U}$.

Through the utilization of the CPM derived from (16(16) $\begin{array}{rl}& {\mathrm{U}}_{\mathrm{m}+1}(\varkappa ,\mathsf{t},\mathsf{h})\\ & =(1-\mathsf{h}){\mathrm{U}}_{\mathrm{m}}(\varkappa ,\mathsf{t})+\mathsf{h}{\Psi }_0\left(\varkappa \right)\\ & -\mathsf{h}{\mathcal{L}}^{-1}\left[\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}\mathcal{L}\right\{\mathsf{L}\left[{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right]\\ & +\mathsf{R}\left[{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right]-\nu (\varkappa ,\mathsf{t})\left\}\phantom{\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}}\right].\end{array}$(16) ) to the fractional GQCGLE (33(33) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})(\mathrm{U}{)}^2\stackrel{-}{\mathrm{U}}\\ & +(1+4\mathsf{i})\left(\mathrm{U}{)}^3\right(\stackrel{-}{\mathrm{U}}{)}^2,\end{array}$(33) ), we have identified the subsequent recursive relationships: (34) $\begin{array}{rl}& {\mathrm{U}}_{\mathrm{m}+1}(\varkappa ,\mathsf{t},\mathsf{h})\\ & =(1-\mathsf{h}){\mathrm{U}}_{\mathrm{m}}(\varkappa ,\mathsf{t})+\mathsf{h}\mathrm{U}(\varkappa ,0)\\ & -\mathsf{h}{\mathcal{L}}^{-1}\left[\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}\mathcal{L}\right\{(1+\mathsf{i})({\mathrm{U}}_{\mathrm{m}}{)}_{\mathrm{\varkappa \varkappa }}\\ & -3{\mathrm{U}}_{\mathrm{m}}+(1+2\mathsf{i})(\mathrm{U}{)}^2\stackrel{-}{{\mathrm{U}}_{\mathrm{m}}}\\ & +(1+4\mathsf{i})\left({\mathrm{U}}_{\mathrm{m}}{)}^3\right(\stackrel{-}{{\mathrm{U}}_{\mathrm{m}}}{)}^2\left\}\right],\end{array}$(34) the initial approximation is given by: (35) ${\mathrm{U}}_0(\varkappa ,\mathsf{t})=e^{\mathsf{i}\varkappa }.$(35) Setting m = 0, the first approximation solution is: (36) $\begin{array}{rl}& {\mathrm{U}}_1(\varkappa ,\mathsf{t},\mathsf{h})\\ & =(1-\mathsf{h}){\mathrm{U}}_0(\varkappa ,\mathsf{t})+\mathsf{h}\mathrm{U}(\varkappa ,0)\\ & -\mathsf{h}{\mathcal{L}}^{-1}\left[\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}\mathcal{L}\right\{(1+\mathsf{i})({\mathrm{U}}_0{)}_{\mathrm{\varkappa \varkappa }}\\ & -3{\mathrm{U}}_0+(1+2\mathsf{i})(\mathrm{U}{)}^2\stackrel{-}{{\mathrm{U}}_0}+(1+4\mathsf{i}\left)\right({\mathrm{U}}_0{)}^3\left(\stackrel{-}{{\mathrm{U}}_0}{)}^2\right\}].\end{array}$(36) By using Mathematica 11, ${\mathrm{U}}_1(\varkappa ,\mathsf{t},\mathsf{h})$ can be evaluated as: (37) ${\mathrm{U}}_1(\varkappa ,\mathsf{t},\mathsf{h})=e^{\mathsf{i}\varkappa }+\mathsf{h}{e}^{-\mathsf{i}\varkappa }(1-\alpha +\alpha \mathsf{t}).$(37) Setting m = 1, the second approximation solution is: (38) $\begin{array}{rl}& {\mathrm{U}}_2(\varkappa ,\mathsf{t},\mathsf{h})\\ & =(1-\mathsf{h}){\mathrm{U}}_1(\varkappa ,\mathsf{t})+\mathsf{h}\mathrm{U}(\varkappa ,0)\\ & -\mathsf{h}{\mathcal{L}}^{-1}\left[\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}\mathcal{L}\right\{(1+\mathsf{i})({\mathrm{U}}_1{)}_{\mathrm{\varkappa \varkappa }}\\ & -3{\mathrm{U}}_1+(1+2\mathsf{i})(\mathrm{U}{)}^2\stackrel{-}{{\mathrm{U}}_1}+(1+4\mathsf{i}\left)\right({\mathrm{U}}_1{)}^3\left(\stackrel{-}{{\mathrm{U}}_1}{)}^2\right\}].