$ q $-rung logarithmic Pythagorean neutrosophic vague normal aggregating operators and their applications in agricultural robotics

Murugan Palanikumar; Chiranjibe Jana; Biswajit Sarkar; Madhumangal Pal; Murugan Palanikumar; Chiranjibe Jana; Biswajit Sarkar; Madhumangal Pal

doi:10.3934/math.20231544

AIMS Mathematics

2023, Volume 8, Issue 12: 30209-30243. doi: 10.3934/math.20231544

Previous Article Next Article

Research article Special Issues

$q$ -rung logarithmic Pythagorean neutrosophic vague normal aggregating operators and their applications in agricultural robotics

1.
Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai 602105, India
2.
Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, India
3.
Department of Industrial Engineering, Yonsei University, 50 Yonsei-ro, Sinchon-dong, Seodaemun-gu, Seoul 03722, South Korea
4.
Center for Global Health Research, Saveetha Medical College, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Chennai, Tamil Nadu 600077, India

Received: 18 April 2023 Revised: 18 August 2023 Accepted: 11 September 2023 Published: 07 November 2023
MSC : 06D72, 90B50

The article explores multiple attribute decision making problems through the use of the Pythagorean neutrosophic vague normal set (PyNVNS). The PyNVNS can be generalized to the Pythagorean neutrosophic interval valued normal set (PyNIVNS) and vague set. This study discusses $q$ -rung log Pythagorean neutrosophic vague normal weighted averaging ( $q$ -rung log PyNVNWA), $q$ -rung logarithmic Pythagorean neutrosophic vague normal weighted geometric ( $q$ -rung log PyNVNWG), $q$ -rung log generalized Pythagorean neutrosophic vague normal weighted averaging ( $q$ -rung log GPyNVNWA), and $q$ -rung log generalized Pythagorean neutrosophic vague normal weighted geometric ( $q$ -rung log GPyNVNWG) sets. The properties of $q$ -rung log PyNVNSs are discussed based on algebraic operations. The field of agricultural robotics can be described as a fusion of computer science and machine tool technology. In addition to crop harvesting, other agricultural uses are weeding, aerial photography with seed planting, autonomous robot tractors and soil sterilization robots. This study entailed selecting five types of agricultural robotics at random. There are four types of criteria to consider when choosing a robotics system: robot controller features, cheap off-line programming software, safety codes and manufacturer experience and reputation. By comparing expert judgments with the criteria, this study narrows the options down to the most suitable one. Consequently, $q$ has a significant effect on the results of the models.

Keywords:

Citation: Murugan Palanikumar, Chiranjibe Jana, Biswajit Sarkar, Madhumangal Pal. $q$ -rung logarithmic Pythagorean neutrosophic vague normal aggregating operators and their applications in agricultural robotics[J]. AIMS Mathematics, 2023, 8(12): 30209-30243. doi: 10.3934/math.20231544

Related Papers:

[1]	Ali Khalouta, Abdelouahab Kadem . A new computational for approximate analytical solutions of nonlinear time-fractional wave-like equations with variable coefficients. AIMS Mathematics, 2020, 5(1): 1-14. doi: 10.3934/math.2020001
[2]	Musawa Yahya Almusawa, Hassan Almusawa . Numerical analysis of the fractional nonlinear waves of fifth-order KdV and Kawahara equations under Caputo operator. AIMS Mathematics, 2024, 9(11): 31898-31925. doi: 10.3934/math.20241533
[3]	Emad Salah, Ahmad Qazza, Rania Saadeh, Ahmad El-Ajou . A hybrid analytical technique for solving multi-dimensional time-fractional Navier-Stokes system. AIMS Mathematics, 2023, 8(1): 1713-1736. doi: 10.3934/math.2023088
[4]	Humaira Yasmin, Aljawhara H. Almuqrin . Analytical study of time-fractional heat, diffusion, and Burger's equations using Aboodh residual power series and transform iterative methodologies. AIMS Mathematics, 2024, 9(6): 16721-16752. doi: 10.3934/math.2024811
[5]	Mariam Sultana, Muhammad Waqar, Ali Hasan Ali, Alina Alb Lupaş, F. Ghanim, Zaid Ameen Abduljabbar . Numerical investigation of systems of fractional partial differential equations by new transform iterative technique. AIMS Mathematics, 2024, 9(10): 26649-26670. doi: 10.3934/math.20241296
[6]	Qasem M. Tawhari . Mathematical analysis of time-fractional nonlinear Kuramoto-Sivashinsky equation. AIMS Mathematics, 2025, 10(4): 9237-9255. doi: 10.3934/math.2025424
[7]	M. Mossa Al-Sawalha, Khalil Hadi Hakami, Mohammad Alqudah, Qasem M. Tawhari, Hussain Gissy . Novel Laplace-integrated least square methods for solving the fractional nonlinear damped Burgers' equation. AIMS Mathematics, 2025, 10(3): 7099-7126. doi: 10.3934/math.2025324
[8]	Nader Al-Rashidi . Innovative approaches to fractional modeling: Aboodh transform for the Keller-Segel equation. AIMS Mathematics, 2024, 9(6): 14949-14981. doi: 10.3934/math.2024724
[9]	Humaira Yasmin, Aljawhara H. Almuqrin . Efficient solutions for time fractional Sawada-Kotera, Ito, and Kaup-Kupershmidt equations using an analytical technique. AIMS Mathematics, 2024, 9(8): 20441-20466. doi: 10.3934/math.2024994
[10]	Zeliha Korpinar, Mustafa Inc, Dumitru Baleanu . On the fractional model of Fokker-Planck equations with two different operator. AIMS Mathematics, 2020, 5(1): 236-248. doi: 10.3934/math.2020015

Abstract

1. Introduction

Many engineering and scientific issues have been solved using fractional differential equations (FDEs). In the past two decades, fractional differential equations have garnered great attention due to their capacity to simulate numerous events in various scientific and engineering disciplines. Models based on fractional differential equations can depict various physical applications in science and engineering ^[1,2,3]. These models are precious for a wide range of physical problems. These equations are represented by fractional linear and non-linear PDEs, and fractional differential equations must be solved ^[1,2,3]. Most non-linear FDEs need approximate and numerical solutions since they cannot be solved precisely ^[7,8,9]. Variational iteration method ^[10], Adomian decomposition method ^[11], homotopy analysis method ^[12], homotopy perturbation method ^[13], tanh-coth method ^[14], spectral collocation method ^[15], Mittag-Leffler function method ^[16], exp function method ^[17] and differential quadrature method ^[18], are a few of the more recent analytical techniques for non-linear problems ^[19,20,21].

The Gardner equation ^[22] is an amalgamation of KdV and modified KdV equations, and it is generated to demonstrate the description of solitary inner waves in shallow water. The Gardner equation is frequently utilized in many areas of physics, including quantum area theories, plasma physics and fluid physics ^[23,24]. It also discusses many wave phenomena in the plasma and solid states ^[25]. In the current research, we recognize the fractional Gardner (FG) equation of the form ^[26]:

$D^\alpha_\eta u(\omega, \eta)+6\left(u-\lambda^2 u^2\right) u_\omega+u_{\omega \omega \omega} = 0, \; 0 < \alpha \leq 1,$

where $\lambda$ is real constant, here, $u(\omega, \eta)$ is the wave term with scaling variable spaces $(\omega)$ and time $(\eta)$ , the functions $u u_\omega$ and $u^2 u_\omega$ are symbolizes the non-linear wave steepening and $u_{\omega \omega \omega}$ defines the wave dispersive effect.

The Cahn-Hilliard equation, first presented in 1958 by Cahn and Hilliard ^[27], serves as an example of the phase separation of a binary alloy under critical temperature. This equation is a key component of several intriguing physical processes, including spinodal decomposition, phase separation, and phase ordering dynamics ^[28,29]. In this framework, the following fractional Cahn-Hilliard (FCH) equation is taken into consideration ^[26,30] :

$D^\alpha_\eta u(\omega, \eta)-u_\omega-6 u u_\omega^2-\left(3 u^2-1\right) u_{\omega \omega}+u_{\omega \omega \omega \omega} = 0, \; 0 < \alpha \leq 1.$

Several novels and cutting-edge approaches to studying non-linear differential systems with fractional order have been developed over the previous thirty years, concurrently developing new computing techniques and symbolic programming. In the pre-computer age, most complex phenomena, such as solitons, chaos, singular formation, asymptotic characteristics, etc., remained unnoticed or, at best, weakly projected. This revolution in understanding has been sparked by analytical methods, new mathematical theories and computing techniques that let us investigate non-linear complex events. Many techniques have been used, including the F-expansion method ^[31], the q-Homotopy analysis method ^[32], the reduced differential transform method ^[33], the generalized Kudryashov method ^[34], the sub equation method ^[35], the Adomian decomposition method, the homotopy analysis method ^[36], variational iteration method ^[37], and improved (G/G)-expansion method ^[38].

The Jordanian mathematician, Omar Abu Arqub created the residual power series method in 2013 as a technique for quickly calculating the coefficients of the power series solutions for 1st and 2nd-order fuzzy differential equations ^[39]. Without perturbation, linearization, or discretization, the residual power series method provides a powerful and straightforward power series solution for highly linear and non-linear equations. The residual power series method has been used to solve an increasing variety of non-linear ordinary and partial differential equations of various sorts, orders, and classes during the past several years. It has been used to make solitary pattern results for non-linear dispersive fractional partial differential equations and to predict them ^[40], to solve the non-linear singular highly differential equation known as the generalized Lane-Emden equation ^[41], to solve higher-order ordinary differential equations numerically ^[42], to approximate solve the fractional non-linear KdV-Burger equations, to predict and represent The RPSM differs from several other analytical and numerical approaches in some crucial ways ^[43]. First, there is no requirement for a recursion connection or for the RPSM to compare the coefficients of the related terms. Second, by reducing the associated residual error, the RPSM offers a straightforward method to guarantee the convergence of the series solution. Thirdly, the RPSM doesn't suffer from computational rounding mistakes and doesn't use a lot of time or memory. Fourth, the approach may be used immediately to the provided issue by selecting an acceptable starting guess approximation since the residual power series method does not need any converting when transitionary from low-order to higher-order and from simple linearity to complicated nonlinearity ^[44,45,46].

