Solitons unveilings and modulation instability analysis for sixth-order coupled nonlinear Schrödinger equations in fiber bragg gratings

Noha M. Kamel; Hamdy M. Ahmed; Wafaa B. Rabie; Noha M. Kamel; Hamdy M. Ahmed; Wafaa B. Rabie

doi:10.3934/math.2025318

AIMS Mathematics

2025, Volume 10, Issue 3: 6952-6980. doi: 10.3934/math.2025318

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Research article Special Issues

Solitons unveilings and modulation instability analysis for sixth-order coupled nonlinear Schrödinger equations in fiber bragg gratings

1.
Department of Physics and Engineering Mathematics, Faculty of engineering, Ain Shams University, Cairo, Egypt
2.
Department of Physics and Engineering Mathematics, Higher Institute of Engineering, El Shorouk Academy, Cairo, Egypt
3.
Department of Basic Sciences, Higher Institute of Engineering and Technology, Menoufia, Egypt

Received: 24 January 2025 Revised: 16 March 2025 Accepted: 21 March 2025 Published: 26 March 2025
MSC : 35C07, 35C08, 35C09

This work investigated analytical solutions for a coupled system of nonlinear perturbed Schrödinger equations in fiber Bragg gratings (FBGs), characterized by sixth-order dispersion and a combination of Kerr and parabolic nonlocal nonlinear refractive indices. Chromatic dispersion, which restricts wave propagation in standard optical fibers, was effectively compensated using FBGs, making them indispensable in modern optical networks. In this study, the modified Sardar sub-equation technique (MSSE) was applied to the system for the first time. This method was chosen for its advantages, including low computational cost, high consistency, and simplicity in calculations. The novelty of this work lied in the derivation of new analytical solutions, such as exponential, singular periodic, hyperbolic, and rational solutions, which have not been previously reported in the literature. Additionally, bright gap solitons and singular gap solitons, previously studied, were also obtained. All solutions were rigorously verified by direct substitution into the system. Another significant contribution of this work was the derivation of modulation instability (MI) analysis using linear stability analysis. For the first time in the literature, an analytical expression for the MI gain spectrum was derived. This gain spectrum depended on key parameters such as normalized power, perturbation wave number, dispersion coefficients, phase modulation coefficients, and nonlinearity coefficients. The study also included 2D and 3D graphical representations of selected exact solutions, with parameters chosen to satisfy specific limiting conditions, as well as visual illustrations of the MI gain spectrum. The solutions derived in this work have profound implications for optical communication systems. Exponential and hyperbolic solutions can model pulse propagation in FBGs with high accuracy, enabling better design of dispersion-compensating devices and improving signal integrity over long distances. Singular periodic and rational solutions provided insights into the behavior of nonlinear waves in FBGs, which can be exploited for advanced signal processing applications, such as pulse shaping and wavelength conversion. Bright and singular-gap solitons were crucial for maintaining stable signal transmission in FBG-based systems, particularly in high-power scenarios where nonlinear effects were significant. The MI analysis further enhanced the practical relevance of this work. By understanding the conditions under which MI occured, engineers can design FBG systems that minimize signal degradation and optimize performance. The MI gain spectrum provided a tool to predict and control instability, ensuring robust and efficient optical communication networks. This work not only advanced the theoretical understanding of nonlinear wave dynamics in FBGs but also offered practical tools and solutions for improving optical communication systems. The derived solutions and MI analysis have direct applications in enhancing signal stability, dispersion management, and overall system performance, making this research highly relevant to the field of photonics and optical engineering.

Keywords:

perturbed nonlinear Schrödinger equations,
fiber Bragg gratings,
modulation instability analysis,
Kerr nonlinearity,
gap solitons

Citation: Noha M. Kamel, Hamdy M. Ahmed, Wafaa B. Rabie. Solitons unveilings and modulation instability analysis for sixth-order coupled nonlinear Schrödinger equations in fiber bragg gratings[J]. AIMS Mathematics, 2025, 10(3): 6952-6980. doi: 10.3934/math.2025318

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Abstract

1. Introduction

The ordinary differential equations (ODEs) and partial differential equations (PDEs) are widely used to represent physical phenomena in mathematical language in science and technology. The mathematical form of the physical phenomena easily explains the whole scenario of the phenomena and makes them openly understandable and investigated straightforwardly. Initially, these phenomena were not only modeled accurately by using integer-order differential equations but later on fractional differential equations have over come the deficiencies and comparatively provide the best and adequate modelling of the given problems. Fractional order ODEs and PDEs describe some phenomena more accurately than non-fractional order ODES and PDEs and have numerous applications in applied sciences. It is shown that the fractional-order ODEs and PDEs are non-local and imply that the next state of a system depends on its current state and its previous states. Therefore, the fractional-order derivatives and integration have numerous applications such as the nonlinear oscillation of earth quack is molded with fractional-order derivatives ^[1], chaos theory ^[2], fractional diabetes model ^[3], fractional order Covid-19 Model ^[4], optics ^[5], fractional model of cancer chemotherapy ^[6], effect of fractional order on ferromagnetic fluid ^[7], the fractional-order fluid dynamic traffic model ^[2], signal processing phenomena ^[9], electrodynamics ^[10], fractional model for the dynamics of Hepatitis B Virus ^[11], fractional model for tuberculosis ^[12], fractional-order pine wilt disease model ^[13], and some others references therein ^{[14,15,16,17,18]}.

The Korteweg-De Vries (KdV) equation is a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an exactly solvable model that is, a non-linear partial differential equation whose solutions can be exactly and precisely specified. KdV can be solved by means of the inverse scattering transform. The mathematical theory behind the KdV equation is a topic of active research. The KdV equation was first introduced by Boussinesq (1877) and rediscovered by Diederik Korteweg and Gustav de Vries (1895) ^[19].

Among these applications, we have considered Fokker-Planck equations of fractional order of the general from

$\begin{equation} \psi_t^{\delta}(\vartheta, \tau) = L\left(\psi_{\vartheta} (\vartheta, \tau)+\psi_{\vartheta \vartheta} (\vartheta, \tau)\right)+N \psi_{\vartheta \vartheta}(\vartheta, \tau), \ \ \vartheta, \tau > 0, \ \delta \in (0, 1], \end{equation}$

(1.1)

with initial source

$\psi (\vartheta, 0) = \hbar(\vartheta).$

The Fokker-Planck equation (1.1) was introduced by Fokker and Planck to describe the brownian motion of partiles ^[20]. The Fokker-Plank equation represents the change in probability of a random function in time and space, which explain solute transport. Many phenomena, such as wave propagation, continuous random walk, charge carrier transport in amorphous semiconductors, anomalous diffusion, the motion of ribosomes along mRNA and pattern formation, polymeric networks are modelled by PDEs of both time and space fractional order ^[21].

Gorge Adomian was the first who proposed Adomian Decomposition Method (ADM) 1980. The crucial aspect of the method is employment of the "Adomian polynomials" which allow for solution convergence of the nonlinear portion of the equation, without simply linearizing the system. ADM is a type of method that uses a decomposition technique to generate approximation and even accurate solutions for non-linear systems with valid initial data. ADM have been used effectively for the solution of partial differential equations (PDE's) and fractional partial differential equations (FPDE's). In recent years, more and more researchers have applied this method to solving non-linear systems ^[24]. ADM is preferred over other techniques because of its simple implementation for wide range of nonlinear systems, including ordinary and partial differential equations with fractional derivatives. Moreover, ADM can easily handle the solutions of nonlinear fractional problems which is not the case with other techniques. It can be extended easily for higher dimensions problems.

The targeted solutions of fractional order ODEs and PDEs are also the main research focus of the mathematicians. There are many techniques that have been used for the solutions of fractional order Fokker-Planck equations such as Generalized finite differences method (GFDM) ^[22], radial basis functions finite difference (RBF-FD) ^[23], adomian decomposition method ^[25], homotopy perturbation transform method ^[26], iterative Laplace transform method ^[27], residual power method ^[28], finite element method ^[29], fractional variational iteration method ^[30], fractional reduced differential transform method ^[31].

In this article, we have implemented new approximate analytical method (NAAM) for the solution these type Fokker-Planck equations of fractional order. The NAAM is a analytical procedure which provide series form solution. The NAAM technique is easy and straight forward approach. The obtained results are fastly convergent towards the exact solution of each problem. The graphical analysis of NAAM solutions are demonstrated which has good agreement with exact solution of the problems.The remaining paper is systematized as Section 2, shows Preliminary concepts. In Section 3, signifies Procedure of NAAM. Section 4, we explained the implementation of NAAM on Fokker-Planck equations. Section 5, we summarized the obtained results.

2. Preliminaries and basic concepts

In this section, the related definitions and preliminary concepts of fractional calculus and the procedure of the NAAM are presented.

2.1. Riemann-Liouuille integral operator ^[32]

The Riemann-Liouuille fractional partial integral denoted by $I_{\tau}^{\delta},$ where, $\delta\in N, \delta \geq0$ , which is define as under

$\begin{equation} I_{\tau}^{\delta}\psi(\vartheta, \tau) = \left\{ \begin{array}{c c} \frac{1}{\Gamma(\delta)}\int_{0}^{\tau}\psi(\vartheta, \tau)d\tau, \ \ \vartheta, \tau > 0, \\ \psi(\vartheta, \tau), \ \ \vartheta = 0, \tau > 0, \end{array} \right. \end{equation}$

(2.1)

where, $\Gamma$ is represent gamma function.

2.2. Some properties of Riemann-Liouuille fractional partial integral

Let $\delta, \beta \in R, \setminus N, \ \ \delta, \gamma > 0, \rho > -1,$ then for the function $\psi(\vartheta, \tau)$ the operator $I_{\tau}^{\delta}$ has the following properties.