\end{array}$(38) Hence, the function ${\mathrm{U}}_2(\varkappa ,\mathsf{t},\mathsf{h})$ is: (39) $\begin{array}{rl}& {\mathrm{U}}_2(\varkappa ,\mathsf{t},\mathsf{h})\\ & =\mathsf{h}{e}^{\mathsf{i}\varkappa }+(1-\mathsf{h})(e^{\mathsf{i}\varkappa }+\mathsf{h}{e}^{-\mathsf{i}\varkappa }(1-\alpha +\alpha \mathsf{t}\left)\right)\\ & -\mathsf{h}{e}^{-\mathsf{i}\varkappa }(\mathsf{i}\alpha +(4+\mathsf{i}){\mathsf{h}}^5+(1-4\mathsf{i}){\mathsf{h}}^4\phantom{\frac{2}{3}}\\ & +(6+3\mathsf{i}){\mathsf{h}}^3+(3-6\mathsf{i}){\mathsf{h}}^2-2\alpha \mathsf{h}+\mathsf{h}-\mathsf{i}\\ & -(9-18\mathsf{i})\alpha {\mathsf{h}}^2-(24+12\mathsf{i})\alpha {\mathsf{h}}^3\\ & -(5-20\mathsf{i})\alpha {\mathsf{h}}^4-(24+6\mathsf{i})\alpha {\mathsf{h}}^5+{\alpha }^2\mathsf{h}\\ & +(9-18\mathsf{i}){\alpha }^2{\mathsf{h}}^2+(36+18\mathsf{i}){\alpha }^2{\mathsf{h}}^3\\ & +(10-40\mathsf{i}){\alpha }^2{\mathsf{h}}^4+{\alpha }^3-(10-40\mathsf{i}){\alpha }^3{\mathsf{h}}^4\\ & -(24+12\mathsf{i}){\alpha }^3{\mathsf{h}}^3(60+15\mathsf{i}){\alpha }^4{\mathsf{h}}^5\\ & -(80+20\mathsf{i}){\alpha }^3{\mathsf{h}}^5-(1-4\mathsf{i}){\alpha }^5{\mathsf{h}}^4\\ & +(5-20\mathsf{i}){\alpha }^4{\mathsf{h}}^4+(6+3\mathsf{i}){\alpha }^4{\mathsf{h}}^3-(4+\mathsf{i}){\alpha }^6{\mathsf{h}}^5\end{array}$(39) $\begin{array}{rl}& -(24+6\mathsf{i}){\alpha }^5{\mathsf{h}}^5+(\frac{2}{3}+\frac{\mathsf{i}}{6}){\alpha }^6{\mathsf{h}}^5{\mathsf{t}}^6\\ & +\mathsf{t}(-\mathsf{i}\alpha +2\alpha \mathsf{h}+(9-18\mathsf{i})\alpha {\mathsf{h}}^2+(24+12\mathsf{i})\alpha {\mathsf{h}}^3\\ & +\cdots -(24+6\mathsf{i}){\alpha }^6{\mathsf{h}}^5+(120+30\mathsf{i}){\alpha }^5{\mathsf{h}}^5\\ & +(-(240+60\mathsf{i}\left)\right){\alpha }^4{\mathsf{h}}^5+(5-20\mathsf{i}){\alpha }^5{\mathsf{h}}^4)\\ & -(\frac{4}{5}+\frac{\mathsf{i}}{5}){\mathsf{t}}^5(10{\alpha }^6{\mathsf{h}}^5-10{\alpha }^5{\mathsf{h}}^5+\mathsf{i}{\alpha }^5{\mathsf{h}}^4)\\ & +\frac{1}{2}{\mathsf{t}}^2({\alpha }^2\mathsf{h}+(12-24\mathsf{i}){\alpha }^2{\mathsf{h}}^2+(54+27\mathsf{i}){\alpha }^2{\mathsf{h}}^3\\ & +(16-64\mathsf{i}){\alpha }^2{\mathsf{h}}^4+\cdots (100+25\mathsf{i}){\alpha }^6{\mathsf{h}}^5\end{array}$ $\begin{array}{rl}& -(400+100\mathsf{i}){\alpha }^5{\mathsf{h}}^5+(-(16-64\mathsf{i}\left)\right){\alpha }^5{\mathsf{h}}^4)\\ & +\frac{{\mathsf{t}}^4}{4}\left(\right(120+30\mathsf{i}){\alpha }^4{\mathsf{h}}^5-(8-32\mathsf{i}){\alpha }^5{\mathsf{h}}^4\\ & +(8-32\mathsf{i}){\alpha }^4{\mathsf{h}}^4+(6+3\mathsf{i}){\alpha }^4{\mathsf{h}}^3\\ & -(240+60\mathsf{i}){\alpha }^5{\mathsf{h}}^5+(120+30\mathsf{i}){\alpha }^6{\mathsf{h}}^5)\\ & +\frac{{\mathsf{t}}^3}{3}(-(18-72\mathsf{i}){\alpha }^3{\mathsf{h}}^4-(36+18\mathsf{i}){\alpha }^3{\mathsf{h}}^3\\ & +(-3+6\mathsf{i}){\alpha }^3{\mathsf{h}}^2+\cdots +(480+120\mathsf{i}){\alpha }^4{\mathsf{h}}^5\\ & -(18-72\mathsf{i}){\alpha }^5{\mathsf{h}}^4-(160+40\mathsf{i}){\alpha }^6{\mathsf{h}}^5\\ & +(480+120\mathsf{i}){\alpha }^5{\mathsf{h}}^5\left)\phantom{\frac{2}{3}}\right).\end{array}$The approximate solution can be found by continuing the same procedure for various values of m, such as m = 2, 3, and so on. But for now, we will only perform the calculation ${\mathrm{U}}_2(\varkappa ,\mathsf{t},\mathsf{h})$.