This article uses the Laplace residual power series technique to achieve the definitive solution of the fractional-order non-linear Cahn-Hilliard and Gardner equations. The Laplace transformation efficiently integrates the RPSM for the renewability algorithmic technique. The fractional Caputo derivative explains quantitative categorizations of the Gardner and Cahn-Hilliard equations. The offered methodology is well demonstrated in modeling and calculation investigation.

2. Preliminaries

Definition 2.1. The fractional Caputo derivative of a function ${u}({\omega}, {\eta})$ of order ${\alpha}$ is given as

$\begin{equation} ^{C}D_{{\eta}}^{{\alpha}}{u}({\omega}, {\eta}) = J_{{\eta}}^{m-{\alpha}}{u}^{{m}}({\omega}, {\eta}), \; m-1 < \alpha\leq m, \; \eta > 0, \end{equation}$

(2.1)

where $m\in N$ and $J_{{\eta}}^{\alpha}$ is the fractional Riemann-Liouville integral of ${u}({\omega}, {\eta})$ of fractional-order $\alpha$ is define as

$\begin{equation} J_{{\eta}}^{\alpha}{u}({\omega}, {\eta}) = \frac{1}{\Gamma(\alpha)}\int_{0}^{{\eta}}({t}-\omega){u}({\omega} , {t})d{t}, \; \alpha > 0, \ \end{equation}$

(2.2)

assuming that the given integral exists.

Definition 2.2. Assume that the continuous piecewise function ${u}(\omega, {\eta})$ is expressed as:

$\begin{equation} {u}(\omega, {\upsilon}) = {{\rm{£}}}_{{\eta}}[{u}(\omega , {\eta})] = \int_{0}^{\infty}e^{-{\upsilon}{\eta}}{u}(\omega, {\eta})d{\eta}, \; {\upsilon} > {\alpha}, \end{equation}$

(2.3)

where the inverse Laplace transform is expressed as

$\begin{equation} {u}(\omega, {\eta}) = {{\rm{£}}}_{{\upsilon}}^{-1}[{u}](\omega , {\upsilon})] = \int_{l-i\infty}^{l+i\infty}e^{{\upsilon}{\eta}}{u}(\omega, {\upsilon})d{\upsilon}, \; l = Re({\upsilon}) > {0}. \end{equation}$

(2.4)

Lemma 2.1. Suppose that ${u}(\omega, {\eta})$ is a piecewise continuous term and of exponential-order ${\psi}$ and ${u}(\omega, {\upsilon}) = {{\rm{£}}}_{\eta}[{u}(\omega, {\eta})]$ , we get

(1) ${{\rm{£}}}_{{\eta}}[J_{\eta}^{\beta}{u}(\omega, {\eta})] = \frac{{u}(\omega, {\upsilon})}{{\upsilon}^{\beta}}, \; \beta > 0.$

(2) ${{\rm{£}}}_{{\eta}}[D_{\eta}^{\psi}{u}(\omega, {\eta})] = {\upsilon}^{\psi}{u}(\omega, {\upsilon})-\sum_{k = 0}^{m-1}{\upsilon}^{\psi-{k}-1}{u}^{{k}}(\omega, {0}), \; m-1 < \psi\leq m.$

(3) ${{\rm{£}}}_{{\eta}}[D_{\eta}^{n\psi}{u}(\omega, {\eta})] = {\upsilon}^{n\psi}{u}(\omega, {\upsilon})-\sum_{{k} = 0}^{n-1}{\upsilon}^{(n-{k})\psi-1}D_{\eta}^{{k}\psi}{u}(\omega, {0}), \; 0 < \psi\leq1.$

Proof. The proof are in ^[1,2,3,47].

Theorem 2.1. Let us assume that ${u}(\omega, {\eta})$ is a continuous piecewise on ${\bf I}\times[0, \infty)$ . Consider that ${u}(\omega, {\upsilon}) = {{\rm{£}}}_{\eta}[{u}(\omega, {\eta})]$ has fractional power series (FPS) representation:

$\begin{equation} {u}(\omega, {\upsilon}) = \sum\limits_{i = 0}^{\infty}\frac{f_{i}(\omega )}{{\upsilon}^{1+i{\alpha}}}, \; 0 < \zeta\leq1, \; \omega\in{\bf I}, \; {\upsilon} > {\psi}. \end{equation}$

(2.5)

Then, $f_{i}(\omega) = D_{\eta}^{n{\alpha}}{u}(\omega, {0})$ .

Proof. For proof, see Ref. ^[47].

Remark 2.1. The inverse Laplace transform of the Eq (2.5) represented as:

$\begin{equation} {u}(\omega, {\eta}) = \sum\limits_{i = 0}^{\infty}\frac{D_{{\eta}}^{\psi}{u}(\omega, {0})}{\Gamma(1+i{\psi})}{\eta}^{i(\psi)}, \; 0 < \psi\leq1, \; {\eta}\geq0. \end{equation}$

(2.6)

which is the same as Taylor's formula for fractions, that can be found in ^[48].

Theorem 2.2. Suppose that ${u}(\omega, {\eta})$ is piecewise continuous on ${\bf I}\times[0, \infty)$ and of order $\psi$ . As shown in Theorem 2.1, ${u}(\omega, {\upsilon}) = {{\rm{£}}}_{\eta}[{u}(\omega, {\eta})]$ . Taylor's formula can be written in its new form. If $\left|{\upsilon}{{\rm{£}}}_{\eta}[D_{\eta}^{i{\alpha}+1}{u}(\omega, {\eta})]\right|\leq M(\omega)$ , on ${\bf I}\times({\psi}, \gamma]$ , where $0 < {\alpha}\leq1,$ then $R_{i}(\omega, {\upsilon})$ the rest of the new way of writing fractions. The following inequality is true about Taylor's formula in Theorem 2.1:

$\begin{equation} |R_{i}(\omega, {\upsilon})|\leq\frac{M(\omega)}{S^{1+(i+1){\alpha}}}, \; \omega \in{\bf I}, \; {\psi} < {\upsilon}\leq\gamma. \end{equation}$

(2.7)

Proof. Let us consider that ${{\rm{£}}}_{\eta}\left[D_{\eta}^{{k}{\alpha}}{u}(\omega, {\eta})\right]({\upsilon})$ on interval ${\bf I}\times({\psi}, \gamma]$ for ${k} = 0, 1, 2, 3, \cdots, i+1$ , suppose that

$\begin{equation} \left|{\upsilon}{{\rm{£}}}_{\eta}[D_{\eta}^{i{\alpha}+1}{u}(\omega, {\eta})]\right|\leq M(\omega), \; \omega\in{\bf I}, \; {\psi} < {\upsilon}\leq\gamma. \end{equation}$

(2.8)

Using the definition of remainder, $R_{i}(\omega, {\upsilon}) = {u}(\omega, {\upsilon})-\sum_{{k} = 0}^{i}\frac{D_{\eta}^{{k}{\alpha}}{u}(\omega, {0})}{{\upsilon}^{1+{k}{\alpha}}}$ , we can obtain

$\begin{equation} \begin{split} S^{1+(i+1){\alpha}}R_{i}(\omega, {\upsilon}) = &{\upsilon}^{1+(i+1){\alpha}}{u}(\omega, {\upsilon})-\sum\limits_{{k} = 0}^{i}{\upsilon}^{(i+1-{k}){\alpha}}D_{\eta}^{{k}{\alpha}}{u}(\omega, {0})\\ = &{\upsilon}\left({\upsilon}^{(i+1){\alpha}}{u}(\omega, {\upsilon})-\sum\limits_{{k} = 0}^{i}{\upsilon}^{(i+1-{k}){\alpha}-1}D_{\eta}^{{k}{\alpha}}{u}(\omega, {0})\right)\\ = &{\upsilon}{{\rm{£}}}_{\eta}\left[D_{\eta}^{(n+1)\zeta}{u}(\omega, {\eta})\right]. \end{split} \end{equation}$

(2.9)

From Eqs (2.8) and (2.9) that $|{\upsilon}^{1+(i+1){\alpha}}R_{i}(\omega, {\upsilon})|\leq M(\omega)$ . Thus,

$\begin{equation} -M(\omega)\leq {\upsilon}^{1+(i+1){\alpha}}R_{i}(\omega, {\upsilon})\leq M(\omega), \; \omega\in{\bf I}, \; {\psi} < s\leq\gamma. \end{equation}$

(2.10)

The proof of Theorem 2.2 is completed.