$\begin{equation} \left\{ \begin{array}{c c} I_{\tau}^{\delta}\psi(\vartheta, \tau) I_t^{\gamma}\psi(\vartheta, \tau) = I_{\tau}^{\delta+\gamma}\psi(\vartheta, \tau), \\ I_{\tau}^{\delta}\psi(\vartheta, \tau) I_{\tau}^{\gamma}\psi(\vartheta, \tau) = I_{\tau}^{\gamma}\psi(\vartheta, \tau)I_{\tau}^{\delta}\psi(\vartheta, \tau), \\ I_{\tau}^{\delta}\tau^{\rho} = \frac{\Gamma(\rho+1)}{\Gamma(\delta+\rho+1)}\tau^{\delta+\rho}. \end{array} \right. \end{equation}$

(2.2)

2.3. Caputo operator of fractional partial derivative ^[36]

$\begin{equation} D_{\tau}^\delta \psi(\vartheta, \tau) = \frac{ \partial^{\delta}\psi(\vartheta, \tau)}{\partial {\tau}^{\delta}} = \left\{ \begin{array}{c c} I^{n-\delta}[\frac{ \partial^{\delta}{\psi(\vartheta, \tau)}}{\partial {\tau}^{\delta}}], \ \ n-1 < \delta < n, \ \ n\in N, \\ \frac{ \partial^{\delta}\psi(\vartheta, \tau)}{\partial {\tau}^{\delta}}, \ \ \ n = \delta. \end{array} \right. \end{equation}$

(2.3)

2.4. Theorem ^[36]

Let $\vartheta, \tau \in R, \ \tau > 0,$ and $m-1 < \rho < m\in N,$ then

$\begin{equation} \begin{split} &I_{\tau}^{\delta} D_{\tau}^\delta \psi(\vartheta, \tau) = \psi(\vartheta, \tau)-\sum\limits_{k = 0}^{m-1}\frac{\tau^k}{k!}\frac{\partial^k \psi(\vartheta, 0^{+}) }{\partial \tau^k}, \\ &D_{\tau}^\vartheta I_{\tau}^{\vartheta} \psi(\vartheta, \tau) = \psi(\vartheta, \tau). \end{split} \end{equation}$

(2.4)

2.5. Theorem

If there exists a constant $0 < \gamma < 1 such that:$

$\begin{equation} ||u_{n+1}(\vartheta, \tau)||\leq\gamma||u_{n}(\vartheta, \tau)||, \ \ n\in N, \ \ \vartheta\in \mathrm{I}\subset\mathbb{R}, \ \ 0\leq t < R, \end{equation}$

(2.5)

then the sequence of approximate solution converges to the exact solution.

Proof. See ^[35].

3. NAAM procedure for the solution of problems^[37]

Here, we have analyzed the analytical procedure for the solution of fractional order Fokker-Planck equations by introducing approximate analytical method.

Consider a general form of Fokker-Planck equation as

(3.1)

with initial source

$\psi (\vartheta, 0) = \hbar(\vartheta),$

where $L, N$ are linear and non-linear operator respectively.

Before computational procedure, we have defined some mandatory procedural results below.

3.1. Lemma ^[37]

For $L\psi(\vartheta, \tau) = \sum_{0}^{\infty}\gamma^k L\psi_k(\vartheta, \tau),$ the linear term $L\psi (\vartheta, \tau)$ satisfy the following property

$\begin{equation} L\psi (\vartheta, \tau) = \left(L\sum\limits_{k = 0}^{\infty}\gamma^k \psi_k(\vartheta, \tau)\right) = \sum\limits_{k = 0}^{\infty}\left(\gamma^k L \psi_k(\vartheta, \tau)\right). \end{equation}$

(3.2)

3.2. Theorem ^[37]

The nonlinear operator $N\psi(\vartheta, \tau),$ for the parameter $\lambda,$ we define $\psi_{\lambda}(\vartheta, \tau) = \sum_{0}^{\infty}\lambda^k \psi_k(\vartheta, \tau),$ compensate the following property

$\begin{equation} N\left( \psi_\lambda\right) = N\left(\sum\limits_{0}^{\infty}\lambda^k \psi_k(\vartheta, \tau)\right) = \sum\limits_{0}^{\infty}\left[\frac{1}{n!}\frac{d^n}{d\lambda^n}\left[N\left(\sum\limits_{0}^{\infty}\lambda^k \psi_k(\vartheta, \tau)\right)\right]_{\lambda = 0}\right]\lambda^n. \end{equation}$

(3.3)

3.3. Definition ^[1]

The polynomial $\chi_n = \chi_n(\psi_0, \psi_1, \cdots, \psi_n),$ can be calculated as

$\begin{equation} \chi_n(\psi_0, \psi_1, \cdots, \psi_n) = \frac{1}{n!}\frac{d^n}{d\lambda^n}\left[\aleph\left(\sum\limits_{0}^{\infty}\lambda^k \psi_k(\vartheta, \tau)\right)\right]_{\lambda = 0} \end{equation}$

(3.4)

3.4. Definition ^[31]

For $\chi_n = \chi_n(\psi_0, \psi_1, \cdots, \psi_n),$ the non-linear term $N\left(\psi_\lambda\right),$ with using definition (3.4), is represented as

$\begin{equation} N\left( \psi_\lambda\right) = \sum\limits_{0}^{\infty}\lambda^k \chi_k \end{equation}$

(3.5)

3.5. The existence and uniqueness of NAAM ^[37]

The following result briefly explain the exitance of NAAM.

Theorem

Let $N\left(\psi_\lambda\right), \psi(\vartheta, \tau)$ are define for $\epsilon-1 < \delta < \epsilon$ , in (3.1). The Fokker-Planck model (3.1), the unique solution is given as

$\begin{equation} \psi(\vartheta, \tau) = \psi(\vartheta, 0)+\sum\limits_{k = 1}^{\infty}\left[L_{\tau}^{-\delta}(\psi_{(k-1)})+\chi_{(k-1)t}^{-\delta}\right], \end{equation}$

(3.6)

where, $L_{\tau}^{-\delta}(\psi_{(k-1)})$ and $\chi_{(k-1)\tau}^{-\delta}$ represent the fractional partial integral of order $\delta$ for $L(\psi_{k-1})$ and $\chi_{(k-1)}$ with respect to $\tau$ .

Proof. The solution of Fokker-Planck equation $\psi(\vartheta, \tau),$ is obtained by substituting the following expansion

$\begin{equation} \psi(\vartheta, \tau) = \sum\limits_{k = 0}^{\infty}\psi_k(\vartheta, \tau). \end{equation}$

(3.7)

To numerate the Eq (3.1), we can investigate as

$\begin{equation} \psi_{\tau\lambda} ^{\delta}(\vartheta, \tau) = \lambda[L (\psi_{\vartheta} (\vartheta, \tau)+\psi_{\vartheta \vartheta} (\vartheta, \tau))+N \psi_{\vartheta \vartheta}(\vartheta, \tau)], \ \ \lambda \in (0, 1], \end{equation}$

(3.8)

along with initial sources

$\begin{equation} \psi(\vartheta, 0) = \hbar (\vartheta). \end{equation}$

(3.9)

Additionally, the solution of Eq (3.6) is estimated as

$\begin{equation} \psi_{\lambda}(\vartheta, \tau) = \sum\limits_{0}^{\infty}\lambda^k\psi_{\lambda}(\vartheta, \tau), \end{equation}$

(3.10)

using the Caupto-Riemann property on Eq (3.8), we have

$\begin{equation} \psi_{\lambda} (\vartheta, \tau) = \psi (\vartheta, 0)+\lambda[L (\psi_{\vartheta} (\vartheta, \tau)+\psi_{\vartheta \vartheta} (\vartheta, \tau))+N \psi_{\vartheta \vartheta}(\vartheta, \tau)], \end{equation}$

(3.11)

assuming Eq (3.9), the initial source, Eq (3.11), become as

$\begin{equation} \psi_{\lambda} (\vartheta, \tau) = \hbar (\vartheta)+\lambda[L (\psi_{\vartheta} (\vartheta, \tau)+\psi_{\vartheta \vartheta} (\vartheta, \tau))+N \psi_{\vartheta \vartheta}(\vartheta, \tau)]. \end{equation}$

(3.12)

By substituting, Eq (3.10), in Eq (3.12), we get

$\begin{equation} \sum\limits_{k = 0}^{\infty}\lambda^{k} \psi_{\lambda}(\vartheta, \tau) = \hbar (\vartheta)+ \lambda I_{t}^{\delta }\left[L\left(\sum\limits_{k = 0}^{\infty}\lambda^{k} (\psi_{\vartheta} (\vartheta, \tau)+\psi_{\vartheta \vartheta} (\vartheta, \tau))\right)+N\left( \sum\limits_{k = 0}^{\infty}\lambda^{k} \psi_{\vartheta \vartheta}(\vartheta, \tau)\right)\right], \end{equation}$

(3.13)

with the help of Theorem (3.4), and Lemma (3.1), the Eq (3.13) become as

$\begin{equation} \sum\limits_{k = 0}^{\infty}\lambda^{k} \psi_{\lambda}(\vartheta, \tau) = \hbar (\vartheta)+ \lambda I_{t}^{\delta }\left[L\left(\sum\limits_{k = 0}^{\infty}\lambda^{k} (\psi_{\vartheta} (\vartheta, \tau)+\psi_{\vartheta \vartheta} (\vartheta, \tau))\right)\right]+\lambda I_{t}^{\delta }\left[N\left( \sum\limits_{k = 0}^{\infty}\lambda^{k} \chi_n \right)\right], \end{equation}$

(3.14)

the iterative scheme is obtained by comparing the identical power of $\lambda$ in Eq (3.14), as

$\begin{equation} \left\{ \begin{array}{c c} \psi_0(\vartheta, \tau) = \hbar(\vartheta), \\ \psi_1(\vartheta, \tau) = L_{\tau}^{-\delta}\psi_0+\chi_{\tau 0}^{-\delta}, \\ \psi_k(\vartheta, \tau) = L_{\tau}^{-\delta}\psi_{(k-1)}+\chi_{\tau(k-1)}^{-\delta}, \ \ k = 2, 3, \cdots \end{array} \right. \end{equation}$

(3.15)

4. NAAM implementation and results discussion ^[37]

In this section, we have tested the validity and applicability of NAAM by solving some Fokker-Planck equations.