Following is the precise solution of Equation (31(31) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})\mathrm{U}|\mathrm{U}{|}^2\\ & +(1-4\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4=0,0<\alpha ⩽1,\end{array}$(31) ) at $\alpha =1$ : (40) $\mathrm{U}(\varkappa ,\mathsf{t})=e^{\mathsf{i}(\varkappa +\mathsf{t})},$(40) which is proposed in [22].

The auxiliary parameter $\mathsf{h}$ indeed plays a crucial role in achieving a series solution that converges, denoted as ${\mathrm{U}}_{\mathrm{m}+1}(\varkappa ,\mathsf{t},\mathsf{h})$, as we increase the value of m. We have drawn the $\mathsf{h}$ curves for several values of alpha at $\varkappa =1$ and $\mathsf{t}=0.2$ in order to identify the region of converge for the solution. Figure 1(e) illustrates that the region of convergence falls within the range of $-0.2\le \mathsf{h}\le 0.6$.

Figure 1. 2D representation of $\mathrm{U}(\varkappa ,\mathsf{t})$ (a) $\mathsf{Re}\left(\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right)$ for Equation (31(31) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})\mathrm{U}|\mathrm{U}{|}^2\\ & +(1-4\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4=0,0<\alpha ⩽1,\end{array}$(31) ) at $\alpha =1,\alpha =0.75,\alpha =0.5$ and $\mathsf{t}=2$. (b)Real part of Equation (31(31) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})\mathrm{U}|\mathrm{U}{|}^2\\ & +(1-4\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4=0,0<\alpha ⩽1,\end{array}$(31) ) at different steps of time and $\alpha =1$. (c) $|\mathrm{Error}|$ for the fractional GQCGLE (31(31) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})\mathrm{U}|\mathrm{U}{|}^2\\ & +(1-4\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4=0,0<\alpha ⩽1,\end{array}$(31) ) at $\mathsf{h}=0.01$ (d) $|\mathrm{Error}|$ for the fractional GQCGLE (31(31) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})\mathrm{U}|\mathrm{U}{|}^2\\ & +(1-4\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4=0,0<\alpha ⩽1,\end{array}$(31) ) at $\mathsf{h}=0.001$ (e) The $\mathsf{h}$ curve of Equation (31(31) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})\mathrm{U}|\mathrm{U}{|}^2\\ & +(1-4\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4=0,0<\alpha ⩽1,\end{array}$(31) ) at $\varkappa =1$ and $\mathsf{t}=0.2$ for different varying α values.

5.2. Example 2

Let's consider the following fractional GQCGLE [23]: (41) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+3\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4\\ & +(1-6\mathsf{i})\mathrm{U}|\mathrm{U}{|}^8=0,0<\alpha ⩽1,\end{array}$(41) subject to the starting guess: (42) $\mathrm{U}(\varkappa ,0)=e^{-\mathsf{i}\varkappa }.$(42) Upon simplification, Equation (41(41) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+3\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4\\ & +(1-6\mathsf{i})\mathrm{U}|\mathrm{U}{|}^8=0,0<\alpha ⩽1,\end{array}$(41) ) can be rewritten as: (43) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+3\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+\mathsf{i})\left(\mathrm{U}{)}^3\right(\stackrel{-}{\mathrm{U}}{)}^2\\ & +(1-6\mathsf{i})\left(\mathrm{U}{)}^5\right(\stackrel{-}{\mathrm{U}}{)}^4,\end{array}$(43) where $|\mathrm{U}{|}^2=\mathrm{U}\stackrel{-}{\mathrm{U}}$, $\stackrel{-}{\mathrm{U}}$ is conjugate of $\mathrm{U}$.