3. General implementation of Laplace residual power series method

$\begin{equation} D_{\eta}^{\alpha}u(\omega, {\eta})-u_{\omega}+6uu^{2}_{\omega}-\left(3u^{2}-1\right)u_{\omega\omega}+u_{\omega\omega\omega\omega} = 0, \; 0 < \alpha\leq1, \end{equation}$

(3.1)

with the initial condition,

$\begin{equation} u(\omega, {\eta}) = f_{0}(\omega), \end{equation}$

(3.2)

where $a$ and $c$ are free constants and $D_{\eta}^{\alpha}$ is the Caputo-fractional derivative. First, we use the Laplace transformation to Eq (3.1),

$\begin{equation} {{\rm{£}}}\left[D_{\eta}^{\alpha} u(\omega, {\eta})\right] = -{{\rm{£}}}\Big[u_{\omega}+6uu^{2}_{\omega}-\left(3u^{2}-1\right)u_{\omega\omega}+u_{\omega\omega\omega\omega}\Big]. \end{equation}$

(3.3)

By the fact that ${{\rm{£}}}\left[D_{\eta}^{\alpha} u(\omega, {\eta})\right] = {\upsilon}^{a} {{\rm{£}}}[u(\omega, {\eta})]-{\upsilon}^{a-1} u(\omega, 0)$ and using the initial condition (3.2), we rewrite (3.3) as

$\begin{equation} U(\omega, {\upsilon}) = \frac{f_{0}(\omega)}{\upsilon}-\frac{b}{{\upsilon}^{\alpha}} {{\rm{£}}}\left[\left({{\rm{£}}}^{-1}{{\rm{£}}}\Big[U_{\omega}+6UU^{2}_{\omega}-\left(3U^{2}-1\right)U_{\omega\omega}+U_{\omega\omega\omega\omega}\Big]\right)\right], \end{equation}$

(3.4)

where $U(\omega, {\upsilon}) = {{\rm{£}}}[u(\omega, {\eta})]$ .

Second, we define the transform term $U(\omega, {\upsilon})$ as the following expression:

$\begin{equation} U(\omega, {\upsilon}) = \sum\limits_{n = 0}^{\infty} \frac{f_{\upsilon}(\omega)}{{\upsilon}^{n\alpha+1}} . \end{equation}$

(3.5)

The series form of kth-truncated of Eq (3.5):

$\begin{equation} U_{k}(\omega, {\upsilon}) = \sum\limits_{n = 0}^{k} \frac{f_{\upsilon}(\omega)}{{\upsilon}^{n\alpha+1}} = \frac{f_{o}(\omega)}{\upsilon}+\sum\limits_{n = 1}^{k} \frac{f_{k}(\omega)}{{\upsilon}^{n\alpha+1}} . \end{equation}$

(3.6)

The laplace residual function to (3.5) is

$\begin{equation} \begin{split} {{{\rm{£}}} R e s}_{k}(\omega, {\upsilon}) = & U_{k}(\omega, {\upsilon})-\frac{f_{0}(\omega)}{\upsilon}-\frac{b}{{\upsilon}^{\alpha}} {{\rm{£}}}\left[\left({{\rm{£}}}^{-1}{{\rm{£}}}\Big[U_{\omega}+6UU^{2}_{\omega}-\left(3U^{2}-1\right)U_{\omega\omega}+U_{\omega\omega\omega\omega}\Big]\right)\right]. \end{split} \end{equation}$

(3.7)

Third, we use a few properties that come up in the standard RPSM to point out certain facts: ${{\rm{£}}} \operatorname{Res}(\omega, {\upsilon}) = 0$ and $\lim _{k \rightarrow \infty} {{\rm{£}}} \operatorname{Res} {u}_{k}(\omega, {\upsilon}) = {{\rm{£}}} \operatorname{Res}(\omega, {\upsilon})$ for each ${\upsilon} > 0$ ; $\lim _{{\upsilon} \rightarrow \infty} {u} {{\rm{£}}} \operatorname{Res}(\omega, {\upsilon}) = 0 \Rightarrow \lim _{{\upsilon} \rightarrow \infty} {u} {{\rm{£}}} \operatorname{Res}(\omega, {\upsilon}) = 0$ ; $\lim _{{\upsilon} \rightarrow \infty} {u}^{k \alpha+1} {{\rm{£}}} \operatorname{Res}(\omega, {\upsilon}) = \lim _{{\upsilon} \rightarrow \infty} {u}^{k \alpha+1} {{\rm{£}}} \operatorname{Res}_{k}(\omega, {\upsilon}) = 0, 0 < \alpha \leq$ $1, k = 1, 2, 3, \ldots.$

So, to find the co-efficient functions $f_n(\omega),$ we solve the following scheme successively:

$\lim _{{\upsilon} \rightarrow \infty}\left({u}^{k a+1} {{{\rm{£}}}Res}_{k}(\omega, {\upsilon})\right) = 0, \; 0 < \alpha \leq 1, \; k = 1, 2, 3, \ldots.$

Finally, we use the inverse Laplace to $U_{k}(\omega, {\upsilon})$ , to achieved the kth approximated supportive result $u_{k}(\omega, {\eta})$ .

Next, we investigate the efficiency of the suggested above technique by investigating a numerical problem of the Cahn-Hilliard and Gardner models.

Example 3.1. Consider fractional-order Cahn-Hilliard equation,

$\begin{equation} \begin{split} D_{\eta}^{\alpha}u(\omega, {\eta})-u_{\omega}+6uu^{2}_{\omega}-\left(3u^{2}-1\right)u_{\omega\omega}+u_{\omega\omega\omega\omega} = 0, \; 0 < \alpha\leq1, \end{split} \end{equation}$

(3.8)

with initial condition,

$\begin{equation} \begin{split} u(\omega, 0) = \tanh\left(\frac{\sqrt{2}}{2}\omega\right). \end{split} \end{equation}$

(3.9)

The exact result when $\alpha = 1$ is

$\begin{equation} \begin{split} u(\omega, \eta) = \tanh\left(\frac{\sqrt{2}}{2}(\omega+\eta)\right). \end{split} \end{equation}$

(3.10)

Applying laplace transform to (3.8) and using the initial condition (3.9), we get

$\begin{equation} \begin{split} U(\omega, {\upsilon})& = \frac{\tanh\left(\frac{\sqrt{2}}{2}\omega\right)}{\upsilon}-\frac{1}{{\upsilon}^{\alpha}}{{\rm{£}}}_{\eta}\Big[{{\rm{£}}}_{\eta}^{-1}\left(U(\omega, {\upsilon})\right)-6{{\rm{£}}}_{\eta}^{-1}\left(U(\omega, {\upsilon})\right){{\rm{£}}}_{\eta}^{-1}\left(U^{2}_{\omega}(\omega, {\upsilon})\right)\\ &-3{{\rm{£}}}_{\eta}^{-1}\left(U(\omega, {\upsilon})\right){{\rm{£}}}_{\eta}^{-1}\left(U_{\omega\omega}(\omega, {\upsilon})\right)+{{\rm{£}}}_{\eta}^{-1}\left(U_{\omega\omega}(\omega, {\upsilon})\right)+{{\rm{£}}}_{\eta}^{-1}\left(U_{\omega\omega\omega\omega}(\omega, {\upsilon})\right)\Big]. \end{split} \end{equation}$

(3.11)

The k-th truncated term series of (3.20) is

$\begin{equation} \begin{split} U(\omega, {\upsilon}) = \frac{\tanh\left(\frac{\sqrt{2}}{2}\omega\right)}{\upsilon}+\sum\limits_{n = 1}^{k}\frac{f_{n}(\omega)}{{\upsilon}^{n\alpha+1}}, \end{split} \end{equation}$

(3.12)

and the Laplace residual k-th term is

$\begin{equation} \begin{split} {{\rm{£}}}_{\eta}Res_{k}(\omega, {\upsilon})& = U(\omega, {\upsilon})-\frac{\tanh\left(\frac{\sqrt{2}}{2}\omega\right)}{\upsilon}+\frac{1}{{\upsilon}^{\alpha}}{{\rm{£}}}_{\eta}\Big[{{\rm{£}}}_{\eta}^{-1}\left(U(\omega, {\upsilon})\right)-6{{\rm{£}}}_{\eta}^{-1}\left(U(\omega, {\upsilon})\right){{\rm{£}}}_{\eta}^{-1}\left(U^{2}_{\omega}(\omega, {\upsilon})\right)\\ &-3{{\rm{£}}}_{\eta}^{-1}\left(U(\omega, {\upsilon})\right){{\rm{£}}}_{\eta}^{-1}\left(U_{\omega\omega}(\omega, {\upsilon})\right)+{{\rm{£}}}_{\eta}^{-1}\left(U_{\omega\omega}(\omega, {\upsilon})\right)+{{\rm{£}}}_{\eta}^{-1}\left(U_{\omega\omega\omega\omega}(\omega, {\upsilon})\right)\Big]. \end{split} \end{equation}$

(3.13)

Now, to determine $f_{k}(x)$ , $k = 1, 2, 3, \cdots$ , we put the $k$ th-truncated series (3.12) into the $k$ th-Laplace residual term (3.13), multiply the solution of equation by ${\upsilon}^{k\alpha+1}$ , and then solve recursively the relation $\lim_{{\upsilon}\rightarrow \infty}[{\upsilon}^{k\alpha+1}Res_{k}(x, {\upsilon})] = 0$ , $k = 1, 2, 3, \cdots$ for $f_{k}$ . Following are the first some components of the sequences $f_{k}(x)$ :

$\begin{equation} \begin{split} f_{1}(\omega)& = \frac{{\rm{sech}}\left(\frac{\omega}{\sqrt{2}}\right)^{2}}{\sqrt{2}}, \\ f_{2}(\omega)& = -{\rm{sech}}\left(\frac{\omega}{\sqrt{2}}\right)^{2}\tanh\left(\frac{\omega}{\sqrt{2}}\right), \\ f_{3}(\omega)& = \frac{1}{8}{\rm{sech}}\left(\frac{\omega}{\sqrt{2}}\right)^{6}\left(-4\sqrt{2}+(264-96\cosh(\sqrt{2}\omega)+\sqrt{2}\sinh(2\sqrt{2}\omega))\tanh\left(\frac{\omega}{\sqrt{2}}\right)\right)\\ &+\left(\frac{-21}{2}{\rm{sech}}\left(\frac{\omega}{\sqrt{2}}\right)^{6}\tanh\left(\frac{\omega}{\sqrt{2}}\right)+12{\rm{sech}}\left(\frac{\omega}{\sqrt{2}}\right)^{4}\tanh\left(\frac{\omega}{\sqrt{2}}\right)^{3}\right)\frac{\Gamma(1+2\alpha)}{\Gamma(1+\alpha)^{2}}, \\ &\vdots \end{split} \end{equation}$