Problem 4.1. Consider a Fokker-Planck equation of time fractional order in the form ^[36]:

$\begin{equation} \frac{\partial^\delta}{\partial \tau^{\delta}}\left( \psi (\vartheta, \tau)\right)+\frac{\partial}{\partial \vartheta}\left(\frac{\vartheta}{6} \psi (\vartheta, \tau)\right)-\frac{\partial^2}{\partial \vartheta^2}\left(\frac{{\vartheta}^2}{12} \psi (\vartheta, \tau)\right) = 0, \ \ \vartheta, \tau > 0, \ \delta \in (0, 1], \end{equation}$

(4.1)

with initial source

$\psi(\vartheta, 0) = \vartheta^2,$

for special value $\delta = 1,$ the exact form solution is

$\psi(\vartheta, \tau) = \vartheta^2 \exp^{\frac{\tau}{2}}.$

To investigate the solution of the Fokker-Planck equation (21), we compare it with the Eq (6), we get

$\begin{equation} \frac{\partial^\delta}{\partial \tau^{\delta}}\left( \psi (\vartheta, \tau)\right) = \frac{\partial}{\partial \vartheta}\left(\frac{\vartheta}{6} \psi (\vartheta, \tau)\right)-\frac{\partial^2}{\partial \vartheta^2}\left(\frac{{\vartheta}^2}{12} \psi (\vartheta, \tau)\right), \ \ \vartheta, \tau > 0, \ \delta \in (0, 1], \end{equation}$

(4.2)

the genral NAAM solution of Eq (4.1), is assume as

$\begin{equation} \psi(\vartheta, \tau) = \sum\limits_{k = 0}^{\infty}\psi_k(\vartheta, \tau). \end{equation}$

(4.3)

For investigating the approximate solution of Eq (4.2), we process as

$\begin{equation} \frac{\partial^\delta}{\partial \tau^{\delta}}\left( \psi (\vartheta, \tau)\right) = \lambda\left[\frac{\partial}{\partial \vartheta}\left(\frac{\vartheta}{6} \psi (\vartheta, \tau)\right)-\frac{\partial^2}{\partial \vartheta^2}\left(\frac{{\vartheta}^2}{12} \psi (\vartheta, \tau)\right)\right], \ \ \ \ \delta \in (0, 1], \end{equation}$

(4.4)

where the initial condition is given as

$\begin{equation} \psi(\vartheta, 0) = \vartheta^2. \end{equation}$

(4.5)

The assume solution of Eq (4.4), is in the form

$\begin{equation} \psi_\lambda(\vartheta, \tau) = \sum\limits_{k = 0}^{\infty} \lambda^k \psi_k(\vartheta, \tau). \end{equation}$

(4.6)

Using the Caupto-Riemann property on Eq (3.8), we have

$\begin{equation} \psi_\lambda(\vartheta, \tau) = \psi(\vartheta, 0)+ \lambda I_t^{\delta}\left[\frac{\partial}{\partial \vartheta}\left(\frac{\vartheta}{6} \psi (\vartheta, \tau)\right)-\frac{\partial^2}{\partial \vartheta^2}\left(\frac{{\vartheta}^2}{12} \psi (\vartheta, \tau)\right)\right], \end{equation}$

(4.7)

assuming the initial condition (3.5), Eq (4.6), become as

$\begin{equation} \psi_\lambda(\vartheta, \tau) = \vartheta^2+ \lambda I_t^{\delta}\left[\frac{\partial}{\partial \vartheta}\left(\frac{\vartheta}{6} \psi (\vartheta, \tau)\right)-\frac{\partial^2}{\partial \vartheta^2}\left(\frac{{\vartheta}^2}{12} \psi (\vartheta, \tau)\right)\right], \end{equation}$

(4.8)

by substituting Eq (3.10), in Eq (4.8), we get

$\begin{equation} \sum\limits_{k = 0}^{\infty} \lambda^k \psi_k(\vartheta, \tau) = \vartheta^2+ \lambda I_t^{\delta}\left[ \sum\limits_{k = 0}^{\infty} \lambda^k (\frac{\partial}{\partial \vartheta}\left(\frac{\vartheta}{6} \psi_k (\vartheta, \tau)\right))- \sum\limits_{k = 0}^{\infty} \lambda^k\frac{\partial^2}{\partial \vartheta^2}\left(\frac{{\vartheta}^2}{12} \psi_k (\vartheta, \tau)\right)\right], \end{equation}$

(4.9)

the iterative scheme is obtained by comparing the identical power of $\lambda$ in Eq (4.9), as

$\begin{equation} \left\{ \begin{array}{c c} \psi_0(\vartheta, \tau) = \vartheta^2, \\ \psi_1(\vartheta, \tau) = I_t^{\delta}\left[\frac{\partial}{\partial \vartheta}\left(\frac{\vartheta}{6} \psi_0 (\vartheta, \tau)\right)-\frac{\partial^2}{\partial \vartheta^2}\left(\frac{{\vartheta}^2}{12} \psi_0 (\vartheta, \tau)\right)\right], \\ \psi_k(\vartheta, \tau) = I_t^{\delta}\left[\frac{\partial}{\partial \vartheta}\left(\frac{\vartheta}{6} \psi_{k-1} (\vartheta, \tau)\right)-\frac{\partial^2}{\partial \vartheta^2}\left(\frac{{\vartheta}^2}{12} \psi_{k-1} (\vartheta, \tau)\right)\right]. \end{array} \right. \end{equation}$

(4.10)

Consequently, we get

$\begin{equation} \psi_0(\vartheta, \tau) = \vartheta^2, \end{equation}$

(4.11)

$\begin{equation} \psi_1(\vartheta, \tau) = \vartheta^2 \frac{t^\delta}{2 \Gamma(\delta+1)}, \end{equation}$

(4.12)

$\begin{equation} \psi_2(\vartheta, \tau) = \vartheta^2 \frac{t^{2\delta}}{8 \Gamma(2\delta+1)}, \end{equation}$

(4.13)

$\begin{equation} \psi_3(\vartheta, \tau) = \vartheta^2 \frac{t^{3\delta}}{8 \Gamma(3\delta+1)}, \end{equation}$

(4.14)

$\vdots.$

The NAAM solution is

$\begin{equation} \psi(\vartheta, \tau) = \psi_0(\vartheta, \tau)+ \psi_1(\vartheta, \tau)+ \psi_2(\vartheta, \tau)+ \psi_3(\vartheta, \tau)+\cdots, \end{equation}$

(4.15)

subsisting Eqs (4.11)–(4.14), in Eq (4.15), we get

$\begin{equation} \psi(\vartheta, \tau) = \vartheta^2+ \vartheta^2 \frac{t^\delta}{2 \Gamma(\delta+1)}+\vartheta^2 \frac{t^{2\delta}}{8 \Gamma(2\delta+1)}+\vartheta^2 \frac{t^{3\delta}}{8 \Gamma(3\delta+1)}+\cdots, \end{equation}$

(4.16)

Specifically for $\delta = 1,$ the solution converted to

$\begin{equation} \psi(\vartheta, \tau) = \vartheta^2+ \vartheta^2 \frac{t}{2 \Gamma(2)}+\vartheta^2 \frac{t^{2}}{8 \Gamma(3)}+\vartheta^2 \frac{t^{3}}{8 \Gamma(4)}+\cdots, \end{equation}$

(4.17)

which converge to exact solution as

$\begin{equation} \psi(\vartheta, \tau) = \vartheta^2 \exp^{\frac{\tau}{2}}. \end{equation}$

(4.18)

Problem 4.2.

Consider a Fokker-Planck equation of time fractional order in the form ^[36];

$\begin{equation} \frac{\partial}{\partial \tau^{\delta}}\left( \psi (\vartheta, \tau)\right)+\frac{\partial}{\partial \vartheta}\left(\vartheta \psi (\vartheta, \tau)\right)-\frac{\partial^2}{\partial \vartheta^2}\left(\frac{{\vartheta}^2}{2} \psi (\vartheta, \tau)\right) = 0, \ \ \vartheta, \tau > 0, \ \delta \in (0, 1], \end{equation}$

(4.19)

with initial source

$\begin{equation} \psi(\vartheta, 0) = \vartheta, \end{equation}$

(4.20)

for special value $\delta = 1,$ the exact form solution is

$\psi(\vartheta, \tau) = \vartheta \exp^{\tau}.$

To investigate the solution of the Fokker-Planck equation (4.19), we compare it with the Eq (3.1), we get

$\begin{equation} \frac{\partial^\delta}{\partial \tau^{\delta}}\left( \psi (\vartheta, \tau)\right) = -\frac{\partial}{\partial \vartheta}\left(\vartheta \psi (\vartheta, \tau)\right)+\frac{\partial^2}{\partial \vartheta^2}\left(\frac{{\vartheta}^2}{2} \psi (\vartheta, \tau)\right), \ \ \vartheta, \tau > 0, \ \delta \in (0, 1], \end{equation}$

(4.21)

the general NAAM solution of Eq (4.19), is assume as

$\begin{equation} \psi(\vartheta, \tau) = \sum\limits_{k = 0}^{\infty}\psi_k(\vartheta, \tau). \end{equation}$

(4.22)

For investigating the approximate solution of Eq (4.20), we process as

$\begin{equation} \frac{\partial^\delta}{\partial \tau^{\delta}}\left( \psi (\vartheta, \tau)\right) = \lambda\left[-\frac{\partial}{\partial \vartheta}\left(\vartheta \psi (\vartheta, \tau)\right)+\frac{\partial^2}{\partial \vartheta^2}\left(\frac{{\vartheta}^2}{2} \psi (\vartheta, \tau)\right)\right], \ \ \ \ \delta \in (0, 1], \end{equation}$

(4.23)

where the initial condition is given as

$\begin{equation} \psi(\vartheta, 0) = \vartheta. \end{equation}$

(4.24)

The assume solution of Eq (4.23), is in the form

$\begin{equation} \psi_\lambda(\vartheta, \tau) = \sum\limits_{k = 0}^{\infty} \lambda^k \psi_k(\vartheta, \tau). \end{equation}$

(4.25)

Using the Caupto-Riemann property on Eq (4.23), we have

$\begin{equation} \psi_\lambda(\vartheta, \tau) = \psi(\vartheta, 0)+ \lambda I_t^{\delta}\left[-\frac{\partial}{\partial \vartheta}\left(\vartheta \psi (\vartheta, \tau)\right)+\frac{\partial^2}{\partial \vartheta^2}\left(\frac{{\vartheta}^2}{2} \psi (\vartheta, \tau)\right)\right], \end{equation}$