By utilizing the controlled Picard's transform method derived from Equation (16(16) $\begin{array}{rl}& {\mathrm{U}}_{\mathrm{m}+1}(\varkappa ,\mathsf{t},\mathsf{h})\\ & =(1-\mathsf{h}){\mathrm{U}}_{\mathrm{m}}(\varkappa ,\mathsf{t})+\mathsf{h}{\Psi }_0\left(\varkappa \right)\\ & -\mathsf{h}{\mathcal{L}}^{-1}\left[\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}\mathcal{L}\right\{\mathsf{L}\left[{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right]\\ & +\mathsf{R}\left[{\mathrm{U}}_{\mathrm{m}}\right(\varkappa ,\mathsf{t}\left)\right]-\nu (\varkappa ,\mathsf{t})\left\}\phantom{\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}}\right].\end{array}$(16) ) on the fractional GQCGL Equation (43(43) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+3\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+\mathsf{i})\left(\mathrm{U}{)}^3\right(\stackrel{-}{\mathrm{U}}{)}^2\\ & +(1-6\mathsf{i})\left(\mathrm{U}{)}^5\right(\stackrel{-}{\mathrm{U}}{)}^4,\end{array}$(43) ), we derived the subsequent recurrent relationships. (44) $\begin{array}{rl}& {\mathrm{U}}_{\mathrm{m}+1}(\varkappa ,\mathsf{t},\mathsf{h})\\ & =(1-\mathsf{h}){\mathrm{U}}_{\mathrm{m}}(\varkappa ,\mathsf{t})+\mathsf{h}\mathrm{U}(\varkappa ,0)\\ & -\mathsf{h}{\mathcal{L}}^{-1}\left[\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}\mathcal{L}\right\{(1+3\mathsf{i})({\mathrm{U}}_{\mathrm{m}}{)}_{\mathrm{\varkappa \varkappa }}\\ & -3{\mathrm{U}}_{\mathrm{m}}+(1+\mathsf{i})\left(\mathrm{U}{)}^3\right(\stackrel{-}{{\mathrm{U}}_{\mathrm{m}}}{)}^2\\ & +(1-6\mathsf{i})\left({\mathrm{U}}_{\mathrm{m}}{)}^5\right(\stackrel{-}{{\mathrm{U}}_{\mathrm{m}}}{)}^4\left\}\right].\end{array}$(44) The initial approximation in this case is provided by: (45) ${\mathrm{U}}_0(\varkappa ,\mathsf{t})=e^{-\mathsf{i}\varkappa }.$(45) Setting m = 0, the system described in Equation (44(44) $\begin{array}{rl}& {\mathrm{U}}_{\mathrm{m}+1}(\varkappa ,\mathsf{t},\mathsf{h})\\ & =(1-\mathsf{h}){\mathrm{U}}_{\mathrm{m}}(\varkappa ,\mathsf{t})+\mathsf{h}\mathrm{U}(\varkappa ,0)\\ & -\mathsf{h}{\mathcal{L}}^{-1}\left[\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}\mathcal{L}\right\{(1+3\mathsf{i})({\mathrm{U}}_{\mathrm{m}}{)}_{\mathrm{\varkappa \varkappa }}\\ & -3{\mathrm{U}}_{\mathrm{m}}+(1+\mathsf{i})\left(\mathrm{U}{)}^3\right(\stackrel{-}{{\mathrm{U}}_{\mathrm{m}}}{)}^2\\ & +(1-6\mathsf{i})\left({\mathrm{U}}_{\mathrm{m}}{)}^5\right(\stackrel{-}{{\mathrm{U}}_{\mathrm{m}}}{)}^4\left\}\right].\end{array}$(44) ) simplifies to the following form: (46) $\begin{array}{rl}& {\mathrm{U}}_1(\varkappa ,\mathsf{t},\mathsf{h})\\ & =(1-\mathsf{h}){\mathsf{U}}_{\mathsf{0}}(\varkappa ,\mathsf{t})+\mathsf{h}\mathrm{U}(\varkappa ,0)\\ & -\mathsf{h}{\mathcal{L}}^{-1}\left[\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}\mathcal{L}\right\{(1+3\mathsf{i})({\mathrm{U}}_0{)}_{\mathrm{\varkappa \varkappa }}\\ & -3{\mathrm{U}}_0+(1+\mathsf{i})\left(\mathrm{U}{)}^3\right(\stackrel{-}{{\mathrm{U}}_0}{)}^2\\ & +(1-6\mathsf{i})\left({\mathrm{U}}_0{)}^5\right(\stackrel{-}{{\mathrm{U}}_0}{)}^4\left\}\right].\end{array}$(46) By utilizing Mathematica 11, we can evaluate ${\mathrm{U}}_1(\varkappa ,\mathsf{t},\mathsf{h})$ as: (47) ${\mathrm{U}}_1(\varkappa ,\mathsf{t},\mathsf{h})=e^{-\mathsf{i}\varkappa }+2i\mathsf{h}{e}^{-\mathsf{i}\varkappa }(1-\alpha +\alpha \mathsf{t}).$(47) Setting m = 1, the second approximation solution will be: (48) $\begin{array}{rl}& {\mathrm{U}}_2(\varkappa ,\mathsf{t},\mathsf{h})\\ & =(1-\mathsf{h}){\mathrm{U}}_1(\varkappa ,\mathsf{t})+\mathsf{h}\mathrm{U}(\varkappa ,0)\\ & -\mathsf{h}{\mathcal{L}}^{-1}\left[\frac{\mathsf{s}+(1-\alpha )\mathsf{s}}{\mathsf{s}}\mathcal{L}\right\{(1+3\mathsf{i})({\mathrm{U}}_1{)}_{\mathrm{\varkappa \varkappa }}\\ & -3{\mathrm{U}}_1+(1+\mathsf{i})\left(\mathrm{U}{)}^3\right(\stackrel{-}{{\mathrm{U}}_1}{)}^2\\ & +(1-6\mathsf{i})\left({\mathrm{U}}_1{)}^5\right(\stackrel{-}{{\mathrm{U}}_1}{)}^4\left\}\right].