Fractional-order partial differential equations describing propagation of shallow water waves depending on power and Mittag-Leffler memory

Maysaa Al Qurashi; Saima Rashid; Sobia Sultana; Fahd Jarad; Abdullah M. Alsharif; Maysaa Al Qurashi; Saima Rashid; Sobia Sultana; Fahd Jarad; Abdullah M. Alsharif

doi:10.3934/math.2022697

AIMS Mathematics

2022, Volume 7, Issue 7: 12587-12619. doi: 10.3934/math.2022697

Previous Article Next Article

Research article Special Issues

Fractional-order partial differential equations describing propagation of shallow water waves depending on power and Mittag-Leffler memory

1.
Department of Mathematics, King Saud University, P. O. Box 22452, Riyadh 11495, Saudi Arabia
2.
Department of Mathematics, Government College University, Faisalabad, Pakistan
3.
Department of Mathematics, Imam Mohammad Ibn Saud Islamic University, Riyadh 12211, Saudi Arabia
4.
Department of Mathematics, Cankaya University, Ankara, Turkey
5.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
6.
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
7.
Department of Mathematics and Statistics, College of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia

Received: 26 February 2022 Revised: 14 April 2022 Accepted: 19 April 2022 Published: 28 April 2022
MSC : 46S40, 47H10, 54H25

In this research, the $\bar{\mathbf{q}}$ -homotopy analysis transform method ( $\bar{\mathbf{q}}$ -HATM) is employed to identify fractional-order Whitham–Broer–Kaup equation (WBKE) solutions. The WBKE is extensively employed to examine tsunami waves. With the aid of Caputo and Atangana-Baleanu fractional derivative operators, to obtain the analytical findings of WBKE, the predicted algorithm employs a combination of $\bar{\mathbf{q}}$ -HAM and the Aboodh transform. The fractional operators are applied in this work to show how important they are in generalizing the frameworks connected with kernels of singularity and non-singularity. To demonstrate the applicability of the suggested methodology, various relevant problems are solved. Graphical and tabular results are used to display and assess the findings of the suggested approach. In addition, the findings of our recommended approach were analyzed in relation to existing methods. The projected approach has fewer processing requirements and a better accuracy rate. Ultimately, the obtained results reveal that the improved strategy is both trustworthy and meticulous when it comes to assessing the influence of nonlinear systems of both integer and fractional order.

Keywords:

Citation: Maysaa Al Qurashi, Saima Rashid, Sobia Sultana, Fahd Jarad, Abdullah M. Alsharif. Fractional-order partial differential equations describing propagation of shallow water waves depending on power and Mittag-Leffler memory[J]. AIMS Mathematics, 2022, 7(7): 12587-12619. doi: 10.3934/math.2022697

Related Papers:

[1]	Aslı Alkan, Halil Anaç . The novel numerical solutions for time-fractional Fornberg-Whitham equation by using fractional natural transform decomposition method. AIMS Mathematics, 2024, 9(9): 25333-25359. doi: 10.3934/math.20241237
[2]	Hussain Gissy, Abdullah Ali H. Ahmadini, Ali H. Hakami . The travelling wave phenomena of the space-time fractional Whitham-Broer-Kaup equation. AIMS Mathematics, 2025, 10(2): 2492-2508. doi: 10.3934/math.2025116
[3]	Aslı Alkan, Halil Anaç . A new study on the Newell-Whitehead-Segel equation with Caputo-Fabrizio fractional derivative. AIMS Mathematics, 2024, 9(10): 27979-27997. doi: 10.3934/math.20241358
[4]	Qasem M. Tawhari . Advanced analytical techniques for fractional Schrödinger and Korteweg-de Vries equations. AIMS Mathematics, 2025, 10(5): 11708-11731. doi: 10.3934/math.2025530
[5]	M. Mossa Al-Sawalha, Osama Y. Ababneh, Rasool Shah, Amjad khan, Kamsing Nonlaopon . Numerical analysis of fractional-order Whitham-Broer-Kaup equations with non-singular kernel operators. AIMS Mathematics, 2023, 8(1): 2308-2336. doi: 10.3934/math.2023120
[6]	Faeza Lafta Hasan, Mohamed A. Abdoon, Rania Saadeh, Ahmad Qazza, Dalal Khalid Almutairi . Exploring analytical results for (2+1) dimensional breaking soliton equation and stochastic fractional Broer-Kaup system. AIMS Mathematics, 2024, 9(5): 11622-11643. doi: 10.3934/math.2024570
[7]	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
[8]	Thongchai Botmart, Ravi P. Agarwal, Muhammed Naeem, Adnan Khan, Rasool Shah . On the solution of fractional modified Boussinesq and approximate long wave equations with non-singular kernel operators. AIMS Mathematics, 2022, 7(7): 12483-12513. doi: 10.3934/math.2022693
[9]	Mohammad Alqudah, Safyan Mukhtar, Albandari W. Alrowaily, Sherif. M. E. Ismaeel, S. A. El-Tantawy, Fazal Ghani . Breather patterns and other soliton dynamics in (2+1)-dimensional conformable Broer-Kaup-Kupershmit system. AIMS Mathematics, 2024, 9(6): 13712-13749. doi: 10.3934/math.2024669
[10]	Muhammad Farman, Ali Akgül, Kottakkaran Sooppy Nisar, Dilshad Ahmad, Aqeel Ahmad, Sarfaraz Kamangar, C Ahamed Saleel . Epidemiological analysis of fractional order COVID-19 model with Mittag-Leffler kernel. AIMS Mathematics, 2022, 7(1): 756-783. doi: 10.3934/math.2022046