(4.26)

assuming the initial condition (4.20), Eq (4.26), become as

$\begin{equation} \psi_\lambda(\vartheta, \tau) = \vartheta+ \lambda I_t^{\delta}\left[-\frac{\partial}{\partial \vartheta}\left(\vartheta \psi (\vartheta, \tau)\right)+\frac{\partial^2}{\partial \vartheta^2}\left(\frac{{\vartheta}^2}{2} \psi (\vartheta, \tau)\right)\right], \end{equation}$

(4.27)

by substituting Eq (4.25), in Eq (4.27), we get

$\begin{equation} \sum\limits_{k = 0}^{\infty} \lambda^k \psi_k(\vartheta, \tau) = \vartheta+ \lambda I_t^{\delta}\left[ \sum\limits_{k = 0}^{\infty} \lambda^k (-\frac{\partial}{\partial \vartheta}\left(\vartheta \psi (\vartheta, \tau)\right))+ \sum\limits_{k = 0}^{\infty} \lambda^k\frac{\partial^2}{\partial \vartheta^2}\left(\frac{\partial^2}{\partial \vartheta^2}\left(\frac{{\vartheta}^2}{2} \psi (\vartheta, \tau)\right)\right)\right], \end{equation}$

(4.28)

the iterative scheme is obtained by comparing the identical power of $\lambda$ in Eq (4.28), as

$\begin{equation} \left\{ \begin{array}{c c} \psi_0(\vartheta, \tau) = \vartheta, \\ \psi_1(\vartheta, \tau) = I_t^{\delta}\left[-\frac{\partial}{\partial \vartheta}\left(\vartheta \psi_0 (\vartheta, \tau)\right)+\frac{\partial^2}{\partial \vartheta^2}\left(\frac{{\vartheta}^2}{2} \psi_0 (\vartheta, \tau)\right)\right], \\ \psi_k(\vartheta, \tau) = I_t^{\delta}\left[-\frac{\partial}{\partial \vartheta}\left(\vartheta \psi_{k-1} (\vartheta, \tau)\right)+\frac{\partial^2}{\partial \vartheta^2}\left(\frac{{\vartheta}^2}{2} \psi_{k-1} (\vartheta, \tau)\right)\right]. \end{array} \right. \end{equation}$

(4.29)

Consequently, we get

$\begin{equation} \psi_0(\vartheta, \tau) = \vartheta, \end{equation}$

(4.30)

$\begin{equation} \psi_1(\vartheta, \tau) = \vartheta \frac{t^\delta}{\Gamma(\delta+1)}, \end{equation}$

(4.31)

$\begin{equation} \psi_2(\vartheta, \tau) = \vartheta \frac{t^{2\delta}}{ \Gamma(2\delta+1)}, \end{equation}$

(4.32)

$\begin{equation} \psi_3(\vartheta, \tau) = \vartheta \frac{t^{3\delta}}{ \Gamma(3\delta+1)}, \end{equation}$

(4.33)

$\vdots$

The NAAM solution is

$\begin{equation} \psi(\vartheta, \tau) = \psi_0(\vartheta, \tau)+ \psi_1(\vartheta, \tau)+ \psi_2(\vartheta, \tau)+ \psi_3(\vartheta, \tau)+\cdots, \end{equation}$

(4.34)

subsisting Eqs (4.30)–(4.33), in Eq (4.34), we get

$\begin{equation} \psi(\vartheta, \tau) = \vartheta+ \vartheta \frac{t^\delta}{ \Gamma(\delta+1)}+\vartheta \frac{t^{2\delta}}{ \Gamma(2\delta+1)}+\vartheta \frac{t^{3\delta}}{\Gamma(3\delta+1)}+\cdots. \end{equation}$

(4.35)

Specifically for $\delta = 1,$ the solution converted to

$\begin{equation} \psi(\vartheta, \tau) = \vartheta+ \vartheta \frac{t}{ \Gamma(2)}+\vartheta \frac{t^{2}}{ \Gamma(3)}+\vartheta \frac{t^{3}}{ \Gamma(4)}+\cdots, \end{equation}$

(4.36)

which converge to exact solution as

$\begin{equation} \psi(\vartheta, \tau) = \vartheta \exp^{\tau}. \end{equation}$

(4.37)

Problem 4.3.

Consider a Fokker-Planck equation of time fractional order in the form ^[36];

$\begin{equation} \frac{\partial}{\partial \tau^{\delta}}\left( \psi (\vartheta, \tau)\right)+\frac{\partial}{\partial \vartheta}\left(\frac{4}{\vartheta} \psi^2 (\vartheta, \tau)\right)-\frac{\partial}{\partial \vartheta}\left(\frac{{\vartheta}}{3} \psi (\vartheta, \tau)\right)-\frac{\partial^2}{\partial \vartheta^2}\left( \psi^2 (\vartheta, \tau)\right) = 0, \ \ \vartheta, \tau > 0, \ \delta \in (0, 1], \end{equation}$

(4.38)

where initial condition, given as

$\begin{equation} \psi(\vartheta, 0) = \vartheta^2. \end{equation}$

(4.39)

For special value $\delta = 1,$ the exact form solution is

$\psi(\vartheta, \tau) = \vartheta^2 \exp^{\tau}.$

To investigate the solution of the Fokker-Planck equation (4.38), we compare it with the Eq (3.1), we get

$\begin{equation} \frac{\partial^\delta}{\partial \tau^{\delta}}\left( \psi (\vartheta, \tau)\right) = \frac{\partial}{\partial \vartheta}\left(\frac{{\vartheta}}{3} \psi (\vartheta, \tau)\right)+N \left(\psi (\vartheta, \tau)\right), \ \ \vartheta, \tau > 0, \ \delta \in (0, 1], \end{equation}$

(4.40)

where, the non-linear term $N \left(\psi (\vartheta, \tau)\right) = \frac{\partial^2}{\partial \vartheta^2}\left(\psi^2 (\vartheta, \tau)\right)- \frac{\partial}{\partial \vartheta}\left(\frac{4}{\vartheta} \psi^2 (\vartheta, \tau)\right).$

The general NAAM solution of Eq (4.38), is assume as

$\begin{equation} \psi(\vartheta, \tau) = \sum\limits_{k = 0}^{\infty}\psi_k(\vartheta, \tau). \end{equation}$

(4.41)

For investigating the approximate solution of Eq (4.40), we process as

$\begin{equation} \frac{\partial^\delta}{\partial \tau^{\delta}}\left( \psi (\vartheta, \tau)\right) = \lambda\left[\frac{\partial}{\partial \vartheta}\left(\frac{{\vartheta}}{3} \psi (\vartheta, \tau)\right)+N \left(\psi (\vartheta, \tau)\right)\right], \ \ \ \ \delta \in (0, 1], \end{equation}$

(4.42)

The assume solution of Eq (4.42), is in the form

$\begin{equation} \psi_\lambda(\vartheta, \tau) = \sum\limits_{k = 0}^{\infty} \lambda^k \psi_k(\vartheta, \tau). \end{equation}$

(4.43)

Using the Caupto-Riemann property on Eq (4.42), we have

(4.44)

assuming the initial condition 4.39, Eq (4.44), become as

$\begin{equation} \psi_\lambda(\vartheta, \tau) = \vartheta^2+ \lambda I_t^{\delta}\left[\frac{\partial}{\partial \vartheta}\left(\frac{{\vartheta}}{3} \psi (\vartheta, \tau)\right)+N \left(\psi (\vartheta, \tau)\right)\right], \end{equation}$

(4.45)

by substituting Eq (4.43), in Eq (4.45), we get

$\begin{equation} \sum\limits_{k = 0}^{\infty} \lambda^k \psi_k(\vartheta, \tau) = \vartheta^2+ \lambda I_t^{\delta}\left[ \sum\limits_{k = 0}^{\infty} \lambda^k (\frac{\partial}{\partial \vartheta}\left(\frac{{\vartheta}}{3} \psi (\vartheta, \tau)\right))+ \sum\limits_{k = 0}^{\infty} \lambda^k (N \left(\psi (\vartheta, \tau)\right))\right], \end{equation}$

(4.46)

the nonlinear operator $N \left(\psi (\vartheta, \tau)\right)$ is evaluated by using definition (3.4).

The iterative scheme is obtained by comparing the identical power of $\lambda$ in Eq (4.46), as

$\begin{equation} \left\{ \begin{array}{c c} \psi_0(\vartheta, \tau) = \vartheta^2, \\ \psi_1(\vartheta, \tau) = L_{\tau}^{\frac{\partial}{\partial \vartheta}}\left(\frac{{\vartheta}}{3} \psi_0 (\vartheta, \tau)\right)+\chi_{\tau 0}^{-\delta}, \\ \psi_k(\vartheta, \tau) = L_{\tau}^{-\delta}\frac{\partial}{\partial \vartheta}\left(\frac{{\vartheta}}{3} \psi_{k-1} (\vartheta, \tau)\right)+\chi_{\tau(k-1)}^{-\delta}, \ \ k = 2, 3, \cdots. \end{array} \right. \end{equation}$

(4.47)

Consequently, we get

$\begin{equation} \psi_0(\vartheta, \tau) = \vartheta^2, \end{equation}$

(4.48)

$\begin{equation} \psi_1(\vartheta, \tau) = \vartheta^2 \frac{t^\delta}{\Gamma(\delta+1)}, \end{equation}$

(4.49)

$\begin{equation} \psi_2(\vartheta, \tau) = \vartheta^2 \frac{t^{2\delta}}{ \Gamma(2\delta+1)}, \end{equation}$

(4.50)

$\begin{equation} \psi_3(\vartheta, \tau) = \vartheta^2 \frac{t^{3\delta}}{ \Gamma(3\delta+1)}, \end{equation}$

(4.51)

$\vdots.$

The NAAM solution is

$\begin{equation} \psi(\vartheta, \tau) = \psi_0(\vartheta, \tau)+ \psi_1(\vartheta, \tau)+ \psi_2(\vartheta, \tau)+ \psi_3(\vartheta, \tau)+\cdots, \end{equation}$

(4.52)

subsisting Eqs (4.48)–(4.51), in Eq (4.52), we get

$\begin{equation} \psi(\vartheta, \tau) = \vartheta^2+ \vartheta^{2} \frac{t^\delta}{ \Gamma(\delta+1)}+\vartheta^2 \frac{t^{2\delta}}{ \Gamma(2\delta+1)}+\vartheta^2 \frac{t^{3\delta}}{\Gamma(3\delta+1)}+\cdots. \end{equation}$

(4.53)

Specifically for $\delta = 1,$ the solution converted to

$\begin{equation} \psi(\vartheta, \tau) = \vartheta^2+ \vartheta^{2} \frac{t}{ \Gamma(2)}+\vartheta^2 \frac{t^{2}}{ \Gamma(3)}+\vartheta^2 \frac{t^{3}}{ \Gamma(4)}+\cdots \end{equation}$

(4.54)

which converge to exact solution as

$\begin{equation} \psi(\vartheta, \tau) = \vartheta^2 \exp^{\tau}. \end{equation}$

(4.55)

Problem 4.4.