(3.14)

Putting the values of $f_{n}(\omega)$ , $(n\geq1)$ in Eq (3.12), we have

$\begin{array}{l} U(\omega, {\upsilon}) = \frac{\tanh\left(\frac{\sqrt{2}}{2}\omega\right)}{\upsilon}+\frac{{\rm{sech}}\left(\frac{\omega}{\sqrt{2}}\right)^{2}}{\sqrt{2}}\frac{1}{{\upsilon}^{\alpha+1}}-{\rm{sech}}\left(\frac{\omega}{\sqrt{2}}\right)^{2}\tanh\left(\frac{\omega}{\sqrt{2}}\right)\frac{1}{{\upsilon}^{2\alpha+1}}\\ +\frac{1}{8}{\rm{sech}}\left(\frac{\omega}{\sqrt{2}}\right)^{6}\left(-4\sqrt{2}+(264-96\cosh(\sqrt{2}\omega)+\sqrt{2}\sinh(2\sqrt{2}\omega))\tanh\left(\frac{\omega}{\sqrt{2}}\right)\right)\\ +\left(\frac{-21}{2}{\rm{sech}}\left(\frac{\omega}{\sqrt{2}}\right)^{6}\tanh\left(\frac{\omega}{\sqrt{2}}\right)+12{\rm{sech}}\left(\frac{\omega}{\sqrt{2}}\right)^{4}\tanh\left(\frac{\omega}{\sqrt{2}}\right)^{3}\right)\\ \frac{\Gamma(1+2\alpha)}{\Gamma(1+\alpha)^{2}}\frac{1}{{\upsilon}^{3\alpha+1}}\\ +\cdots. \end{array}$

(3.15)

Applying inverse Laplace transform, we get

$\begin{array}{l} u(\omega, {\eta}) = \tanh\left(\frac{\sqrt{2}}{2}\omega\right)+\frac{{\rm{sech}}\left(\frac{\omega}{\sqrt{2}}\right)^{2}}{\sqrt{2}}\frac{\eta^{\alpha}}{\Gamma(\alpha+1)}-{\rm{sech}}\left(\frac{\omega}{\sqrt{2}}\right)^{2}\tanh\left(\frac{\omega}{\sqrt{2}}\right)\frac{{\eta}^{2\alpha}}{\Gamma(2\alpha+1)}\\ +\frac{1}{8}{\rm{sech}}\left(\frac{\omega}{\sqrt{2}}\right)^{6}\left(-4\sqrt{2}+(264-96\cosh(\sqrt{2}\omega)+\sqrt{2}\sinh(2\sqrt{2}\omega))\tanh\left(\frac{\omega}{\sqrt{2}}\right)\right)\\ +\left(\frac{-21}{2}{\rm{sech}}\left(\frac{\omega}{\sqrt{2}}\right)^{6}\tanh\left(\frac{\omega}{\sqrt{2}}\right)+12{\rm{sech}}\left(\frac{\omega}{\sqrt{2}}\right)^{4}\tanh\left(\frac{\omega}{\sqrt{2}}\right)^{3}\right)\\ \frac{\Gamma(1+2\alpha)}{\Gamma(1+\alpha)^{2}}\frac{{\eta}^{3\alpha}}{\Gamma(3\alpha+1)}\\ +\cdots. \end{array}$

(3.16)

Throughout this investigation, the method are being employed to assess the precise analytical solution of fractional-order Cahn-Hilliard equation. For various spatial and temporal parameters, the Caputo fractional derivative operators in facilitate appropriate numerical findings for the Cahn-Hilliard equation option revenue framework utilizing multiple orders. In , actual and approximate solutions graph and second fractional order $\alpha = 0.8$ of Example 3.1 at $\alpha = 1$ . In , approximate result graph at $\alpha = 0.6, 0.4$ and , the approximate result at various value of $\alpha$ of Example 3.1.

Figure 1. The exact and approximate results graph and 2nd fractional-order

$\alpha = 0.8$ of Example 3.1 at

$\alpha = 1$ .

DownLoad: Full-Size Img PowerPoint

Figure 2. The approximate result graph at

$\alpha = 0.6, 0.4$ for Example 3.1.

DownLoad: Full-Size Img PowerPoint

Figure 3. The approximate result at various value of

$\alpha$ for Example 3.1.

DownLoad: Full-Size Img PowerPoint

Example 3.2. Consider the homogeneous fractional Gardner equation,

$\begin{equation} \begin{split} D_{\eta}^{\alpha}u(\omega, {\eta})+6\left(u-\epsilon^{2}u^{2}\right)u_{\omega}+u_{\omega\omega\omega} = 0, \; 0 < \alpha\leq1, \end{split} \end{equation}$

(3.17)

with the initial condition,

$\begin{equation} \begin{split} u(\omega, 0) = \frac{1}{2}+\frac{1}{2}\tanh\Big(\frac{\omega}{2}\Big). \end{split} \end{equation}$

(3.18)

The exact result when $\epsilon = 1$ , $\alpha = 1$ is

$\begin{equation} \begin{split} u(\omega, {\eta}) = \frac{1}{2}+\frac{1}{2}\tanh\Big(\frac{\omega-\eta}{2}\Big). \end{split} \end{equation}$

(3.19)

Applying Laplace transform to (3.17) and using the initial condition (3.18), we get

$\begin{equation} \begin{split} U(\omega, {\upsilon})& = \left(\frac{1}{2}+\frac{1}{2}\tanh(\frac{\omega}{2})\right)\frac{1}{\upsilon}-\frac{1}{{\upsilon}^{\alpha}}{{\rm{£}}}_{\eta}\Big[6\Big[{{\rm{£}}}_{\eta}^{-1}\left[U(\omega, {\upsilon})\right]{{\rm{£}}}_{\eta}^{-1}\left[U_{\omega}(\omega, {\upsilon})\right]\\ &-\epsilon^{2}{{\rm{£}}}_{\eta}^{-1}\left[U^{2}(\omega, {\upsilon})\right]{{\rm{£}}}_{\eta}^{-1}\left[U_{\omega}(\omega, {\upsilon})\right]\Big]\Big]-\frac{1}{{\upsilon}^{\alpha}}{{\rm{£}}}_{\eta}\left[{{\rm{£}}}_{\eta}^{-1}\left[U_{\omega\omega\omega}(\omega, {\upsilon})\right]\right]. \end{split} \end{equation}$

(3.20)

The k-th truncated term series of (3.20) is

$\begin{equation} \begin{split} U(\omega, {\upsilon}) = \left(\frac{1}{2}+\frac{1}{2}\tanh(\frac{\omega}{2})\right)\frac{1}{\upsilon}+\sum\limits_{n = 1}^{k}\frac{f_{n}(\omega)}{{\upsilon}^{n\alpha+1}}, \end{split} \end{equation}$

(3.21)

and the k-th Laplace residual function is

$\begin{equation} \begin{split} {{\rm{£}}}_{\eta}Res_{k}(\omega, {\upsilon})& = U(\omega, {\upsilon})-\left(\frac{1}{2}+\frac{1}{2}\tanh(\frac{\omega}{2})\right)\frac{1}{\upsilon}+\frac{1}{{\upsilon}^{\alpha}}{{\rm{£}}}_{\eta}\Big[6\Big[{{\rm{£}}}_{\eta}^{-1}\left[U_{k}(\omega, {\upsilon})\right]{{\rm{£}}}_{\eta}^{-1}\left[U_{k\omega}(\omega, {\upsilon})\right]\\ &-\epsilon^{2}{{\rm{£}}}_{\eta}^{-1}\left[U_{k}^{2}(\omega, {\upsilon})\right]{{\rm{£}}}_{\eta}^{-1}\left[U_{k\omega}(\omega, {\upsilon})\right]\Big]\Big]+\frac{1}{{\upsilon}^{\alpha}}{{\rm{£}}}_{\eta}\left[{{\rm{£}}}_{\eta}^{-1}\left[U_{k\omega\omega\omega}(\omega, {\upsilon})\right]\right]. \end{split} \end{equation}$

(3.22)

Now, to determine $f_{k}(x)$ , $k = 1, 2, 3, \cdots$ , we put the $k$ th-truncate series (3.21) into the residual term of $k$ th-Laplace (3.22), multiply the result equation by ${\upsilon}^{k\alpha+1}$ , and then solve recursively the relations $\lim_{s\rightarrow \infty}[{\upsilon}^{k\alpha+1}Res_{k}(x, {\upsilon})] = 0$ , $k = 1, 2, 3, \cdots$ for $f_{k}$ . First few components of the sequence $f_{k}(x)$ ,