Abstract

1. Introduction

The enrichment of day-to-day existence is the cornerstone of the environment and sentient creatures. Time is constantly a global autonomous factor in the cosmos, and eternity is often referred to as such. Furthermore, scholars are continuously in need of a technology that can assist them in describing the current phase and forecasting the development of many real-world occurrences ^{[1,2,3,4,5,6,7,8]}. In this way, renowned scholars and physicists like Newton and Leibniz separately suggested the method of calculus from their respective individual perspectives. Subsequently, several scholars and practitioners believed that a basic formula might be used to examine practically all theories on the planet rather than any other forms of resources or machines. However, numerous academics from different fields have previously confirmed that the idea of conventional calculus with differential and integral operators underestimates certain crucial and fascinating phenomena ^{[9,10,11,12,13,14,15]}. Others with inherent features, non-Markovian dynamics, and some others in general ^{[16,17,18,19,20]}. Besides that, analysts are indeed completely eager to describe and invent a conceptual technique to address many of the reported flaws.

Fractional calculus (FC) was developed shortly after conventional calculus, but numerous scientists and researchers are now considering the conception and ideas of FC to explain essence in a comprehensive way, especially following the revelation of conventional calculus' restrictions. Various researchers established and cultivated the key foundation ^{[21,22,23,24]} with the help of innovative characteristics and associated ramifications for FC. A special function theory is used to construct novel fractional formulations. Many academics have employed these operators to analyze and illustrate multiple systems and difficulties connected to intriguing behaviours ^{[25,26,27,28]}. In 2016, Atangana and Baleanu suggested a revolutionary operator that overcomes all of the restrictions of the prespecified interpreters using the Mittag-Leffler function ^[29].

Partial differential equations (PDEs) with non-linearities are used to represent a number of scenarios in physical science and engineering, extending from magnetism to complexities. Spatial PDEs are frequently employed in domains including nano-science, bio-engineering, epidemiology, and hydrodynamics to simulate dynamic behaviour events. Fractional-order PDEs have subsequently sparked considerable interest due to their broad implications in a variety of scientific disciplines, including optimization, image reformation, decision theory, signal transmission, remote sensing and network recognition, and hydrodynamics ^{[30,31,32,33]}. Because most design procedures are recognized to be complex, finding a numerical or approximate result is extremely challenging. The most complex strategies can be represented using an adequate collection of PDEs. A considerable effort has gone into inventing a convergent approach that is both convenient and simple. The new iterative method (NIM) ^[34], homotopy analysis method (HAM) ^[35], Lie symmetry analysis (LSA) ^[36], and Laguerre wavelets collocation method (LWCM) ^[37], residue power series method (RPSM) ^[33] are just a few of the latest estimated strategies for getting realistic findings for complex PDEs.