Consider a Fokker-Planck equation of time fractional order in the form ^[36];

$\begin{equation} \frac{\partial}{\partial \tau^{\delta}}\left( \psi (\vartheta, \tau)\right)-\frac{\partial}{\partial \vartheta} \psi (\vartheta, \tau)-\frac{\partial^2}{\partial \vartheta^2} \psi (\vartheta, \tau) = 0, \tau > 0, \ \delta \in (0, 1], \end{equation}$

(4.56)

with initial source

$\begin{equation} \psi(\vartheta, 0) = \vartheta, \end{equation}$

(4.57)

for special value $\delta = 1,$ the exact form solution is

$\psi(\vartheta, \tau) = \vartheta +\tau.$

To investigate the solution of the Fokker-Planck equation (4.56), we compare it with the Eq (3.1), we get

$\begin{equation} \frac{\partial^\delta}{\partial \tau^{\delta}}\left( \psi (\vartheta, \tau)\right) = \frac{\partial}{\partial \vartheta} \psi (\vartheta, \tau)+\frac{\partial^2}{\partial \vartheta^2} \psi (\vartheta, \tau), \ \ \vartheta, \tau > 0, \ \delta \in (0, 1], \end{equation}$

(4.58)

the general NAAM solution of Eq (4.58), is assume as

$\begin{equation} \psi(\vartheta, \tau) = \sum\limits_{k = 0}^{\infty}\psi_k(\vartheta, \tau). \end{equation}$

(4.59)

For investigating the approximate solution of Eq (4.59), we process as

$\begin{equation} \frac{\partial^\delta}{\partial \tau^{\delta}}\left( \psi (\vartheta, \tau)\right) = \lambda\left[\frac{\partial}{\partial \vartheta} \psi (\vartheta, \tau)+\frac{\partial^2}{\partial \vartheta^2} \psi (\vartheta, \tau)\right], \ \ \ \ \delta \in (0, 1], \end{equation}$

(4.60)

where initial condition, given as

$\begin{equation} \psi(\vartheta, 0) = \vartheta. \end{equation}$

(4.61)

The assume solution of Eq (4.60), is in the form

$\begin{equation} \psi_\lambda(\vartheta, \tau) = \sum\limits_{k = 0}^{\infty} \lambda^k \psi_k(\vartheta, \tau). \end{equation}$

(4.62)

Using the Caupto-Riemann property on Eq (4.60), we have

$\begin{equation} \psi_\lambda(\vartheta, \tau) = \psi(\vartheta, 0)+ \lambda I_t^{\delta}\left[\frac{\partial}{\partial \vartheta} \psi (\vartheta, \tau)+\frac{\partial^2}{\partial \vartheta^2} \psi (\vartheta, \tau)\right], \end{equation}$

(4.63)

assuming the initial condition (4.57), Eq (4.63), become as

$\begin{equation} \psi_\lambda(\vartheta, \tau) = \vartheta+ \lambda I_t^{\delta}\left[\frac{\partial}{\partial \vartheta} \psi (\vartheta, \tau)+\frac{\partial^2}{\partial \vartheta^2} \psi (\vartheta, \tau)\right], \end{equation}$

(4.64)

by substituting Eq (4.62), in Eq (4.64), we get

$\begin{equation} \sum\limits_{k = 0}^{\infty} \lambda^k \psi_k(\vartheta, \tau) = \vartheta^2+ \lambda I_t^{\delta}\left[ \sum\limits_{k = 0}^{\infty} \lambda^k (\frac{\partial}{\partial \vartheta} \psi (\vartheta, \tau))+ \sum\limits_{k = 0}^{\infty} \lambda^k (\frac{\partial^2}{\partial \vartheta^2} \psi (\vartheta, \tau))\right], \end{equation}$

(4.65)

the iterative scheme is obtained by comparing the identical power of $\lambda$ in Eq (4.65), as

$\begin{equation} \left\{ \begin{array}{c c} \psi_0(\vartheta, \tau) = \vartheta, \\ \psi_1(\vartheta, \tau) = I_{\tau}^{-\delta}\left[\frac{\partial}{\partial \vartheta} \psi_0 (\vartheta, \tau)+\frac{\partial^2}{\partial \vartheta^2} \psi_0 (\vartheta, \tau)\right], \\ \psi_k(\vartheta, \tau) = I_{\tau}^{-\delta}\left[\frac{\partial}{\partial \vartheta} \psi_{k-1} (\vartheta, \tau)+\frac{\partial^2}{\partial \vartheta^2} \psi_{k-1} (\vartheta, \tau)\right], \ \ k = 2, 3, \cdots \end{array} \right. \end{equation}$

(4.66)

Consequently, we get

$\begin{equation} \psi_0(\vartheta, \tau) = \vartheta, \end{equation}$

(4.67)

$\begin{equation} \psi_1(\vartheta, \tau) = \frac{t^\delta}{\Gamma(\delta+1)}, \end{equation}$

(4.68)

$\begin{equation} \psi_2(\vartheta, \tau) = 0, \end{equation}$

(4.69)

$\begin{equation} \psi_3(\vartheta, \tau) = 0, \end{equation}$

(4.70)

$\vdots.$

The NAAM solution is

$\begin{equation} \psi(\vartheta, \tau) = \psi_0(\vartheta, \tau)+ \psi_1(\vartheta, \tau)+ \psi_2(\vartheta, \tau)+ \psi_3(\vartheta, \tau)+\cdots, \end{equation}$

(4.71)

subsisting Eqs (4.67)–(4.70), in Eq (4.71), we get

$\begin{equation} \psi(\vartheta, \tau) = \vartheta+ \frac{t^\delta}{ \Gamma(\delta+1)}+0+0+\cdots. \end{equation}$

(4.72)

Specifically for $\delta = 1,$ the solution converted to

$\begin{equation} \psi(\vartheta, \tau) = \vartheta+ \frac{\tau}{ \Gamma(2)}, \end{equation}$

(4.73)

which converge to exact solution as

$\begin{equation} \psi(\vartheta, \tau) = \vartheta +\tau. \end{equation}$

(4.74)

Problem 4.5.

Consider a Fokker-Planck equation of time fractional order in the form ^[36];

$\begin{equation} \frac{\partial^\delta}{\partial \tau^{\delta}}\left( \psi (\vartheta, \tau)\right)-(1-\vartheta)\frac{\partial}{\partial \vartheta} \psi (\vartheta, \tau)-(e^{\tau} \vartheta^2)\frac{\partial^2}{\partial \vartheta^2} \psi (\vartheta, \tau) = 0, \ \ \tau > 0, \ \delta \in (0, 1], \end{equation}$

(4.75)

with initial source

$\begin{equation} \psi(\vartheta, 0) = 1+\vartheta, \end{equation}$

(4.76)

for special value $\delta = 1,$ the exact form solution is

$\psi(\vartheta, \tau) = \exp^\tau(1+\vartheta).$

To investigate the solution of the Fokker-Planck equation (4.75), we compare it with the Eq (3.1), we get

$\begin{equation} \frac{\partial^\delta}{\partial \tau^{\delta}}\left( \psi (\vartheta, \tau)\right) = (1-\vartheta)\frac{\partial}{\partial \vartheta} \psi (\vartheta, \tau)+(e^{\tau} \vartheta^2)\frac{\partial^2}{\partial \vartheta^2} \psi (\vartheta, \tau), \ \ \vartheta, \tau > 0, \ \delta \in (0, 1], \end{equation}$

(4.77)

the general NAAM solution of Eq (4.75), is assume as

$\begin{equation} \psi(\vartheta, \tau) = \sum\limits_{k = 0}^{\infty}\psi_k(\vartheta, \tau). \end{equation}$

(4.78)

For investigating the approximate solution of Eq (4.77), we process as

$\begin{equation} \frac{\partial^\delta}{\partial \tau^{\delta}}\left( \psi (\vartheta, \tau)\right) = \lambda\left[(1-\vartheta)\frac{\partial}{\partial \vartheta} \psi (\vartheta, \tau)+(e^{\tau} \vartheta^2)\frac{\partial^2}{\partial \vartheta^2} \psi (\vartheta, \tau)\right], \ \ \ \ \delta \in (0, 1], \end{equation}$

(4.79)

where initial condition, given as

$\begin{equation} \psi(\vartheta, 0) = \vartheta+1. \end{equation}$

(4.80)

The assume solution of Eq (4.79), is in the form

$\begin{equation} \psi_\lambda(\vartheta, \tau) = \sum\limits_{k = 0}^{\infty} \lambda^k \psi_k(\vartheta, \tau). \end{equation}$

(4.81)

Using the Caupto-Riemann property on Eq (4.79), we have

$\begin{equation} \psi_\lambda(\vartheta, \tau) = \psi(\vartheta, 0)+ \lambda I_t^{\delta}\left[(1-\vartheta)\frac{\partial}{\partial \vartheta} \psi (\vartheta, \tau)+(e^{\tau} \vartheta^2)\frac{\partial^2}{\partial \vartheta^2} \psi (\vartheta, \tau)\right], \end{equation}$