$\begin{array}{l} f_{1}(\omega) = \frac{1}{8}{\rm{sech}}\left(\frac{\omega}{4}\right)^{4}\left[-1+(-4+3\epsilon^{3})\cosh(\omega)+3(-1+\epsilon^{2})\sinh(\omega)\right], \\ f_{2}(\omega) = \frac{-1}{64}{\rm{sech}}\left(\frac{\omega}{4}\right)^{7}\Big[-24(-1+\epsilon^{2})\cosh\left(\frac{\omega}{2}\right)-6(22-37\epsilon^{2}+15\epsilon^{4})\cosh\left(\frac{3\omega}{2}\right)\\ +24\cosh\left(\frac{5\omega}{2}\right)\\ -42\epsilon^{2}\cosh\left(\frac{5\omega}{2}\right)+18\epsilon^{4}\cosh\left(\frac{5\omega}{2}\right)+206\sinh\left(\frac{\omega}{2}\right)-204\epsilon^{2}\sinh\left(\frac{\omega}{2}\right)\\-129\sinh\left(\frac{3\omega}{2}\right)\\ +222\epsilon^{2}\sinh\left(\frac{3\omega}{2}\right)-90\epsilon^{4}\sinh\left(\frac{3\omega}{2}\right)+25\sinh\left(\frac{5\omega}{2}\right)-42\epsilon^{2}\sinh\left(\frac{5\omega}{2}\right)\\+18\epsilon^{4}\sinh\left(\frac{5\omega}{2}\right)\Big], \\ \vdots. \end{array}$

(3.23)

Putting the values of $f_{n}(x)$ , $(n\geq1)$ in Eq (3.21), we have

$\begin{array}{l} U(\omega, {\upsilon}) = \left(\frac{1}{2}+\frac{1}{2}\tanh(\frac{\omega}{2})\right)\frac{1}{\upsilon}+\frac{1}{8}{\rm{sech}}\left(\frac{\omega}{4}\right)^{4}\\ \left[-1+(-4+3\epsilon^{3})\cosh(\omega)+3(-1+\epsilon^{2})\sinh(\omega)\right]\frac{1}{{\upsilon}^{\alpha+1}}\\ -\frac{1}{64}{\rm{sech}}\left(\frac{\omega}{4}\right)^{7}\Big[-24(-1+\epsilon^{2})\cosh\left(\frac{\omega}{2}\right)\\ -6(22-37\epsilon^{2}+15\epsilon^{4})\cosh\left(\frac{3\omega}{2}\right)+24\cosh\left(\frac{5\omega}{2}\right)\\ -42\epsilon^{2}\cosh\left(\frac{5\omega}{2}\right)+18\epsilon^{4}\cosh\left(\frac{5\omega}{2}\right)+206\sinh\left(\frac{\omega}{2}\right)\\-204\epsilon^{2}\sinh\left(\frac{\omega}{2}\right)-129\sinh\left(\frac{3\omega}{2}\right)\\ +222\epsilon^{2}\sinh\left(\frac{3\omega}{2}\right)-90\epsilon^{4}\sinh\left(\frac{3\omega}{2}\right)+25\sinh\left(\frac{5\omega}{2}\right)-42\epsilon^{2}\sinh\left(\frac{5\omega}{2}\right)\\ +18\epsilon^{4}\sinh\left(\frac{5\omega}{2}\right)\Big]\frac{1}{{\upsilon}^{2\alpha+1}}+\cdots. \end{array}$

(3.24)

Applying inverse Laplace transform, we get

$\begin{array}{l} u(\omega, {\eta}) = \left(\frac{1}{2}+\frac{1}{2}\tanh(\frac{\omega}{2})\right)+\frac{1}{8}{\rm{sech}}\left(\frac{\omega}{4}\right)^{4}\\ \left[-1+(-4+3\epsilon^{3})\cosh(\omega)+3(-1+\epsilon^{2})\sinh(\omega)\right]\frac{{\eta}^{\alpha}}{\Gamma(\alpha+1)}\\ -\frac{1}{64}{\rm{sech}}\left(\frac{\omega}{4}\right)^{7}\Big[-24(-1+\epsilon^{2})\cosh\left(\frac{\omega}{2}\right)-6(22-37\epsilon^{2}+15\epsilon^{4})\\ \cosh\left(\frac{3\omega}{2}\right)+24\cosh\left(\frac{5\omega}{2}\right)\\ -42\epsilon^{2}\cosh\left(\frac{5\omega}{2}\right)+18\epsilon^{4}\cosh\left(\frac{5\omega}{2}\right)+206\sinh\left(\frac{\omega}{2}\right)-204\epsilon^{2}\sinh\left(\frac{\omega}{2}\right)\\-129\sinh\left(\frac{3\omega}{2}\right)\\ +222\epsilon^{2}\sinh\left(\frac{3\omega}{2}\right)-90\epsilon^{4}\sinh\left(\frac{3\omega}{2}\right)+25\sinh\left(\frac{5\omega}{2}\right)-42\epsilon^{2}\sinh\left(\frac{5\omega}{2}\right)\\ +18\epsilon^{4}\sinh\left(\frac{5\omega}{2}\right)\Big]\frac{{\eta}^{2\alpha}}{\Gamma(2\alpha+1)}+\cdots. \end{array}$

(3.25)

Throughout this investigation, the method are being employed to assess the precise analytical solution of fractional-order Gardner equation. For various spatial and temporal parameters, the Caputo fractional derivative operators in facilitate appropriate numerical findings for the Cahn-Hilliard equation option revenue framework utilizing multiple orders. In , actual and approximate solutions graph and second fractional order $\alpha = 0.8$ of Example 3.2 at $\alpha = 1$ . In , approximate result graph at $\alpha = 0.6, 0.4$ and , the approximate result at various value of $\alpha$ of Example 3.2.

Figure 4. The actual and approximate results graph and second fractional order at

$\alpha = 0.8$ of Example 3.2.

DownLoad: Full-Size Img PowerPoint

Figure 5. The approximate result at

$\alpha = 0.6, 0.4$ of Example 3.2.

DownLoad: Full-Size Img PowerPoint

Figure 6. The approximate result at various value of

$\alpha$ for Example 3.2.

DownLoad: Full-Size Img PowerPoint

4. Conclusions

In this article, significant nonlinear fractional Cahn-Hilliard and Gardner equations are solved utilizing a combination of the Laplace transformation and the residual power series. This study demonstrated that the suggested method, Laplace residual power series, is a straightforward and effective analytical technique for constructing exact and approximation solutions for partial differential equations with suitable initial conditions. The aforementioned method provided us with solutions in the Laplace transform space via a straightforward method for obtaining the expansion series constants with the aid of the limit idea at infinity. With fewer series terms, the approximate solutions are achieved. The proposed technique is used to solve two separate physical models, and its capacity to handle fractional nonlinear equations with high precision and straightforward computing processes has been demonstrated.

Conflict of interest

The authors declare no conflict of interest.