\end{array}$(48) Then, the value of ${\mathrm{U}}_2(\varkappa ,\mathsf{t},\mathsf{h})$ is as follows: (49) $\begin{array}{rl}& {\mathrm{U}}_2(\varkappa ,\mathsf{t},\mathsf{h})\\ & =\mathsf{h}{e}^{\mathsf{i}\varkappa }+(1-\mathsf{h})(e^{-\mathsf{i}\varkappa }+2\mathsf{i}\mathsf{h}{e}^{-\mathsf{i}\varkappa }(1-\alpha +\alpha \mathsf{t}\left)\right)\\ & -\mathsf{h}{e}^{-\mathsf{i}\varkappa }((\frac{1536}{5}+\frac{256\mathsf{i}}{5}){\alpha }^{10}{\mathsf{h}}^9{\mathsf{t}}^{10}\\ & -(\frac{512}{3}+\frac{256\mathsf{i}}{9}){\mathsf{t}}^9(36{\alpha }^{10}{\mathsf{h}}^9-36{\alpha }^9{\mathsf{h}}^9+\mathsf{i}{\alpha }^9{\mathsf{h}}^8)\\ & +(384+64\mathsf{i}){\mathsf{t}}^8({\alpha }^8{\mathsf{h}}^7-8\mathsf{i}{\alpha }^8{\mathsf{h}}^8\\ & +108{\alpha }^8{\mathsf{h}}^9+108{\alpha }^{10}{\mathsf{h}}^9-216{\alpha }^9{\mathsf{h}}^9+8\mathsf{i}{\alpha }^9{\mathsf{h}}^8)\\ & -(\frac{1536}{7}+\frac{256\mathsf{i}}{7}){\mathsf{t}}^7(84\mathsf{i}{\alpha }^7{\mathsf{h}}^8-28{\alpha }^7{\mathsf{h}}^7\\ & +\mathsf{i}{\alpha }^7{\mathsf{h}}^{6}672{\alpha }^{10}{\mathsf{h}}^9-2016{\alpha }^9{\mathsf{h}}^9+2016{\alpha }^8{\mathsf{h}}^9\\ & -672{\alpha }^7{\mathsf{h}}^9+84\mathsf{i}{\alpha }^9{\mathsf{h}}^8-168\mathsf{i}{\alpha }^8{\mathsf{h}}^8+28{\alpha }^8{\mathsf{h}}^7)\\ & +2(-\mathsf{i}+2\mathsf{h}+(12-44\mathsf{i}){\mathsf{h}}^2+(88+24\mathsf{i}){\mathsf{h}}^3\end{array}$(49) $\begin{array}{rl}& +(56-280\mathsf{i}){\mathsf{h}}^4+(560+112\mathsf{i}){\mathsf{h}}^5\\ & +(128-768\mathsf{i})\cdots +(1536+256\mathsf{i}){\alpha }^{10}{\mathsf{h}}^9\\ & -(15360+2560\mathsf{i}){\alpha }^9{\mathsf{h}}^9+\left(\right(69120\\ & +11520\mathsf{i}\left)\right){\alpha }^8{\mathsf{h}}^9-(128-768\mathsf{i}){\alpha }^9{\mathsf{h}}^8)\\ & -2t(\mathsf{i}\alpha -(280-1400\mathsf{i})\alpha {\mathsf{h}}^4-(352+96\mathsf{i})\alpha {\mathsf{h}}^3\\ & -(36-132\mathsf{i})\alpha {\mathsf{h}}^2-4\alpha \mathsf{h}+\mathsf{i}\alpha +\cdots \\ & -(15360+2560\mathsf{i})\alpha {\mathsf{h}}^9+4{\alpha }^2\mathsf{h}\\ & +(72-264\mathsf{i}){\alpha }^2{\mathsf{h}}^2\\ & +(1056+288\mathsf{i}){\alpha }^2{\mathsf{h}}^3+(1120-5600\mathsf{i}){\alpha }^2{\mathsf{h}}^4\\ & +(16800+3360\mathsf{i}){\alpha }^2{\mathsf{h}}^5+(5376-32256\mathsf{i})\cdots \end{array}$ $\begin{array}{rl}& +(9216-55296\mathsf{i}){\alpha }^8{\mathsf{h}}^8+(15360\\ & +2560\mathsf{i}){\alpha }^{10}{\mathsf{h}}^9-(138240+23040\mathsf{i}\left){\alpha }^9{\mathsf{h}}^9\right)\\ & +(\frac{16}{3}-\frac{16i}{3}){\mathsf{t}}^6\left(\right(14+21\mathsf{i}){\alpha }^6{\mathsf{h}}^5\\ & +(336-240\mathsf{i}){\alpha }^6{\mathsf{h}}^6+(2520+3528\mathsf{i}){\alpha }^6{\mathsf{h}}^7\\ & +(6272-4480\mathsf{i}){\alpha }^6{\mathsf{h}}^8+\cdots +(25200\\ & +35280\mathsf{i}){\alpha }^{10}{\mathsf{h}}^9-(100800+141120\mathsf{i}\left){\alpha }^9{\mathsf{h}}^9\right)\\ & -(\frac{8}{3}-\frac{8\mathsf{i}}{3}){\mathsf{t}}^3\left(\right(756-504\mathsf{i}){\alpha }^3{\mathsf{h}}^4\\ & -(48+84\mathsf{i}){\alpha }^3{\mathsf{h}}^3+(-7+4i){\alpha }^3{\mathsf{h}}^2\\ & -(2240+3360i)\cdots -(46080+64512\mathsf{i}){\alpha }^{10}{\mathsf{h}}^9\end{array}$ $\begin{array}{rl}& -(322560+451584\mathsf{i}){\alpha }^9{\mathsf{h}}^9\\ & +(-(21952-15680\mathsf{i}\left)\right){\alpha }^9{\mathsf{h}}^8)\\ & +(4-4\mathsf{i}){\mathsf{t}}^4\left(\right(4+7\mathsf{i}){\alpha }^4{\mathsf{h}}^3+(168-112\mathsf{i}){\alpha }^4{\mathsf{h}}^4\\ & +(840+1260\mathsf{i}){\alpha }^4{\mathsf{h}}^5\\ & +\cdots (47040+65856\mathsf{i}){\alpha }^{10}{\mathsf{h}}^9-(282240\\ & +395136\mathsf{i}){\alpha }^9{\mathsf{h}}^9+(-(18816-13440\mathsf{i})\left){\alpha }^9{\mathsf{h}}^8\right)\\ & -(\frac{16}{5}-\frac{16\mathsf{i}}{5}){\mathsf{t}}^5(-(2520-1800\mathsf{i}){\alpha }^5{\mathsf{h}}^6\\ & -(280+420\mathsf{i}){\alpha }^5{\mathsf{h}}^5+(-21+14\mathsf{i}){\alpha }^5{\mathsf{h}}^4\\ & -(60480+84672\mathsf{i}){\alpha }^{10}{\mathsf{h}}^9-(302400\\ & +423360\mathsf{i}\left){\alpha }^9{\mathsf{h}}^9\right)+2{\mathsf{t}}^2({\alpha }^2\mathsf{h}+(24-88\mathsf{i}){\alpha }^2{\mathsf{h}}^2\\ & +\cdots -(62208+10368\mathsf{i}){\alpha }^{10}{\mathsf{h}}^9\\ & -(497664+82944\mathsf{i}){\alpha }^9{\mathsf{h}}^9)\end{array}$By following a similar approach, we can determine an estimated solution for various values of m, including $m=2,3,\dots$. But for now, our focus will be exclusively on computing ${\mathrm{U}}_2(\varkappa ,\mathsf{t},\mathsf{h})$ due to huge calculations.