(4.82)

assuming the initial condition, Eq (4.82), become as

$\begin{equation} \psi_\lambda(\vartheta, \tau) = 1+\vartheta+ \lambda I_t^{\delta}\left[(1-\vartheta)\frac{\partial}{\partial \vartheta} \psi (\vartheta, \tau)+(e^{\tau} \vartheta^2)\frac{\partial^2}{\partial \vartheta^2} \psi (\vartheta, \tau)\right], \end{equation}$

(4.83)

by substituting Eq (4.81), in Eq (4.83), we get

$\begin{equation} \sum\limits_{k = 0}^{\infty} \lambda^k \psi_k(\vartheta, \tau) = 1+\vartheta+ \lambda I_t^{\delta}\left[ \sum\limits_{k = 0}^{\infty} \lambda^k ((1-\vartheta)\frac{\partial}{\partial \vartheta} \psi (\vartheta, \tau))+ \sum\limits_{k = 0}^{\infty} \lambda^k ((e^{\tau} \vartheta^2)\frac{\partial^2}{\partial \vartheta^2} \psi (\vartheta, \tau))\right], \end{equation}$

(4.84)

the iterative scheme is obtained by comparing the identical power of $\lambda$ in Eq (4.84), as

$\begin{equation} \left\{ \begin{array}{c c} \psi_0(\vartheta, \tau) = 1+\vartheta, \\ \psi_1(\vartheta, \tau) = I_{\tau}^{-\delta}\left[(1-\vartheta)\frac{\partial}{\partial \vartheta} \psi_0 (\vartheta, \tau)+(e^{\tau} \vartheta^2)\frac{\partial^2}{\partial \vartheta^2} \psi_0 (\vartheta, \tau)\right], \\ \psi_k(\vartheta, \tau) = I_{\tau}^{-\delta}\left[(1-\vartheta)\frac{\partial}{\partial \vartheta} \psi_{k-1} (\vartheta, \tau)+(e^{\tau} \vartheta^2)\frac{\partial^2}{\partial \vartheta^2} \psi_{k-1} (\vartheta, \tau)\right], \ \ k = 2, 3, \cdots \end{array} \right. \end{equation}$

(4.85)

Consequently, we get

$\begin{equation} \psi_0(\vartheta, \tau) = 1+\vartheta, \end{equation}$

(4.86)

$\begin{equation} \psi_1(\vartheta, \tau) = (1+\vartheta)\frac{\tau^\delta}{\Gamma(\delta+1)}, \end{equation}$

(4.87)

$\begin{equation} \psi_2(\vartheta, \tau) = (1+\vartheta)\frac{\tau^{2\delta}}{\Gamma(2\delta+1)}, \end{equation}$

(4.88)

$\begin{equation} \psi_3(\vartheta, \tau) = (1+\vartheta)\frac{\tau^{3\delta}}{\Gamma(3\delta+1)}, \end{equation}$

(4.89)

$\vdots.$

The NAAM solution is

$\begin{equation} \psi(\vartheta, \tau) = \psi_0(\vartheta, \tau)+ \psi_1(\vartheta, \tau)+ \psi_2(\vartheta, \tau)+ \psi_3(\vartheta, \tau)+\cdots, \end{equation}$

(4.90)

subsisting Eqs (4.86)–(4.89), in Eq (4.90), we get

$\begin{equation} \psi(\vartheta, \tau) = 1+\vartheta+ (1+\vartheta) \frac{t^\delta}{ \Gamma(\delta+1)}+(1+\vartheta)\frac{\tau^{2\delta}}{\Gamma(2\delta+1)}+(1+\vartheta)\frac{\tau^{3\delta}}{\Gamma(3\delta+1)}+\cdots. \end{equation}$

(4.91)

Specifically for $\delta = 1,$ the solution converted to

$\begin{equation} \psi(\vartheta, \tau) = 1+ \vartheta+ (1+ \vartheta) \frac{\tau}{ \Gamma(2)}+(1+ \vartheta) \frac{\tau^2}{ \Gamma(3)}+(1+ \vartheta) \frac{\tau^3}{ \Gamma(4)}+\cdots, \end{equation}$

(4.92)

which converge to exact solution as

$\begin{equation} \psi(\vartheta, \tau) = \exp^\tau(1+\vartheta). \end{equation}$

(4.93)

Figures 1 and 2 show the comparison of exact and NAAM solutions while Figures 3 and 4 represent the 2D and 3D solutions graphs respectively of Problem 4.1 at different fractional orders. Figures 5 and 6 show the comparison of exact and NAAM solutions while Figures 7 and 8 represent the 2D and 3D solutions graphs respectively of Problem 4.2 at different fractional orders. Figures 9 and 10 show the comparison of exact and NAAM solutions while Figures 11 and 12 represent the 2D and 3D solutions graphs respectively of Problem 4.3 at different fractional orders. Figures 13 and 14 show the comparison of exact and NAAM solutions while Figures 15 and 16 represent the 2D and 3D solutions graphs respectively of Problem 4.4 at different fractional orders. Figures 17 and 18 show the comparison of exact and NAAM solutions while Figures 19 and 20 represent the 2D and 3D solutions graphs respectively of Problem 4.5 at different fractional orders. Table 1, display the solutions comparison of NAAM with HPM, ADM and exact solutions of Problem 4.2. Table 2, display the solutions comparison of NAAM with HPM, ADM and exact solutions of Problem 4.3. Table 3, display the solutions comparison of NAAM with HPM, ADM and exact solutions of Problem 4.4. Table 4, display the solutions comparison of NAAM with HPM, ADM and exact solutions of Problem 4.5.

Figure 1. Exact solution graph of Problem 4.1.

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Figure 2. NAAM solution graph of Problem 4.1.

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Figure 3. 3D fractional order solution graphs of Problem 4.1.

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Figure 4. 2D fractional order solution graph of Problem 4.1.

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Figure 5. Exact solution graph of Problem 4.2.

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Figure 6. NAAM solution graph of Problem 4.2.

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Figure 7. 3D Fractional order solution graphs of Problem 4.2.

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Figure 8. 2D Fractional order solution graph of Problem 4.2.

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Figure 9. Exact solution graph of Problem 4.3.

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Figure 10. NAAM solution graph of Problem 4.3.

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Figure 11. 3D fractional order solution graphs of Problem 4.3.

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Figure 12. 2D fractional order solution graph of Problem 4.3.

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Figure 13. Exact solution graph of Problem 4.4.

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Figure 14. NAAM solution graph of Problem 4.4.

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Figure 15. 3D Fractional order solution graphs of Problem 4.4.

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Figure 16. 2D Fractional order solution graph of Problem 4.4.

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Figure 17. Exact solution graph of Problem: 4.5.

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Figure 18. NAAM solution graph of Problem 4.5.

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Figure 19. 3D Fractional order solution graphs of Problem 4.5.

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Figure 20. 2D Fractional order solution graph of Problem 4.5.

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Table 1. Comparison study for solution of Problem 4.2.

${\tau}$	$\vartheta$	NAAM	HPM ^[33]	ADM ^[34]	Exact
0.2	0.25	0.305333	0.305333	0.305333	0.305351
0.2	0.50	0.610667	0.610667	0.610667	0.610701
0.2	0.75	0.916000	0.916000	0.916000	0.916052
0.2	1.00	1.221333	1.221333	1.221333	1.221403
0.4	0.25	0.372667	0.372667	0.372667	0.372956
0.4	0.50	0.745333	0.745333	0.745333	0.745912
0.4	0.75	1.118000	1.118000	1.118000	1.118869
0.4	1.00	1.490667	1.490667	1.490667	1.491825
0.6	0.25	0.454000	0.454000	0.454000	0.455530
0.6	0.50	0.908000	0.908000	0.908000	0.911059
0.6	0.75	1.362000	1.362000	1.362000	1.366589
0.6	1.00	1.816000	1.816000	1.816000	1.822119

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Table 2. Comparison study for solution of Problem 4.3.

${\tau}$	$\vartheta$	NAAM	HPM ^[33]	ADM ^[34]	Exact
0.2	0.25	0.076333	0.076333	0.076333	0.076338
	0.50	0.305333	0.305333	0.305333	0.305351
	0.75	0.687000	0.687000	0.687000	0.687039
	1.00	1.221333	1.221333	1.221333	1.221403
0.4	0.25	0.093167	0.093167	0.093167	0.093239
	0.50	0.372667	0.372667	0.372667	0.372956
	0.75	0.838500	0.838500	0.838500	0.839151
	1.00	1.490667	1.490667	1.490667	1.491825
0.6	0.25	0.113500	0.113500	0.11350	0 0.113882
	0.50	0.454000	0.454000	0.454000	0.455530
	0.75	1.021500	1.021500	1.021500	1.024942
	1.00	1.816000	1.816000	1.816000	1.822119

| Show Table

DownLoad: CSV

5. Conclusions

In these research notes, we have applied an analytical procedure known as the new approximate analytical method (NAAM). The NAAM provides power series solutions in the form of convergent series with fractional derivatives. It is more powerful than other analytical methods due to its less computational work. It provides the fractional order solution directly using the Caputo-Riemann property. We have tested some Fokker-Planck equations in linear and non-linear cases. The series form solution and graphical representation reflect the applicability and validity. The main advantage of NAAM is that it significantly minimises the numerical computations required to find an analytical solution with fractional order. The fractional order solutions are also found and verified by 3D and 2D representation. Overall, the NAAM provides series form solution with fractional order, which have good agreement with the exact solutions of the problems.

Acknowledgements

Researchers Supporting Project number (RSP-2021/401), King Saud University, Riyadh, Saudi Arabia. This research was funded by National Science, Research and Innovation Fund (NSRF), and King Mongkut's University of Technology North Bangkok with Contract no. KMUTNB-FF-65-24.

Conflict of interest

The authors declare that they have no competing interests.