References

[1]	A. Kaplan, M. Haenlein, Rulers of the world, unite! The challenges and opportunities of artificial intelligence, Bus. Horizons, 63 (2020), 37–50. http://doi.org/10.1016/j.bushor.2019.09.003 doi: 10.1016/j.bushor.2019.09.003
[2]	H. Margetts, C. Dorobantu, Rethink government with AI, Nature, 568 (2019), 163–165. http://doi.org/10.1038/d41586-019-01099-5 doi: 10.1038/d41586-019-01099-5
[3]	K. Cresswell, M. Callaghan, S. Khan, Z. Sheikh, H. Mozaffar, A. Sheikh, Investigating the use of data-driven artificial intelligence in computerised decision support systems for health and social care: A systematic review, Health Inf. J., 26 (2020), 2138–2147. https://doi.org/10.1177/1460458219900452 doi: 10.1177/1460458219900452
[4]	S. A. Yablonsky, Multidimensional data-driven artificial intelligence innovation, Technol. Innov. Magag. Rev., 9 (2019), 16–28.
[5]	J. Klinger, J. Mateos Garcia, K. Stathoulopoulos, Deep learning, deep change? Mapping the development of the artificial intelligence general purpose technology, Scientometrics, 126 (2021), 5589–5621. http://doi.org/10.1007/s11192-021-03936-9 doi: 10.1007/s11192-021-03936-9
[6]	V. Rasskazov, Financial and economic consequences of distribution of artificial intelligence as a general-purpose technology, Financ.: Theory Pract., 24 (2020), 120–132.
[7]	A. Agostini, C. Torras, F. Worgotter, Efficient interactive decision-making framework for robotic applications, Artif. Intell., 247 (2017), 187–212. http://doi.org/10.1016/j.artint.2015.04.004 doi: 10.1016/j.artint.2015.04.004
[8]	L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338–353. http://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
[9]	K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst., 20 (1986), 87–96. http://doi.org/10.1016/S0165-0114(86)80034-3 doi: 10.1016/S0165-0114(86)80034-3
[10]	M. Gorzalczany, A method of inference in approximate reasoning based on interval valued fuzzy sets, Fuzzy Sets Syst., 21 (1987), 1–17. http://doi.org/10.1016/0165-0114(87)90148-5 doi: 10.1016/0165-0114(87)90148-5
[11]	R. Biswas, Vague Groups, Int. J. Comput. Cogn., 4 (2006), 20–23.
[12]	R. R. Yager, Pythagorean membership grades in multi criteria decision-making, IEEE T. Fuzzy Sets Syst., 22 (2014), 958–965. http://doi.org/10.1109/TFUZZ.2013.2278989 doi: 10.1109/TFUZZ.2013.2278989
[13]	X. Peng, Y. Yang, Fundamental properties of interval valued Pythagorean fuzzy aggregation operators, Int. J. Int. Syst., 31 (2016), 444–487.
[14]	S. Ashraf, S. Abdullah, T. Mahmood, F. Ghani, T. Mahmood, Spherical fuzzy sets and their applications in multi-attribute decision making problems, J. Intell. Fuzzy Syst., 36 (2019), 2829–2844. http://doi.org/10.3233/JIFS-172009 doi: 10.3233/JIFS-172009
[15]	A. Nicolescu, M. Teodorescu, A unifying field in logics - Book review, Int. Lett. Soc. Humanistic Sci., 43 (2015), 48–59. http://doi.org/10.18052/www.scipress.com/ILSHS.43.48 doi: 10.18052/www.scipress.com/ILSHS.43.48
[16]	R. N. Xu, Regression prediction for fuzzy time series, Appl. Math. J. Chin. Univ. Ser. B, 16 (2001), 124376140.
[17]	R. R. Yager, Generalized orthopair fuzzy sets, IEEE T. Fuzzy Syst., 25 (2016), 1222–1230. http://doi.org/10.1109/TFUZZ.2016.2604005 doi: 10.1109/TFUZZ.2016.2604005
[18]	B. P. Joshi, A. Singh, P. K. Bhatt, K. S. Vaisla, Interval valued $q$ -rung orthopair fuzzy sets and their properties, J. Intell. Fuzzy Syst., 35 (2018), 5225–5230. http://doi.org/10.3233/JIFS-169806 doi: 10.3233/JIFS-169806
[19]	R. R. Yager, Generalized orthopair fuzzy sets, IEEE T. Fuzzy Syst., 25 (2017), 1222–1230. http://doi.org/10.1109/TFUZZ.2016.2604005 doi: 10.1109/TFUZZ.2016.2604005
[20]	M. S. Habib, O. Asghar, A. Hussain, M. Imran, M. P. Mughal, B. Sarkar, A robust possibilistic programming approach toward animal fat-based biodiesel supply chain network design under uncertain environment, J. Clean. Prod., 278 (2021), 122403. http://doi.org/10.1016/j.jclepro.2020.122403 doi: 10.1016/j.jclepro.2020.122403
[21]	B. Sarkar, M. Tayyab, N. Kim, M. S. Habib, Optimal production delivery policies for supplier and manufacturer in a constrained closed-loop supply chain for returnable transport packaging through metaheuristic approach, Comput. Ind. Eng., 135 (2019), 987–1003. http://doi.org/10.1016/j.cie.2019.05.035 doi: 10.1016/j.cie.2019.05.035
[22]	D. Yadav, R. Kumari, N. Kumar, B. Sarkar, Reduction of waste and carbon emission through the selection of items with crossprice elasticity of demand to form a sustainable supply chain with preservation technology, J. Clean. Prod., 297 (2019), 126298. http://doi.org/10.1016/j.jclepro.2021.126298 doi: 10.1016/j.jclepro.2021.126298
[23]	B. K. Dey, S. Bhuniya, B. Sarkar, Involvement of controllable lead time and variable demand for a smart manufacturing system under a supply chain management, Expert Syst. Appl., 184 (2021), 115464. http://doi.org/10.1016/j.eswa.2021.115464 doi: 10.1016/j.eswa.2021.115464
[24]	B. C. Cuong, V. Kreinovich, Picture fuzzy sets-A new concept for computational intelligence problems, 2013 Third World Congress on Information and Communication Technologies, 2013. http://doi.org/10.1109/WICT.2013.7113099
[25]	M. Akram, W. A. Dudek, F. Ilyas, Group decision making based on Pythagorean fuzzy TOPSIS method, Int. J. Intell. Syst., 34 (2019), 1455–1475. http://doi.org/10.1002/int.22103 doi: 10.1002/int.22103
[26]	M. Akram, W. A. Dudek, J. M. Dar, Pythagorean Dombi Fuzzy Aggregation Operators with Application in Multi-criteria Decision-making, Int. J. Intell. Syst., 34 (2019), 3000–3019. http://doi.org/10.1002/int.22183 doi: 10.1002/int.22183
[27]	Z. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems, (2007), 1179–1187.
[28]	W. F. Liu, J. Chang, X. He, Generalized Pythagorean fuzzy aggregation operators and applications in decision making, Control Decision, 31 (2016), 2280–2286. http://doi.org/10.13195/j.kzyjc.2015.1537 doi: 10.13195/j.kzyjc.2015.1537
[29]	K. Rahman, S. Abdullah, M. Shakeel, M. S. A. Khan, M. Ullah, Interval valued Pythagorean fuzzy geometric aggregation operators and their application to group decision-making problem, Cogent Math., 4 (2017), 1338638. http://doi.org/10.1080/23311835.2017.1338638 doi: 10.1080/23311835.2017.1338638
[30]	K. Rahman, A. Ali, S. Abdullah, F. Amin, Approaches to multi-attribute group decision-making based on induced interval valued Pythagorean fuzzy Einstein aggregation operator, New Math. Natural Comput., 14 (2018), 343–361. https://doi.org/10.1142/S1793005718500217 doi: 10.1142/S1793005718500217
[31]	Z. Yang, J. Chang, Interval-valued Pythagorean normal fuzzy information aggregation operators for multiple attribute decision making approach, IEEE Access, 8 (2020), 51295–51314. http://doi.org/10.1109/ACCESS.2020.2978976 doi: 10.1109/ACCESS.2020.2978976
[32]	K. G. Fatmaa, K. Cengiza, Spherical fuzzy sets and spherical fuzzy TOPSIS method, J. Intell. Fuzzy Syst., 36 (2019), 337–352. http://doi.org/10.3233/JIFS-181401 doi: 10.3233/JIFS-181401
[33]	P. Liu, G. Shahzadi, M. Akram, Specific types of q-rung picture fuzzy Yager aggregation operators for decision-making, Int. J. Comput. Intell. Syst., 13 (2020), 1072–1091. http://doi.org/10.2991/ijcis.d.200717.001 doi: 10.2991/ijcis.d.200717.001
[34]	Z. Yang, X. Li, Z. Cao, J. Li, $q$ -rung orthopair normal fuzzy aggregation operators and their application in multi attribute decision-making, Mathematics, 7 (2019), 1142. http://doi.org/10.3390/math7121142 doi: 10.3390/math7121142
[35]	R. R. Yager, N. Alajlan, Approximate reasoning with generalized orthopair fuzzy sets, Inform. Fusion., 38 (2017), 65–73. http://doi.org/10.1016/j.inffus.2017.02.005 doi: 10.1016/j.inffus.2017.02.005
[36]	M. I. Ali, Another view on q-rung orthopair fuzzy sets, Int. J. Intell. Syst., 33 (2019), 2139–2153. http://doi.org/10.1002/int.22007 doi: 10.1002/int.22007
[37]	P. Liu, P. Wang, Some $q$ -rung orthopair fuzzy aggregation operators and their applications to multiple-attribute decision making, Int. J. Intell. Syst., 33 (2018), 259–280. http://doi.org/10.1002/int.21927 doi: 10.1002/int.21927
[38]	P. Liu, J. Liu, Some $q$ -rung orthopai fuzzy Bonferroni mean operators and their application to multi-attribute group decision making, Int. J. Intell. Syst., 33 (2018), 315–347. http://doi.org/10.1002/int.21933 doi: 10.1002/int.21933
[39]	C. Jana, G. Muhiuddin, M. Pal, Some Dombi aggregation of $q$ -rung orthopair fuzzy numbers in multiple-attribute decision making, Int. J. Intell. Syst., 34 (2019), 3220–3240. http://doi.org/10.1002/int.22191 doi: 10.1002/int.22191
[40]	J. Wang, R. Zhang, X. Zhu, Z. Zhou, X. Shang, W. Li, Some $q$ -rung orthopair fuzzy Muirhead means with their application to multi-attribute group decision making, J. Intell. Fuzzy Syst., 36 (2019), 1599–1614. http://doi.org/10.3233/JIFS-18607 doi: 10.3233/JIFS-18607