Following is the precise solution of Equation (41(41) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+3\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4\\ & +(1-6\mathsf{i})\mathrm{U}|\mathrm{U}{|}^8=0,0<\alpha ⩽1,\end{array}$(41) ) at $\alpha =1$ : (50) $\mathrm{U}(\varkappa ,\mathsf{t})=e^{\mathsf{i}(-\varkappa +2\mathsf{t})},$(50) which is proposed in [22].

To determine the region of convergence for the solution, we have generated $\mathsf{h}$ curves for different α values at $\varkappa =1$ and $\mathsf{t}=0.2$. Figure 3(e) illustrates that the region of convergence is within the range of $-0.8\le \mathsf{h}\le 0.9$.

6. Results and discussion

Due to the intricate and complex nature of real-world issues in fields such as science and engineering, solving corresponding equations or systems was a significant challenge. This study focussed on a specific complex model, the fractional GQCGLE, which exhibited non-equilibrium behaviour and explored the effects of variable orders on the equation's solution. To ensure the accuracy of our analysis, we utilized a reliable and precise method called cntrolled Picard's tansform technique. To validate our approach, we examined two specific cases involving fractional orders in the Caputo-Fabrizio fractional derivative. Through our analysis, we assessed and illustrated the results obtained using the applied solution methodology. We illustrated the approximations of (31(31) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})\mathrm{U}|\mathrm{U}{|}^2\\ & +(1-4\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4=0,0<\alpha ⩽1,\end{array}$(31) ) in Figure 1(a) to prove the effectiveness and feasibility of the suggested approach. This analysis is conducted within the range of $-10\le \varkappa \le 10$ with a fixed time value of $\mathsf{t}=2$, while considering various fractional values of α. Specifically, we investigated cases where α takes on the values of $\alpha =1,0.75,$ and $0.5$, all while keeping $\vartheta =-4,\varrho =1,\delta =2,\xi =-3$, and $\mathsf{h}=0.01$. It should be emphasized that the approximations exhibit a close correspondence in terms of fractional values. Moreover, the accuracy of the results improves as the fractional values approach the conventional case of $\alpha =1$. In addition, as depicted in Figure 1(b), we examined the behaviour of approximate periodic wave solutions for various values of $\mathsf{t}$ within the two-dimensional domain defined by (31(31) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})\mathrm{U}|\mathrm{U}{|}^2\\ & +(1-4\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4=0,0<\alpha ⩽1,\end{array}$(31) ). This visualization provides insight into the system's behaviour for varying $\mathsf{t}$ values. The discrepancies between the precise solution (31(31) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})\mathrm{U}|\mathrm{U}{|}^2\\ & +(1-4\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4=0,0<\alpha ⩽1,\end{array}$(31) ) and the suggested method for various $\mathsf{h}$ values and $\alpha =1$ are depicted in Figure 1(c,d), demonstrating the method's convergence ability. Figure 1(e) displayed the $\mathsf{h}$ curves for the periodical wave solution to problem (31(31) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})\mathrm{U}|\mathrm{U}{|}^2\\ & +(1-4\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4=0,0<\alpha ⩽1,\end{array}$(31) ) at $\varkappa =1$ and $\mathsf{t}=0.2$, with different values of α. The graphic indicated that the correct range for $\mathsf{h}\in [-0.1,0.6]$. Moreover, Figure 2(a,b) depicted 3D graphs in the coordinate space, presented both the real and imaginary components of periodic approximate solutions for Equation (31(31) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})\mathrm{U}|\mathrm{U}{|}^2\\ & +(1-4\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4=0,0<\alpha ⩽1,\end{array}$(31) ). These visualizations are generated over the domain of $-10\le \varkappa \le 10$ and $0\le \mathsf{t}\le 1$, considering different values of the fractional parameter $\alpha =0.75,1$. The primary aim of these graphical representations is to highlight the level of resemblance between the precise and approximate solutions, as illustrated in Figure 2(c). Likewise, in Figure 3, we presented the solutions derived from approximating the fractional GQCGL Equation (41(41) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+3\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4\\ & +(1-6\mathsf{i})\mathrm{U}|\mathrm{U}{|}^8=0,0<\alpha ⩽1,\end{array}$(41) ), considering various values of both α and $\mathsf{t}$. Figure 3(a) displayed the approximate solutions of $\mathrm{U}(\varkappa ,\mathsf{t})$ at $\vartheta =-6,\varrho =3,\delta =1,\xi =-3,\mathsf{t}=2$, and $\mathsf{h}=0.01$ for different values of α. It also illustrated the extent of convergence in the solutions at these specified conditions. Additionally, Figure 3(b) showcases the approximate solutions under specific parameter values, as described previously, when $\alpha =1$ and for varying values of $\mathsf{t}$. The proposed method's convergence ability is demonstrated by the discrepancies between the accurate solution (41(41) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+3\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4\\ & +(1-6\mathsf{i})\mathrm{U}|\mathrm{U}{|}^8=0,0<\alpha ⩽1,\end{array}$(41) ) and the results of the method for different $\mathsf{h}$ values, as illustrated in Figure 3(c,d). Furthermore, the plots in Figure 3(e) presented the $\mathsf{h}$ curve for the periodic wave solution of (41(41) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+3\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4\\ & +(1-6\mathsf{i})\mathrm{U}|\mathrm{U}{|}^8=0,0<\alpha ⩽1,\end{array}$(41) ) at $\varkappa =1$ and $\mathsf{t}=0.2$, while considering different values of α. Upon analyzing the graph, it can be observed that the appropriate range for $\mathsf{h}$ is within the interval $[-0.8,0.9]$. At the end, Figure 4(a,b) displayed 3D graphs in the coordinate space that show both the real and imaginary components of periodic approximate solutions for Equation (41(41) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+3\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4\\ & +(1-6\mathsf{i})\mathrm{U}|\mathrm{U}{|}^8=0,0<\alpha ⩽1,\end{array}$(41) ). These visualizations are generated over the range of $-10\le \varkappa \le 10$ and $0\le \mathsf{t}\le 1$, using different values of the fractional parameter $\alpha =0.75,1$. The main purpose of these graphical representations was to assess the resemblance between the exact and estimated solutions, as shown in Figure 4(c).

Figure 2. 3D visualization of real and imaginary parts of the fractional GQCGLE (31(31) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})\mathrm{U}|\mathrm{U}{|}^2\\ & +(1-4\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4=0,0<\alpha ⩽1,\end{array}$(31) ) at $\vartheta =-4,\varrho =1,\delta =2$ and $\xi =-3$. (a) Real and imaginary parts for Equation (31(31) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})\mathrm{U}|\mathrm{U}{|}^2\\ & +(1-4\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4=0,0<\alpha ⩽1,\end{array}$(31) ) at $\alpha =0.75,\mathsf{h}=0.01$. (b) Real and imaginary parts for Equation (31(31) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})\mathrm{U}|\mathrm{U}{|}^2\\ & +(1-4\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4=0,0<\alpha ⩽1,\end{array}$(31) ) at $\alpha =1$ and $\mathsf{h}=0.01$. (c) The exact solutions of $\mathsf{Re}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]$ and $\mathsf{Im}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]$.