Whitham-Broer-Kaup equations (WBKEs) ^[38] were discovered to characterize the dynamic characteristics of waves that propagate in hydrodynamics. Whitham, Broer, and Kaup ^[39,40,41] established the coupled strategy for the aforementioned model. This formula describes the dispersion of superficial ripples of liquid with a specified permeation family. The classical form of WBKE is presented as follows:

$\begin{eqnarray} &&\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \bar{\mathbf{t}}}+\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\bar{g}\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa} = 0, \\&& \frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \bar{\mathbf{t}}}+\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\bar{p}\frac{\partial^{3}\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{3}}-\bar{g}\frac{\partial^{2}\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}} = 0, \end{eqnarray}$

(1.1)

where $\bar{p}$ and $\bar{g}$ are constants stated in distinct dispersion energies, correspondingly, and $\Phi(\varkappa, \bar{\mathbf{t}})$ is the linear speed, and $\Psi(\varkappa, \bar{\mathbf{t}})$ is the altitude that detracts from the fluid equilibrium condition, respectively. Exploration of complex PDE systems has been a significant subject of concern ^[42,43,44] in past times. Intellectuals have devised a slew of ways to investigate the analytical solutions to nonlinear PDEs. Authors ^[45] have employed the homotopy perturbation method to solve WBKE. Ray ^[46] obtained the travelling wave solution of WBKE. Tian ^[47] provides exact and explicit WBKE solutions in shallow water. Veeresha et al. ^[48] stimulated efficient technique for coupled fractional WBKE describes the propagation of shallow water waves.

Furthermore, $\bar{\mathbf{q}}$ -HATM is taken into account, as was proposed by Singh et al. ^[49]. This method, which is the combination of a conventional method termed HAM (developed by Liao with the help of the basic notion of topology ^[50,51]) and the Aboodh transform ^[52] is employed in this model. The enhanced methodology is presented to solve the shortcomings of the traditional approach, such as large calculation, effectiveness, primary storage, and others. Furthermore, HAM does not necessitate additional convolutions, instabilities, hypotheses, or physical component assessment. As a result, the predicted approach does not demand the aforementioned prerequisites, and it provides various features that may be used to change and regulate the convergence region, as well as increase the reliability of the generated result. Numerous authors state that the proposed framework approach has been implemented in several real-world simulations and situations that encompass various processes ^[53,54,55]. The investigation employed recently developed fractional operators and methods to evaluate a system of nonlinear complex systems that describes significant behaviors.

In the upcoming sections, we shall go over several of the fundamentals briefly. Sections 3 and 4 describe the basic response process and its relevance to the analyzed system, respectively. The key findings for the investigated interacting mechanism are obtained in Section 5, the consistently highlighted are numerically illustrated in Section 6, and we end with several comments on the acquired outcomes in the last portion of Section 7.

2. Preliminaries

In this part, we revisit certain key concepts, ideas, and terminologies connected to fractional derivative formulations involving power law and ML as a kernel, as well as the Aboodh transform's specific ramifications.

Definition 2.1. (^[23]) The fractional derivative of Caputo (CFD) is specifically defined as:

$\begin{eqnarray} \, _{0}^{c}\mathbf{D}_{\bar{\mathbf{t}}}^{\delta}\mathbf{f}(\bar{\mathbf{t}}) = \begin{cases} \frac{1}{\Gamma(r-\delta)}\int\limits_{0}^{\bar{\mathbf{t}}}\frac{\mathbf{f}^{(r)}(\varkappa_{1})}{(\bar{\mathbf{t}}-\varkappa_{1})^{\delta+1-r}}d\varkappa_{1}, \; \; r-1 < \delta < r, \\ \frac{d^{r}}{d\bar{\mathbf{t}}^{r}}\mathbf{f}(\bar{\mathbf{t}}), \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \delta = r. \end{cases} \end{eqnarray}$

(2.1)

Definition 2.2. (^[29]) The $ABC$ derivative operator is specifically defined as:

$\begin{eqnarray} \, _{a_{1}}^{ABC}\mathbf{D}_{\bar{\mathbf{t}}}^{\delta}\big(\mathbf{f}(\bar{\mathbf{t}})\big) = \frac{\mathbb{N}(\delta)}{1-\delta}\int\limits_{a_{1}}^{\bar{\mathbf{t}}}\mathbf{f}^{\prime}(\bar{\mathbf{t}})E_{\delta}\Big[-\frac{\delta(\bar{\mathbf{t}}-\varkappa_{1})^{\delta}}{1-\delta}\Big]d\varkappa_{1}, \end{eqnarray}$

(2.2)

where $\mathbf{f}\in H^{1}(a_{1}, a_{2}) (Sobolev \; space), \; a_{1} < a_{2}, \delta\in[0, 1]$ and $\mathbb{N}(\delta)$ indicates normalization function as $\mathbb{N}(\delta) = \mathbb{N}(0) = \mathbb{N}(1) = 1.$

Definition 2.3. (^[29]) The $ABC$ fractional integral operator is expressed in the form:

$\begin{eqnarray} \, _{a_{1}}^{ABC}\bar{I}_{\bar{\mathbf{t}}}^{\delta}\big(\mathbf{f}(\bar{\mathbf{t}})\big) = \frac{1-\delta}{\mathbb{N}(\delta)}\mathbf{f}(\bar{\mathbf{t}})+\frac{\delta}{\Gamma(\delta)\mathbb{N}(\delta)}\int\limits_{a_{1}}^{\bar{\mathbf{t}}}\mathbf{f}(\varkappa_{1})(\bar{\mathbf{t}}-\varkappa_{1})^{\delta-1}d\varkappa_{1}. \end{eqnarray}$

(2.3)

Definition 2.4. (^[52]) Aboodh transform for mapping $\mathbf{f}(\bar{\mathbf{t}})$ of exponential order over the set of functions is defined as

$\begin{eqnarray} \mathcal{P} = \Big\{\mathbf{f}:\big\vert\mathbf{f}(\bar{\mathbf{t}})\big\vert < B\exp(p_{\jmath}\vert \bar{\mathbf{t}}\vert), \; if\; \bar{\mathbf{t}}\in(-1)^{\jmath}\times[0, \infty), \; \jmath = 1, 2;\; (B, p_{1}, p_{2} > 0)\Big\} \end{eqnarray}$

(2.4)

is expressed as

$\begin{eqnarray} \mathcal{A}[\mathbf{f}(\bar{\mathbf{t}})] = \mathbf{F}(\omega) \end{eqnarray}$

(2.5)

is described as

$\begin{eqnarray} \mathcal{A}[\mathbf{f}(\bar{\mathbf{t}})] = \frac{1}{\omega}\int\limits_{0}^{\infty}\mathbf{f}(\bar{\mathbf{t}})\exp(-\omega \bar{\mathbf{t}})d\bar{\mathbf{t}}, \; \; p_{1}\leq \omega\leq p_{2}. \end{eqnarray}$

(2.6)

Definition 2.5. ^[56] The following is the Aboodh transform of CFD operator:

$\begin{eqnarray} \mathcal{A}\Big\{\, _{0}^{c}\mathbf{D}_{\bar{\mathbf{t}}}^{\delta}\big(\mathbf{f}(\bar{\mathbf{t}})\big), \bar{\mathbf{t}}\Big\} = \omega^{\delta}\mathbf{F}(\omega)-\sum\limits_{\kappa = 0}^{\mathbf{n}-1}\frac{\mathbf{f}^{(\kappa)}(0)}{\omega^{2-\delta+\kappa}}, \; \; \mathbf{n}-1 < \delta < \mathbf{n}, \; \mathbf{n}\in\mathbb{N}. \end{eqnarray}$

(2.7)

Definition 2.6. (^[57]) The $ABC$ fractional derivative operator has the following Aboodh transform:

$\begin{eqnarray} \mathcal{A}\Big\{\, _{0}^{ABC}\mathbf{D}_{\bar{\mathbf{t}}}^{\delta}\big(\mathbf{f}(\bar{\mathbf{t}})\big), \bar{\mathbf{t}}\Big\}(\delta) = \frac{\mathbb{N}(\delta)(\mathbf{F}(\omega)-\omega^{-2}\mathbf{f}(0))}{1-\delta+\delta \omega^{-\delta}}. \end{eqnarray}$

(2.8)

Definition 2.7. (^[58])The ML function for single parameter is defined as

$\begin{eqnarray} E_{\delta}(z) = \sum\limits_{\kappa = 0}^{\infty}\frac{z_{1}^{\kappa}}{\Gamma(\kappa\delta+1)}, \; \; \delta, z_{1}\in\mathbb{C}, \; \Re(\delta)\geq0. \end{eqnarray}$

(2.9)

3. Road map of $\bar{\mathbf{q}}$ -homotopy analysis transform method

The methodology of the proposed approach is presented using Caputo and Atangana-Baleanu fractional derivatives for the model (1.1) of arbitrary order. Here, we surmise a fractional order nonlinear system of the type:

Case I. (Caputo fractional derivative operator)

$\begin{eqnarray} \, _{a_{1}}^{c}\mathbf{D}_{\bar{\mathbf{t}}}^{\delta}\Phi(\varkappa, \bar{\mathbf{t}})+\mathfrak{R}\Phi(\varkappa, \bar{\mathbf{t}})+\mathfrak{N}\Phi(\varkappa, \bar{\mathbf{t}}) = \mathcal{F}(\varkappa, \bar{\mathbf{t}}), \; \; \mathbf{n}-1 < \delta\leq \mathbf{n}, \end{eqnarray}$