[41]	M. S. Yang, C. H. Ko, On a class of fuzzy c-numbers clustering procedures for fuzzy data, Fuzzy Sets. Syst., 84 (1996), 49–60. http://doi.org/10.1016/0165-0114(95)00308-8 doi: 10.1016/0165-0114(95)00308-8
[42]	A. Hussain, M. I. Ali, T. Mahmood, Hesitant $q$ -rung orthopair fuzzy aggregation operators with their applications in multi-criteria decision making, Iran. J. Fuzzy Syst., 17 (2020), 117–134. http://doi.org/10.22111/IJFS.2020.5353 doi: 10.22111/IJFS.2020.5353
[43]	A. Hussain, M. I. Ali, T. Mahmood, M. Munir, Group based generalized $q$ -rung orthopair average aggregation operators and their applications in multi-criteria decision making, Complex Intell. Syst., 7 (2021), 123–144. http://doi.org/10.1007/s40747-020-00176-x doi: 10.1007/s40747-020-00176-x
[44]	F. Smarandache, A unifying field in logics, Neutrosophy neutrosophic probability, set and logic, 2 Eds., Rehoboth: American Research Press, 1999.
[45]	J. Ye, Similarity measures between interval neutrosophic sets and their applications in Multi-criteria decision-making, J. Intell. Fuzzy Syst., 26 (2014), 165–172. http://doi.org/10.3233/IFS-120724 doi: 10.3233/IFS-120724
[46]	H. Zhang, J. Wang, X. Chen, Interval neutrosophic sets and its application in multi-criteria decision making problems, Sci. World J., 2014 (2014), 645953. http://doi.org/10.1155/2014/645953 doi: 10.1155/2014/645953
[47]	H. Bustince, P. Burillo, Vague sets are intuitionistic fuzzy sets, Fuzzy Sets Syst., 79 (1996), 403–405. http://doi.org/10.1016/0165-0114(95)00154-9 doi: 10.1016/0165-0114(95)00154-9
[48]	A. Kumar, S. P. Yadav, S. Kumar, Fuzzy system reliability analysis using $T$ based arithmetic operations on $LR$ type interval valued vague sets, Int. J. Qual. Reliab. Manag., 24 (2007), 846–860. http://doi.org/10.1108/02656710710817126 doi: 10.1108/02656710710817126
[49]	J. Wang, S. Y. Liu, J. Zhang, S. Y. Wang, On the parameterized OWA operators for fuzzy MCDM based on vague set theory, Fuzzy Optim. Decis. Making, 5 (2006), 5–20. http://doi.org/10.1007/s10700-005-4912-2 doi: 10.1007/s10700-005-4912-2
[50]	X. Zhang, Z. Xu, Extension of TOPSIS to multiple criteria decision-making with Pythagorean fuzzy sets, Int. J. Intell. Syst., 29 (2014), 1061–1078. http://doi.org/10.1002/int.21676 doi: 10.1002/int.21676
[51]	C. L. Hwang, K. Yoon, Multiple Attribute Decision Making-Methods and Applications, A State-of-the-Art Survey, Berlin, Heidelberg: Springer, 1981. http://doi.org/10.1007/978-3-642-48318-9
[52]	C. Jana, M. Pal, Application of bipolar intuitionistic fuzzy soft sets in decision-making problem, Int. J. Fuzzy Syst. Appl., 7 (2018), 32–55. http://doi.org/10.4018/IJFSA.2018070103 doi: 10.4018/IJFSA.2018070103
[53]	C. Jana, Multiple attribute group decision-making method based on extended bipolar fuzzy MABAC approach, Comput. Appl. Math., 40 (2021), 227. http://doi.org/10.1007/s40314-021-01606-3 doi: 10.1007/s40314-021-01606-3
[54]	C. Jana, M. Pal, A Robust single valued neutrosophic soft aggregation operators in multi criteria decision-making, Symmetry, 11 (2019), 110. http://doi.org/10.3390/sym11010110 doi: 10.3390/sym11010110
[55]	C. Jana, T. Senapati, M. Pal, Pythagorean fuzzy Dombi aggregation operators and its applications in multiple attribute decision-making, Int. J. Intell. Syst., 34 (2019), 2019–2038. http://doi.org/10.1002/int.22125 doi: 10.1002/int.22125
[56]	K. Ullah, T. Mahmood, Z. Ali, N. Jan, On some distance measures of complex Pythagorean fuzzy sets and their applications in pattern recognition, Comput. Intell. Syst., 6 (2020), 15–27. http://doi.org/10.1007/s40747-019-0103-6 doi: 10.1007/s40747-019-0103-6
[57]	C. Jana, M. Pal, F. Karaaslan, J. Q. Wang, Trapezoidal neutrosophic aggregation operators and their application to the multi-attribute decision-making process, Sci. Iran., 27 (2020), 1655–1673. http://doi.org/10.24200/SCI.2018.51136.2024 doi: 10.24200/SCI.2018.51136.2024
[58]	C. Jana, M. Pal, Multi criteria decision-making process based on some single valued neutrosophic dombi power aggregation operators, Soft Comput., 25 (2021), 5055–5072. http://doi.org/10.1007/s00500-020-05509-z doi: 10.1007/s00500-020-05509-z
[59]	M. Palanikumar, K. Arulmozhi, C. Jana, M. Pal, Multiple‐attribute decision-making spherical vague normal operators and their applications for the selection of farmers, Expert Syst., 40 (2023), e13188. http://doi.org/10.1111/exsy.13188 doi: 10.1111/exsy.13188
[60]	V. Uluçay, Q-neutrosophic soft graphs in operations management and communication network, Soft Comput., 25 (2021), 8441–8459. http://doi.org/10.1007/s00500-021-05772-8 doi: 10.1007/s00500-021-05772-8
[61]	V. Uluçay, Some concepts on interval-valued refined neutrosophic sets and their applications, J. Ambient Intell. Human. Comput., 12 (2021), 7857–7872. http://doi.org/10.1007/s12652-020-02512-y doi: 10.1007/s12652-020-02512-y
[62]	V. Uluçay, I. Deli, M. Şahin, Similarity measures of bipolar neutrosophic sets and their application to multiple criteria decision making, Neural Comput. Appl., 29 (2021), 739–748. http://doi.org/10.1007/s00521-016-2479-1 doi: 10.1007/s00521-016-2479-1
[63]	M. Sahin, N. Olgun, V. Uluçay, H. Acioglu, Some weighted arithmetic operators and geometric operators with SVNSs and their application to multi-criteria decision making problems, New Trends Neutrosophic Theory Appl., 2 (2018), 85–104. http://doi.org/10.5281/zenodo.1237953 doi: 10.5281/zenodo.1237953
[64]	M. Sahin, N. Olgun, V. Uluçay, A. Kargin, F. Smarandache, A new similarity measure based on falsity value between single valued neutrosophic sets based on the centroid points of transformed single valued neutrosophic numbers with applications to pattern recognition, Neutrosophic Sets Syst., 15 (2017), 31–48.
[65]	Y. Lu, Y. Xu, E. H. Viedma, Consensus progress for large-scale group decision making in social networks with incomplete probabilistic hesitant fuzzy information, Appl. Soft Comput., 126 (2022), 109249. http://doi.org/10.1016/j.asoc.2022.109249 doi: 10.1016/j.asoc.2022.109249
[66]	Y. Lu, Y. Xu, J. Huang, J. Wei, E. H. Viedma, Social network clustering and consensus-based distrust behaviors management for large scale group decision-making with incomplete hesitant fuzzy preference relations, Appl. Soft Comput., 117 (2022), 108373. http://doi.org/10.1016/j.asoc.2021.108373 doi: 10.1016/j.asoc.2021.108373
[67]	C. Jana, G. Muhiuddin, M. Pal, Multi-criteria decision-making approach based on SVTrN Dombi aggregation functions. Artif. Intell. Rev., 54 (2021), 3685–3723. http://doi.org/10.1007/s10462-020-09936-0 doi: 10.1007/s10462-020-09936-0
[68]	M. Palanikumar, K. Arulmozhi, C. Jana, Multiple attribute decision-making approach for Pythagorean neutrosophic normal interval-valued aggregation operators, Comput. Appl. Math., 41 (2022), 90. http://doi.org/10.1007/s40314-022-01791-9 doi: 10.1007/s40314-022-01791-9
[69]	X. Peng, H. Yuan, Fundamental properties of Pythagorean fuzzy aggregation operators, Int. J. Intell. Syst. 31 (2016), 444–487. http://doi.org/10.1002/int.21790 doi: 10.1002/int.21790
[70]	B. Sarkar, M. Omair, N. Kim, A cooperative advertising collaboration policy in supply chain management under uncertain conditions, Appl. Soft Comput., 88 (2020), 105948. http://doi.org/10.1016/j.asoc.2019.105948 doi: 10.1016/j.asoc.2019.105948
[71]	U. Mishra, J. Z. Wu, B. Sarkar, Optimum sustainable inventory management with backorder and deterioration under controllable carbon emissions, J. Clean. Prod., 279 (2021), 123699. http://doi.org/10.1016/j.jclepro.2020.123699 doi: 10.1016/j.jclepro.2020.123699
[72]	B. Sarkar, M. Sarkar, B. Ganguly, L. E. Cárdenas-Barrón, Combined effects of carbon emission and production quality improvement for fixed lifetime products in a sustainable supply chain management, Int. J. Prod. Econ., 231 (2021), 107867. http://doi.org/10.1016/j.ijpe.2020.107867 doi: 10.1016/j.ijpe.2020.107867
[73]	U. Mishra, J. Z. Wu, B. Sarkar, A sustainable production-inventory model for a controllable carbon emissions rate under shortages, J. Clean. Prod., 256 (2020), 120268. http://doi.org/10.1016/j.jclepro.2020.120268 doi: 10.1016/j.jclepro.2020.120268
[74]	M. Ullah, B. Sarkar, Recovery-channel selection in a hybrid manufacturing-remanufacturing production model with RFID and product quality, Int. J. Prod. Econ., 219 (2020), 360–374. http://doi.org/10.1016/j.ijpe.2019.07.017 doi: 10.1016/j.ijpe.2019.07.017
[75]	M. Ullah, I. Asghar, M. Zahid, M. Omair, A. AlArjani, B. Sarkar, Ramification of remanufacturing in a sustainable three-echelon closed-loop supply chain management for returnable products, J. Clean. Prod., 290 (2021), 125609. http://doi.org/10.1016/j.jclepro.2020.125609 doi: 10.1016/j.jclepro.2020.125609