Figure 3. 2D displays for real components of the solution Equation (41(41) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+3\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4\\ & +(1-6\mathsf{i})\mathrm{U}|\mathrm{U}{|}^8=0,0<\alpha ⩽1,\end{array}$(41) ). (a) $\mathsf{Re}\left(\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right)$ for Equation (41(41) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+3\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4\\ & +(1-6\mathsf{i})\mathrm{U}|\mathrm{U}{|}^8=0,0<\alpha ⩽1,\end{array}$(41) ) at $\alpha =1,\alpha =0.75,\alpha =0.5$ and $\mathsf{t}=5$. (b) $\mathsf{Re}\left(\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right)$ for Equation (41(41) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+3\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4\\ & +(1-6\mathsf{i})\mathrm{U}|\mathrm{U}{|}^8=0,0<\alpha ⩽1,\end{array}$(41) ) at several stages of time and $\alpha =1$. (c) $|\mathrm{Error}|$ for the fractional GQCGLE (41(41) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+3\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4\\ & +(1-6\mathsf{i})\mathrm{U}|\mathrm{U}{|}^8=0,0<\alpha ⩽1,\end{array}$(41) ) at $\mathsf{h}=0.01$ (d) $|\mathrm{Error}|$ for the fractional GQCGLE (41(41) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+3\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4\\ & +(1-6\mathsf{i})\mathrm{U}|\mathrm{U}{|}^8=0,0<\alpha ⩽1,\end{array}$(41) ) at $\mathsf{h}=0.001$ (e) The $\mathsf{h}$ curve of Equation (41(41) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+3\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4\\ & +(1-6\mathsf{i})\mathrm{U}|\mathrm{U}{|}^8=0,0<\alpha ⩽1,\end{array}$(41) ) at $\varkappa =1$ and $\mathsf{t}=0.2$ for varying α values.

Figure 4. 3D displays for real and imaginary parts for the fractional GQCGLE (41(41) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+3\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4\\ & +(1-6\mathsf{i})\mathrm{U}|\mathrm{U}{|}^8=0,0<\alpha ⩽1,\end{array}$(41) ) at $\vartheta =-6,\varrho =3,\delta =1$ and $\xi =-3$. (a) Real and imaginary parts for Equation (41(41) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+3\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4\\ & +(1-6\mathsf{i})\mathrm{U}|\mathrm{U}{|}^8=0,0<\alpha ⩽1,\end{array}$(41) ) at $\alpha =0.75,\mathsf{h}=0.01$. (b) Real and imaginary parts for Equation (41(41) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+3\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4\\ & +(1-6\mathsf{i})\mathrm{U}|\mathrm{U}{|}^8=0,0<\alpha ⩽1,\end{array}$(41) ) at $\alpha =1$ and $\mathsf{h}=0.01$. (c) The exact solutions of $\mathsf{Re}\left[\mathrm{U}\right(\varkappa ,\mathsf{t})$] and $\mathsf{Im}\left[\mathrm{U}\right(\varkappa ,\mathsf{t}\left)\right]$.

Table 1 illustrates the errors of the current method used in Equation (31(31) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})\mathrm{U}|\mathrm{U}{|}^2\\ & +(1-4\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4=0,0<\alpha ⩽1,\end{array}$(31) ) when $\varkappa =5,\alpha =1$, and various values for $\mathsf{t}$. The table presents the accuracy of the method at different values of $\mathsf{h}$, where it was observed that the error decreases as the value of $\mathsf{h}$ decreases below $0.1$. Table 2 compares the absolute error between the method proposed for solving Equation (31(31) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})\mathrm{U}|\mathrm{U}{|}^2\\ & +(1-4\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4=0,0<\alpha ⩽1,\end{array}$(31) ) and the homotopy analysis method (HAM) presented in [35] at the same parameters $\vartheta =-4,\varrho =1,\delta =2,\xi =-3,\varkappa =1$ and $\alpha =1$ for $\mathsf{h}=0.1\times {10}^{-6}$. The results demonstrate the efficiency of the suggested approach and its similarity to the other method. Furthermore, Table 3 displays the inaccuracies associated with the current technique when applied to Equation (41(41) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+3\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4\\ & +(1-6\mathsf{i})\mathrm{U}|\mathrm{U}{|}^8=0,0<\alpha ⩽1,\end{array}$(41) ) under the conditions of $\varkappa =5$, $\alpha =1$, and various steps of time $\mathsf{t}$. The table illustrates the method's precision across a range of $\mathsf{h}$ values, revealing a noticeable decrease in error as $\mathsf{h}$ decreases below 0.1. Notably, the results are remarkably consistent, indicating the efficiency of the method used. The precision of the present approach is contingent on the actual value of the optimal parameter, represented as $\mathsf{h}$. Notably, as the actual value of $\mathsf{h}$ falls within the convergence region, the accuracy of the method increases. This technique is broadly effective and can be seamlessly applied to a vast array of intricate FDEs, particularly those with initial conditions. Additionally, it provides a comprehensive framework that can be tailored to diverse physical systems.

Table 1. $|\mathrm{Error}|$ for the fractional-GQCGLE (31(31) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})\mathrm{U}|\mathrm{U}{|}^2\\ & +(1-4\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4=0,0<\alpha ⩽1,\end{array}$(31) ) at $\vartheta =-4,\varrho =1,\delta =2,\xi =-3,\alpha =1$ and $\varkappa =5$.