References

[1]	Z. Li, J. Lyu, E. Hussain, Bifurcation, chaotic behaviors and solitary wave solutions for the fractional Twin-Core couplers with Kerr law non-linearity, Sci. Rep., 14 (2024), 22616. https://doi.org/10.1038/s41598-024-74044-w doi: 10.1038/s41598-024-74044-w
[2]	Z. Li, S. Zhao, Bifurcation, chaotic behavior and solitary wave solutions for the Akbota equation, AIMS Mathematics, 9 (2024), 22590–22601. https://doi.org/10.3934/math.20241100 doi: 10.3934/math.20241100
[3]	M. Gu, J. Li, F. Liu, Z. Li, C. Peng, Propagation of traveling wave solution of the strain wave equation in microcrystalline materials, Open Phys., 22 (2024), 20240093. https://doi.org/10.1515/phys-2024-0093 doi: 10.1515/phys-2024-0093
[4]	M. Gu, F. Liu, J. Li, C. Peng, Z. Li, Explicit solutions of the generalized Kudryashov's equation with truncated M-fractional derivative, Sci. Rep., 14 (2024), 21714. https://doi.org/10.1038/s41598-024-72610-w doi: 10.1038/s41598-024-72610-w
[5]	K. Zhang, J. Cao, J. Lyu, Dynamic behavior and modulation instability for a generalized nonlinear Schrödinger equation with nonlocal nonlinearity, Phys. Scr., 100 (2025), 015262. https://doi.org/10.1088/1402-4896/ad9cfa doi: 10.1088/1402-4896/ad9cfa
[6]	J. Atai, B. A. Malomed, Gap solitons in Bragg gratings with dispersive reflectivity, Phys. Lett. A, 342 (2005), 404–412. https://doi.org/10.1016/j.physleta.2005.05.081 doi: 10.1016/j.physleta.2005.05.081
[7]	Y. Han, B. Gao, H. Wen, C. Ma, J. Huo, Y. Li, et al., Pure-high-even-order dispersion bound solitons complexes in ultra-fast fiber lasers, Light Sci. Appl., 13 (2024), 101. https://doi.org/10.1038/s41377-024-01451-z doi: 10.1038/s41377-024-01451-z
[8]	A. Jose, S. Das Chowdhury, S. Balasubramanian, K. Krupa, Z. Wang, B. N. Upadhyay, et al., Noise‐like pulse seeded supercontinuum generation: An in‐depth review for high‐energy flat broadband sources, Laser Photonics Rev., 19 (2025), 2400511. 2400511. https://doi.org/10.1002/lpor.202400511 doi: 10.1002/lpor.202400511
[9]	T. Nagy, P. Simon, L. Veisz, High-energy few-cycle pulses: Post-compression techniques, Adv. Phys. X, 6 (2021), 1845795. https://doi.org/10.1080/23746149.2020.1845795 doi: 10.1080/23746149.2020.1845795
[10]	M. R. Fernández-Ruiz, A. Carballar, Fiber Bragg grating-based optical signal processing: Review and survey, Appl. Sci., 11 (2021), 8189. https://doi.org/10.3390/app11178189 doi: 10.3390/app11178189
[11]	J. Li, Y. Zhang, J. Zeng, Dark gap solitons in one-dimensional nonlinear periodic media with fourth-order dispersion, Chaos Soliton. Fract., 157 (2022), 111950. https://doi.org/10.1016/j.chaos.2022.111950 doi: 10.1016/j.chaos.2022.111950
[12]	N. Pernet, P. St-Jean, D. D. Solnyshkov, G. Malpuech, N. C. Zambon, Q. Fontaine, et al., Gap solitons in a one-dimensional driven-dissipative topological lattice, Nat. Phys., 18 (2022), 678–684. https://doi.org/10.1038/s41567-022-01599-8 doi: 10.1038/s41567-022-01599-8
[13]	E. M. E. Zayed, M. E. M. Alngar, A. Biswas, M. Ekici, S. Khan, A. K. Alzahrani, et al., Optical solitons in fiber Bragg gratings with quadratic-cubic law of nonlinear refractive index and cubic-quartic dispersive reflectivity, P. Est. Acad. Sci., 71 (2022), 165–177. https://doi.org/10.3176/proc.2022.2.05 doi: 10.3176/proc.2022.2.05
[14]	X. Xue, X. Zheng, B. Zhou, Super-efficient temporal solitons in mutually coupled optical cavities, Nat. Photonics, 13 (2019), 616–622. https://doi.org/10.1038/s41566-019-0436-0 doi: 10.1038/s41566-019-0436-0
[15]	G. Arora, R. Rani, H. Emadifar, Soliton: A dispersion-less solution with existence and its types, Heliyon, 8 (2022), e12122. https://doi.org/10.1016/j.heliyon.2022.e12122 doi: 10.1016/j.heliyon.2022.e12122
[16]	A. Bougaud, A. Fernandez, A. Ly, S. Balac, O. Llopis, Complex pulse profile optimization by chromatic dispersion management in coupled opto-electronic oscillator based on semiconductor optical amplifier, IEEE J. Quantum Elect., 60 (2024), 1300209. https://doi.org/10.1109/JQE.2024.3372575 doi: 10.1109/JQE.2024.3372575
[17]	T. Kori, A. Kori, A. Kori, S. Nandi, A Study on fiber Bragg gratings and its recent applications, In: Innovative mobile and internet services in ubiquitous computing, 2020,880–889. https://doi.org/10.1007/978-3-030-22263-5_84
[18]	K. S. Al-Ghafri, M. Sankar, E. V. Krishnan, A. Biswas, A. Asiri, Chirped gap solitons with Kudryashov's law of self-phase modulation having dispersive reflectivity, J. Eur. Opt. Society-Rapid Publ., 19 (2023), 40. https://doi.org/10.1051/jeos/2023038 doi: 10.1051/jeos/2023038
[19]	F. Yang, K. Zhu, X. Yu, T. Liu, K. Lu, Z. Wang, et al., Air gap fiber Bragg grating for simultaneous strain and temperature measurement, Micromachines, 15 (2024), 140. https://doi.org/10.3390/mi15010140 doi: 10.3390/mi15010140
[20]	M. Bonopera, Fiber-Bragg-grating-based displacement sensors: Review of recent advances, Materials, 15 (2022), 5561. https://doi.org/10.3390/ma15165561 doi: 10.3390/ma15165561
[21]	C. V. N. Bhaskar, S. Pal, P. K. Pattnaik, Performance enhancement of optical communication system with cascaded FBGs of varying lengths, J. Opt., 25 (2023), 125701. https://doi.org/10.1088/2040-8986/ad0def doi: 10.1088/2040-8986/ad0def
[22]	A. Theodosiou, Recent advances in fiber Bragg grating sensing, Sensors, 24 (2024), 532. https://doi.org/10.3390/s24020532 doi: 10.3390/s24020532
[23]	I. Samir, H. M. Ahmed, Extracting stochastic solutions for complex Ginzburg–Landau model with chromatic dispersion and Kerr law nonlinearity using improved modified extended tanh technique, Opt. Quant. Electron., 56 (2024), 824. https://doi.org/10.1007/s11082-024-06677-0 doi: 10.1007/s11082-024-06677-0
[24]	Y. Alhojilan, H. M. Ahmed, Investigating the noise effect on the CGL model having parabolic law of nonlinearity, Results Phys., 53 (2023), 106952. https://doi.org/10.1016/j.rinp.2023.106952 doi: 10.1016/j.rinp.2023.106952
[25]	W. B. Rabie, H. M. Ahmed, Optical solitons for multiple-core couplers with polynomial law of nonlinearity using the modified extended direct algebraic method, Optik, 258 (2022), 168848. https://doi.org/10.1016/j.ijleo.2022.168848 doi: 10.1016/j.ijleo.2022.168848
[26]	E. Zayed, R. Shohib, B. Anjan, Y. Yakup, A. Maggie, P. Seithuti, et al., Gap solitons with cubic-quartic dispersive reflectivity and parabolic law of nonlinear refractive index, Ukr. J. Phys. Opt., 24 (2023), 4030–4045.
[27]	M. S. Ahmed, A. A. S. Zaghrout, H. M. Ahmed, Exploration new solitons in fiber Bragg gratings with cubic–quartic dispersive reflectivity using improved modified extended tanh-function method, Eur. Phys. J. Plus, 138 (2023), 32. https://doi.org/10.1140/epjp/s13360-023-03666-2 doi: 10.1140/epjp/s13360-023-03666-2
[28]	A. Biswas, J. Vega-Guzman, M. F. Mahmood, S. Khan, Q. Zhou, S. P. Moshokoa, et al., Solitons in optical fiber Bragg gratings with dispersive reflectivity, Optik, 182 (2019), 119–123. https://doi.org/10.1016/j.ijleo.2018.12.180 doi: 10.1016/j.ijleo.2018.12.180
[29]	L. Zeng, J. Shi, M. R. Belić, D. Mihalache, J. Chen, J. Li, et al., Surface gap solitons in the Schrödinger equation with quintic nonlinearity and a lattice potential, Opt. Express, 31 (2023), 35471–35483. https://doi.org/10.1364/OE.497973 doi: 10.1364/OE.497973
[30]	C. M. de Sterke, J. E. Sipe, Gap solitons, Prog. Optics, 33 (1994), 203–260. https://doi.org/10.1016/S0079-6638(08)70515-8 doi: 10.1016/S0079-6638(08)70515-8
[31]	K. S. Al-Ghafri, M. Sankar, E. V. Krishnan, S. Khan, A. Biswas, Chirped gap solitons in fiber Bragg gratings with polynomial law of nonlinear refractive index, J. Eur. Opt. Society-Rapid Publ., 19 (2023), 30. https://doi.org/10.1051/jeos/2023025 doi: 10.1051/jeos/2023025
[32]	A. S. Khalifa, N. M. Badra, H. M. Ahmed, W. B. Rabie, Retrieval of optical solitons in fiber Bragg gratings for high-order coupled system with arbitrary refractive index, Optik, 287 (2023), 171116. https://doi.org/10.1016/j.ijleo.2023.171116 doi: 10.1016/j.ijleo.2023.171116
[33]	E. M. E. Zayed, R. M. A. Shohib, A. Biswas, M. Ekici, H. Triki, A. K. Alzahrani, et al., Optical solitons with fiber Bragg gratings and dispersive reflectivity having parabolic–nonlocal combo nonlinearity via three prolific integration architectures, Optik, 208 (2020), 164065. https://doi.org/10.1016/j.ijleo.2019.164065 doi: 10.1016/j.ijleo.2019.164065
[34]	E. M. E. Zayed, R. M. A. Shohib, A. Biswas, O. González-Gaxiola, Y. Yıldırım, A. K. Alzahrani, et al., Optical solitons in fiber Bragg gratings with generalized anti-cubic nonlinearity by extended auxiliary equation, Chinese J. Phys., 65 (2020), 613–628. https://doi.org/10.1016/j.cjph.2020.03.017 doi: 10.1016/j.cjph.2020.03.017
[35]	E. M. E. Zayed, M. E. M. Alngar, A. Biswas, M. Ekici, A. K. Alzahrani, M. R. Belic, Chirped and chirp-free optical solitons in fiber Bragg gratings with Kudryashov's model in presence of dispersive reflectivity, J. Commun. Technol. Electron., 65 (2020), 1267–1287. https://doi.org/10.1134/S1064226920110200 doi: 10.1134/S1064226920110200
[36]	N. A. Kudryashov, Periodic and solitary waves in optical fiber Bragg gratings with dispersive reflectivity, Chinese J. Phys., 66 (2020), 401–405. https://doi.org/10.1016/j.cjph.2020.06.006 doi: 10.1016/j.cjph.2020.06.006
[37]	H. Rezazadeh, M. Inc, D. Baleanu, New solitary wave solutions for variants of (3+1)-dimensional Wazwaz-Benjamin-Bona-Mahony equations, Front. Phys., 8 (2020), 332. https://doi.org/10.3389/fphy.2020.00332 doi: 10.3389/fphy.2020.00332
[38]	L. Akinyemi, U. Akpan, P. Veeresha, H. Rezazadeh, M. Inc, Computational techniques to study the dynamics of generalized unstable nonlinear Schrödinger equation, J. Ocean Eng. Sci., 2022. In press. https://doi.org/10.1016/j.joes.2022.02.011
[39]	S. Ibrahim, T. A. Sulaiman, A. Yusuf, A. S. Alshomrani, D. Baleanu, Families of optical soliton solutions for the nonlinear Hirota-Schrodinger equation, Opt. Quant. Electron., 54 (2022), 722. https://doi.org/10.1007/s11082-022-04149-x doi: 10.1007/s11082-022-04149-x
[40]	N. M. Kamel, H. M. Ahmed, W. B. Rabie, Retrieval of soliton solutions for $4th$ -order $(2+1)$ -dimensional Schrödinger equation with higher-order odd and even terms by modified Sardar sub-equation method, Ain Shams Eng. J., 15 (2024), 102808. https://doi.org/10.1016/j.asej.2024.102808 doi: 10.1016/j.asej.2024.102808
[41]	E. M. E. Zayed, M. E. M. Alngar, R. M. A. Shohib, A. Biswas, Y. Yildirim, C. M. B. Dragomir, et al., Highly dispersive gap solitons in optical fibers with dispersive reflectivity having parabolic nonlocal nonlinearity, Ukr. J. Phys. Opt., 25 (2024), 01033–01044. https://doi.org/10.3116/16091833/Ukr.J.Phys.Opt.2024.01033 doi: 10.3116/16091833/Ukr.J.Phys.Opt.2024.01033
[42]	M. Soliman, H. M. Ahmed, N. Badra, T. A. Nofal, I. Samir, Highly dispersive gap solitons for conformable fractional model in optical fibers with dispersive reflectivity solutions using the modified extended direct algebraic method, AIMS Mathematics, 9 (2024), 25205–25222. https://doi.org/10.3934/math.20241229 doi: 10.3934/math.20241229
[43]	A. J. M. Jawad, Y. Yildirim, L. Hussein, A. Biswas, Sequel to "Highly dispersive optical solitons with differential group delay for kerr law of self—phase modulation by sardar sub—equation approach": non—local law of self—phase modulation, J. Opt., 2024. https://doi.org/10.1007/s12596-024-02185-2
[44]	E. M. E. Zayed, R. M. A. Shohib, M. E. M. Alngar, M. El-Shater, A. Biswas, Y. Yildirim, et al., Alshomrani, Highly dispersive optical gap solitons with Kundu-Eckhaus equation having multiplicative white noise, Contemp. Math., 5 (2024), 1949–1965. https://doi.org/10.37256/cm.5220244141 doi: 10.37256/cm.5220244141
[45]	A. Patel, V. Kumar, Modulation instability analysis of a nonautonomous (3+1)-dimensional coupled nonlinear Schrödinger equation, Nonlinear Dyn., 104 (2021), 4355–4365. https://doi.org/10.1007/s11071-021-06558-1 doi: 10.1007/s11071-021-06558-1
[46]	K. Zhang, Z. Li, J. Cao, Qualitative analysis and modulation instability for the extended (3+1)-dimensional nonlinear Schrödinger equation with conformable derivative, Results Phys., 61 (2024), 107713. https://doi.org/10.1016/j.rinp.2024.107713 doi: 10.1016/j.rinp.2024.107713
[47]	Y. Chahlaoui, A. Ali, S. Javed, Study the behavior of soliton solution, modulation instability and sensitive analysis to fractional nonlinear Schrödinger model with Kerr Law nonlinearity, Ain Shams Eng. J., 15 (2024), 102567. https://doi.org/10.1016/j.asej.2023.102567 doi: 10.1016/j.asej.2023.102567
[48]	H. Ahmad, K. U. Tariq, S. M. R. Kazmi, Stability, modulation instability and traveling wave solutions of (3+1)dimensional Schrödinger model in physics, Opt. Quant. Electron., 56 (2024), 1237. https://doi.org/10.1007/s11082-024-07031-0 doi: 10.1007/s11082-024-07031-0
[49]	A. K. Chakrabarty, S. Akter, M. Uddin, M. M. Roshid, A. Abdeljabbar, H. Or-Roshid, Modulation instability analysis, and characterize time-dependent variable coefficient solutions in electromagnetic transmission and biological field, Part. Differ. Equ. Appl. Math., 11 (2024), 100765. https://doi.org/10.1016/j.padiff.2024.100765 doi: 10.1016/j.padiff.2024.100765
[50]	K. K. Ali, S. Tarla, M. R. Ali, A. Yusuf, Modulation instability analysis and optical solutions of an extended (2+1)-dimensional perturbed nonlinear Schrödinger equation, Results Phys., 45 (2023), 106255. https://doi.org/10.1016/j.rinp.2023.106255 doi: 10.1016/j.rinp.2023.106255
[51]	H. Alsaud, M. Youssoufa, M. Inc, I. E. Inan, H. Bicer, Some optical solitons and modulation instability analysis of (3+1)-dimensional nonlinear Schrödinger and coupled nonlinear Helmholtz equations, Opt. Quant. Electron., 56 (2024), 1138. https://doi.org/10.1007/s11082-024-06851-4 doi: 10.1007/s11082-024-06851-4
[52]	H. Ur. Rehman, A. U. Awan, A. M. Hassan, S. Razzaq, Analytical soliton solutions and wave profiles of the (3+1)-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation, Results Phys., 52 (2023), 106769. https://doi.org/10.1016/j.rinp.2023.106769 doi: 10.1016/j.rinp.2023.106769
[53]	S. Li, X. Zhao, Y. He, L. Wang, W. Fan, X. Gao, et al., Er-doped fiber lasers with all-fiber dispersion management based on Cr $_2$ Sn $_2$ Te $_6$ saturable absorbers, Opt. Laser Technol., 175 (2024), 110729. https://doi.org/10.1016/j.optlastec.2024.110729 doi: 10.1016/j.optlastec.2024.110729
[54]	S. Peng, Z. Wang, F. Hu, Z. Li, Q. Zhang, P. Lu, 260 fs, 403 W coherently combined fiber laser with precise high-order dispersion management, Front. Optoelectron., 17 (2024), 3. https://doi.org/10.1007/s12200-024-00107-5 doi: 10.1007/s12200-024-00107-5
[55]	X. Cao, X. Li, S. Li, Z. Cheng, Y. Xiong, Y. Feng, et al., All-fiber low-noise $1.06 \mu m$ optical frequency comb generated by a figure-9 laser with chirped fiber Bragg grating, Opt. Fiber Technol., 86 (2024), 103853. https://doi.org/10.57760/sciencedb.15246 doi: 10.57760/sciencedb.15246