This article has been cited by:

1.	Muhammad Imran Liaqat, Ali Akgül, Hanaa Abu-Zinadah, Analytical Investigation of Some Time-Fractional Black–Scholes Models by the Aboodh Residual Power Series Method, 2023, 11, 2227-7390, 276, 10.3390/math11020276
2.	M. A. El-Shorbagy, Mati ur Rahman, Maryam Ahmed Alyami, On the analysis of the fractional model of COVID-19 under the piecewise global operators, 2023, 20, 1551-0018, 6134, 10.3934/mbe.2023265
3.	Muhammad Imran Liaqat, Eric Okyere, Arzu Akbulut, The Fractional Series Solutions for the Conformable Time-Fractional Swift-Hohenberg Equation through the Conformable Shehu Daftardar-Jafari Approach with Comparative Analysis, 2022, 2022, 2314-4785, 1, 10.1155/2022/3295076
4.	Muhammad Imran Liaqat, Eric Okyere, Watcharaporn Cholamjiak, Comparative Analysis of the Time-Fractional Black–Scholes Option Pricing Equations (BSOPE) by the Laplace Residual Power Series Method (LRPSM), 2023, 2023, 2314-4785, 1, 10.1155/2023/6092283
5.	Muhammad Imran Liaqat, Ali Akgül, Manuel De la Sen, Mustafa Bayram, Approximate and Exact Solutions in the Sense of Conformable Derivatives of Quantum Mechanics Models Using a Novel Algorithm, 2023, 15, 2073-8994, 744, 10.3390/sym15030744
6.	Muhammad Imran Liaqat, Ali Akgül, Eugenii Yurevich Prosviryakov, An efficient method for the analytical study of linear and nonlinear time-fractional partial differential equations with variable coefficients, 2023, 27, 1991-8615, 214, 10.14498/vsgtu2009
7.	Maalee Almheidat, Humaira Yasmin, Maryam Al Huwayz, Rasool Shah, Samir A. El-Tantawy, A novel investigation into time-fractional multi-dimensional Navier–Stokes equations within Aboodh transform, 2024, 22, 2391-5471, 10.1515/phys-2024-0081
8.	Muhammad Imran Liaqat, Adnan Khan, Alia Irshad, Ali Akgül, Eugenii Yurevich Prosviryakov, Approximate analytical solutions of the nonlinear fractional order financial model by two efficient methods with a comparison study, 2024, 28, 1991-8615, 223, 10.14498/vsgtu2055
9.	Muhammad Nadeem, Mohamed Sharaf, Saipunidzam Mahamad, Numerical investigation of two-dimensional fractional Helmholtz equation using Aboodh transform scheme, 2024, 0961-5539, 10.1108/HFF-07-2024-0543
10.	Abdul Hamid Ganie, Humaira Yasmin, A A Alderremy, Rasool Shah, Shaban Aly, An efficient semi-analytical techniques for the fractional-order system of Drinfeld-Sokolov-Wilson equation, 2024, 99, 0031-8949, 015253, 10.1088/1402-4896/ad1796
11.	Mati ur Rahman, Yeliz Karaca, Ravi P. Agarwal, Sergio Adriani David, Mathematical modelling with computational fractional order for the unfolding dynamics of the communicable diseases, 2024, 32, 2769-0911, 10.1080/27690911.2023.2300330
12.	Rania Saadeh, Ahmad Qazza, Abdelilah Kamal Sedeeg, A Generalized Hybrid Method for Handling Fractional Caputo Partial Differential Equations via Homotopy Perturbed Analysis, 2023, 22, 2224-2880, 988, 10.37394/23206.2023.22.108
13.	Saima Noor, Wedad Albalawi, Rasool Shah, M. Mossa Al-Sawalha, Sherif M. E. Ismaeel, Mathematical frameworks for investigating fractional nonlinear coupled Korteweg-de Vries and Burger’s equations, 2024, 12, 2296-424X, 10.3389/fphy.2024.1374452
14.	Yousef Jawarneh, Zainab Alsheekhhussain, M. Mossa Al-Sawalha, Fractional View Analysis System of Korteweg–de Vries Equations Using an Analytical Method, 2024, 8, 2504-3110, 40, 10.3390/fractalfract8010040
15.	Sanjeev Yadav, Ramesh Kumar Vats, Anjali Rao, Constructing the fractional series solutions for time-fractional K-dV equation using Laplace residual power series technique, 2024, 56, 1572-817X, 10.1007/s11082-024-06412-9
16.	Muhammad Imran Liaqat, Fahim Ud Din, Wedad Albalawi, Kottakkaran Sooppy Nisar, Abdel-Haleem Abdel-Aty, Analysis of stochastic delay differential equations in the framework of conformable fractional derivatives, 2024, 9, 2473-6988, 11194, 10.3934/math.2024549
17.	Muhammad Imran Liaqat, Adnan Khan, Hafiz Muhammad Anjum, Gregory Abe-I-Kpeng, Emad E. Mahmoud, S. A. Edalatpanah, A Novel Efficient Approach for Solving Nonlinear Caputo Fractional Differential Equations, 2024, 2024, 1687-9120, 10.1155/2024/1971059
18.	Abdul Hamid Ganie, Saima Noor, Maryam Al Huwayz, Ahmad Shafee, Samir A. El-Tantawy, Numerical simulations for fractional Hirota–Satsuma coupled Korteweg–de Vries systems, 2024, 22, 2391-5471, 10.1515/phys-2024-0008
19.	Sajad Iqbal, Jun Wang, Utilizing an imaginative approach to examine a fractional Newell–Whitehead–Segel equation based on the Mohand HPA, 2024, 148, 0022-0833, 10.1007/s10665-024-10381-z
20.	Saima Noor, Wedad Albalawi, Rasool Shah, M. Mossa Al-Sawalha, Sherif M. E. Ismaeel, S. A. El-Tantawy, On the approximations to fractional nonlinear damped Burger’s-type equations that arise in fluids and plasmas using Aboodh residual power series and Aboodh transform iteration methods, 2024, 12, 2296-424X, 10.3389/fphy.2024.1374481
21.	Adnan Khan, Muhammad Imran Liaqat, Asma Mushtaq, Analytical Analysis for Space Fractional Helmholtz Equations by Using The Hybrid Efficient Approach, 2024, 18, 2300-5319, 116, 10.2478/ama-2024-0065
22.	Musawa Yahya Almusawa, Hassan Almusawa, Numerical analysis of the fractional nonlinear waves of fifth-order KdV and Kawahara equations under Caputo operator, 2024, 9, 2473-6988, 31898, 10.3934/math.20241533
23.	Saima Noor, Wedad Albalawi, Rasool Shah, Ahmad Shafee, Sherif M. E. Ismaeel, S. A. El-Tantawy, A comparative analytical investigation for some linear and nonlinear time-fractional partial differential equations in the framework of the Aboodh transformation, 2024, 12, 2296-424X, 10.3389/fphy.2024.1374049
24.	Sandipa Bhattacharya, Biswajit Sarkar, Mitali Sarkar, Arka Mukherjee, Hospitality for prime consumers and others under the retail management, 2024, 80, 09696989, 103849, 10.1016/j.jretconser.2024.103849
25.	Nader Al-Rashidi, Innovative approaches to fractional modeling: Aboodh transform for the Keller-Segel equation, 2024, 9, 2473-6988, 14949, 10.3934/math.2024724
26.	Azzh Saad Alshehry, Humaira Yasmin, Rasool Shah, Amjid Ali, Imran Khan, Fractional-order view analysis of Fisher’s and foam drainage equations within Aboodh transform, 2024, 41, 0264-4401, 489, 10.1108/EC-08-2023-0475
27.	Abdelhamid Mohammed Djaouti, Zareen A. Khan, Muhammad Imran Liaqat, Ashraf Al-Quran, A Study of Some Generalized Results of Neutral Stochastic Differential Equations in the Framework of Caputo–Katugampola Fractional Derivatives, 2024, 12, 2227-7390, 1654, 10.3390/math12111654
28.	Isaac Addai, Benedict Barnes, Isaac Kwame Dontwi, Kwaku Forkuoh Darkwah, Modified Fractional Power Series Method for solving fractional partial differential equations, 2024, 24682276, e02467, 10.1016/j.sciaf.2024.e02467
29.	ALROWAILY ALBANDARI W, SHAH RASOOL , SALAS ALVARO H , ALHEJAILI WEAAM , TIOFACK C. G. L. , ISMAEEL SHERIF M. E. , EL-TANTAWY S. A. , Analysis of fractional Swift-Hohenberg models using highly accurate techniques within the Caputo operator framework, 2024, 76, 12211451, 112, 10.59277/RomRepPhys.2024.76.112
30.	Haifa A. Alyousef, Rasool Shah, C. G. L. Tiofack, Alvaro H. Salas, Weaam Alhejaili, Sherif M. E. Ismaeel, S. A. El-Tantawy, Novel Approximations to the Third- and Fifth-Order Fractional KdV-Type Equations and Modeling Nonlinear Structures in Plasmas and Fluids, 2025, 55, 0103-9733, 10.1007/s13538-024-01660-2
31.	Zareen A. Khan, Muhammad Bilal Riaz, Muhammad Imran Liaqat, Ali Akgül, Goutam Saha, A novel technique using integral transforms and residual functions for nonlinear partial fractional differential equations involving Caputo derivatives, 2024, 19, 1932-6203, e0313860, 10.1371/journal.pone.0313860
32.	Mudassir Shams, Bruno Carpentieri, Chaos in Inverse Parallel Schemes for Solving Nonlinear Engineering Models, 2024, 13, 2227-7390, 67, 10.3390/math13010067
33.	Yousuf Alkhezi, Ahmad Shafee, Analytical Techniques for Studying Fractional-Order Jaulent–Miodek System Within Algebraic Context, 2025, 9, 2504-3110, 50, 10.3390/fractalfract9010050
34.	Azzh Saad Alshehry, Rasool Shah, Exploring Fractional Damped Burgers’ Equation: A Comparative Analysis of Analytical Methods, 2025, 9, 2504-3110, 107, 10.3390/fractalfract9020107
35.	Aljawhara H. Almuqrin, C. G. L. Tiofack, A. Mohamadou, Alim Alim, Sherif M. E. Ismaeel, Weaam Alhejaili, Samir A. El-Tantawy, On the “Tantawy Technique” and other methods for analyzing the family of fractional Burgers’ equations: Applications to plasma physics, 2025, 1461-3484, 10.1177/14613484251314580
36.	Yu-Ming Chu, Sobia Sultana, Shazia Karim, Saima Rashid, Mohammed Shaaf Alharthi, A New Scheme of the ARA Transform for Solving Fractional-Order Waves-Like Equations Involving Variable Coefficients, 2024, 138, 1526-1506, 761, 10.32604/cmes.2023.028600
37.	Wedad Albalawi, Muhammad Imran Liaqat, Fahim Ud Din, Kottakkaran Sooppy Nisar, Abdel-Haleem Abdel-Aty, Significant results in the $\mathrm{p}$ th moment for Hilfer fractional stochastic delay differential equations, 2025, 10, 2473-6988, 9852, 10.3934/math.2025451

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)