$\mathsf{t}$	$|\mathrm{Error}|$ at $\mathsf{h}=0.1$	$|\mathrm{Error}|$ at $\mathsf{h}=0.01$	$|\mathrm{Error}|$ at $\mathsf{h}=0.001$
0.1	1.29516$\times {10}^{-4}$	1.47905$\times {10}^{-6}$	1.49790$\times {10}^{-8}$
0.2	5.14252$\times {10}^{-4}$	5.91222$\times {10}^{-6}$	5.99120$\times {10}^{-8}$
0.3	1.14878$\times {10}^{-3}$	1.32936$\times {10}^{-5}$	1.34793$\times {10}^{-7}$
0.4	2.02805$\times {10}^{-3}$	2.36172$\times {10}^{-5}$	2.39616$\times {10}^{-7}$
0.5	3.14742$\times {10}^{-3}$	3.68772$\times {10}^{-5}$	3.74375$\times {10}^{-7}$

Table 2. Comparison of $|\mathrm{Error}|$ for the fractional-GQCGLE (31(31) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})\mathrm{U}|\mathrm{U}{|}^2\\ & +(1-4\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4=0,0<\alpha ⩽1,\end{array}$(31) ) using the suggested approach and HAM described in [35] at $\vartheta =-4,\varrho =1,\delta =2,\xi =-3$, $\mathsf{h}=0.1\times {10}^{-6},\varkappa =1$ and $\alpha =1$.

$\mathsf{t}$	$|\mathrm{Error}|$ controlled Picard method	$|\mathrm{Error}|$ HAM in [35]
0.2	6.66134$\times {10}^{-16}$	1.11020$\times {10}^{-16}$
0.4	2.44249$\times {10}^{-15}$	1.11020$\times {10}^{-16}$
0.6	5.66214$\times {10}^{-15}$	1.49830$\times {10}^{-16}$
0.8	9.76996$\times {10}^{-15}$	1.48950$\times {10}^{-15}$
1	1.50990$\times {10}^{-14}$	4.77440$\times {10}^{-14}$

Table 3. $|\mathrm{Error}|$ for the fractional-GQCGLE (41(41) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+3\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4\\ & +(1-6\mathsf{i})\mathrm{U}|\mathrm{U}{|}^8=0,0<\alpha ⩽1,\end{array}$(41) ) at $\vartheta =-4,\varrho =1,\delta =2,\xi =-3,\alpha =1$ and $\varkappa =5$.

$\mathsf{t}$	$|\mathrm{Error}|$ at $\mathsf{h}=0.1$	$|\mathrm{Error}|$ at $\mathsf{h}=0.01$	$|\mathrm{Error}|$ at $\mathsf{h}=0.001$
0.1	5.14558$\times {10}^{-4}$	5.91226$\times {10}^{-6}$	5.99120$\times {10}^{-8}$
0.2	2.03293$\times {10}^{-3}$	2.36177$\times {10}^{-5}$	2.39616$\times {10}^{-7}$
0.3	4.52733$\times {10}^{-3}$	5.30704$\times {10}^{-5}$	5.39065$\times {10}^{-7}$
0.4	7.98388$\times {10}^{-3}$	9.42258$\times {10}^{-5}$	9.58210$\times {10}^{-7}$
0.5	1.24034$\times {10}^{-2}$	1.47041$\times {10}^{-4}$	1.49700$\times {10}^{-6}$

Table 1 presents the results of the absolute error obtained while solving the fractional GQCGLE (31(31) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+2\mathsf{i})\mathrm{U}|\mathrm{U}{|}^2\\ & +(1-4\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4=0,0<\alpha ⩽1,\end{array}$(31) ) for different values of $\mathsf{h}$ at $\varkappa =2$, $\vartheta =-4$, $\varrho =1$, $\delta =2$, $\xi =-3,\varkappa =5$, and $\alpha =1$ for various values of $\mathsf{t}$.

The results in Table 3 illustrate the accuracy of the solution we obtained compared with the solutions using HAM presented in reference [35].

Table 3 displays the absolute error outcomes obtained when solving the the fractional-GQCGLE (41(41) $\begin{array}{rl}& {\mathsf{D}}_{\mathsf{t}}^{\alpha }\mathrm{U}\left(\varkappa ,\mathsf{t}\right)-(1+3\mathsf{i}){\mathrm{U}}_{\mathrm{\varkappa \varkappa }}-3\mathrm{U}+(1+\mathsf{i})\mathrm{U}|\mathrm{U}{|}^4\\ & +(1-6\mathsf{i})\mathrm{U}|\mathrm{U}{|}^8=0,0<\alpha ⩽1,\end{array}$(41) ) for various values of $\mathsf{h}$, where $\varkappa =5$ takes the values $\vartheta =-6,\varrho =3,\delta =1,\xi =-3$ and $\alpha =1$, along with different values of $\mathsf{t}$.

7. Conclusion

In this paper, a recent approach called the controlled Picard technique was introduced, along with the LT, to approximate the solution of the fractional-GQCGLE, which had applications in various fields such as quantum field theory, weakly complex dispersive water waves, and dynamic optical technology. The suggested approach combined the advantages of the CPM and the LT to solve nonlinear fractional problems. Unlike traditional methods that relied on Lagrange multipliers or Adomian polynomials, the proposed approach offered a highly efficient, accurate, and straightforward solution methodology. The graphs of the solution and the tables that presented the absolute error clarified the accuracy of the obtained solutions.

The proposed approach provided a valuable tool for solving nonlinear fractional equations in various fields of study and opened up new possibilities for further research in the area of fractional differential equations in quantum mechanics (for example, fractional Klein-Gordon equation), fluid dynamics (for example, fractional Navier-Stokes equation and fractional Cahn-Hilliard equation for phase separation in fluids), and optics (for example, fractional nonlinear Schrödinger equation used in optical solitons and fractional Helmholtz equation describing wave propagation in optical systems.

Acknowledgements

All authors thank the editor-in-chief of the journal and those responsible for it. The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability

The datasets used and/or analyzed during the current study are available from the corresponding author upon reasonable request.