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Abstract

1. Introduction

2. Preliminaries and basic concepts

2.1. Riemann-Liouuille integral operator [32]

2.2. Some properties of Riemann-Liouuille fractional partial integral

2.3. Caputo operator of fractional partial derivative [36]

2.4. Theorem [36]

2.5. Theorem

3. NAAM procedure for the solution of problems[37]

3.1. Lemma [37]

3.2. Theorem [37]

3.3. Definition [1]

3.4. Definition [31]

3.5. The existence and uniqueness of NAAM [37]

4. NAAM implementation and results discussion [37]

5. Conclusions

Acknowledgements

Conflict of interest

References

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Abstract

1. Introduction

2. Preliminaries and basic concepts

2.1. Riemann-Liouuille integral operator [32]

2.2. Some properties of Riemann-Liouuille fractional partial integral

2.3. Caputo operator of fractional partial derivative [36]

2.4. Theorem [36]

2.5. Theorem

3. NAAM procedure for the solution of problems[37]

3.1. Lemma [37]

3.2. Theorem [37]

3.3. Definition [1]

3.4. Definition [31]

3.5. The existence and uniqueness of NAAM [37]

4. NAAM implementation and results discussion [37]

5. Conclusions

Acknowledgements

Conflict of interest

References

2.1. Riemann-Liouuille integral operator ^[32]

2.3. Caputo operator of fractional partial derivative ^[36]

2.4. Theorem ^[36]

3. NAAM procedure for the solution of problems^[37]

3.1. Lemma ^[37]

3.2. Theorem ^[37]

3.3. Definition ^[1]

3.4. Definition ^[31]

3.5. The existence and uniqueness of NAAM ^[37]

4. NAAM implementation and results discussion ^[37]

2.1. Riemann-Liouuille integral operator ^[32]

2.3. Caputo operator of fractional partial derivative ^[36]

2.4. Theorem ^[36]

3. NAAM procedure for the solution of problems^[37]

3.1. Lemma ^[37]

3.2. Theorem ^[37]

3.3. Definition ^[1]

3.4. Definition ^[31]

3.5. The existence and uniqueness of NAAM ^[37]

4. NAAM implementation and results discussion ^[37]