(3.1)

supplemented with the initial conditions (ICs)

$\begin{eqnarray} \Phi(\varkappa, 0) = \Phi_{0}(\varkappa), \end{eqnarray}$

(3.2)

where $\, _{a_{1}}^{c}\mathbf{D}_{\bar{\mathbf{t}}}^{\delta}$ indicates the Caputo derivative of $\Phi(\varkappa, \bar{\mathbf{t}}).$

In view of the differentiation rule of Aboodh transform on (3.1), we obtain

$\begin{eqnarray} \omega^{\delta}\mathcal{A}[\Phi(\varkappa, \bar{\mathbf{t}})]-\sum\limits_{\kappa = 0}^{\mathbf{n}-1}\frac{\Phi^{(\kappa)}(\varkappa, 0)}{\omega^{2-\delta+\kappa}}+\mathcal{A}\big[\mathfrak{R}\Phi(\varkappa, \bar{\mathbf{t}})+\mathfrak{N}\Phi(\varkappa, \bar{\mathbf{t}})\big] = \mathcal{A}\big[\mathcal{F}(\varkappa, \bar{\mathbf{t}})\big]. \end{eqnarray}$

(3.3)

After simplification, we have

$\begin{eqnarray} \mathcal{A}[\Phi(\varkappa, \bar{\mathbf{t}})]-\frac{1}{\omega^{\delta}}\sum\limits_{\kappa = 0}^{\mathbf{n}-1}\frac{\Phi^{(\kappa)}(\varkappa, 0)}{\omega^{2-\delta+\kappa}}+\mathcal{A}\big[\mathfrak{R}\Phi(\varkappa, \bar{\mathbf{t}})+\mathfrak{N}\Phi(\varkappa, \bar{\mathbf{t}})\big] = \frac{1}{\omega^{\delta}}\mathcal{A}\big[\mathcal{F}(\varkappa, \bar{\mathbf{t}})\big]. \end{eqnarray}$

(3.4)

The non-linearity factor can be described as

$\begin{eqnarray} \mathfrak{N}\big[\varphi(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})\big]&& = \mathcal{A}\big[\varphi(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})\big]-\frac{1}{\omega^{\delta}}\sum\limits_{\kappa = 0}^{\mathbf{n}-1}\frac{\Phi^{(\kappa)}(\varkappa, 0)}{\omega^{2-\delta+\kappa}}+\frac{1}{\omega^{\delta}}\mathcal{A}\big[\mathfrak{R}\varphi(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})\big]+\frac{1}{\omega^{\delta}}\mathcal{A}\big[\mathfrak{N}\varphi(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})\big]\\&&\quad-\frac{1}{\omega^{\delta}}\mathcal{A}\big[\mathcal{F}(\varkappa, \bar{\mathbf{t}})\big], \end{eqnarray}$

(3.5)

where $\bar{\mathbf{q}}\in[0, 1/\mathbf{n}].$ Then, we demonstrate the non-zero auxiliary and embedding factor $\hbar,$ and $\bar{\mathbf{q}},$ respectively, by the classical approach as below

$\begin{eqnarray} (1-\mathbf{n}\bar{\mathbf{q}})\mathcal{A}\big[\varphi(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})-\Phi_{0}(\varkappa, \bar{\mathbf{t}})\big] = \hbar \bar{\mathbf{q}}\mathcal{H}(\varkappa, \bar{\mathbf{t}})\mathfrak{N}\big[\varphi(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})\big], \end{eqnarray}$

(3.6)

where $\mathcal{A}$ symbolize Aboodh transform, $\mathcal{H}(\varkappa, \bar{\mathbf{t}})$ represents the nonzero auxiliary mapping and $\varphi(\varkappa, \bar{\mathbf{t}}; \bar{\mathbf{q}})$ is an unknown mapping. Also, $\Phi_{0}(\varkappa, \bar{\mathbf{t}})$ is the IC of $\Phi(\varkappa, \bar{\mathbf{t}}).$

Furthermore, for $\bar{\mathbf{q}} = 0$ and $\bar{\mathbf{q}} = 1/\mathbf{n}$ , then the subsequent assumptions hold true

$\begin{eqnarray} \varphi(\varkappa, \bar{\mathbf{t}};0) = \Phi_{0}(\varkappa, \bar{\mathbf{t}}), \; \; \; \; \; \; \; \; \; \varphi\Big(\varkappa, \bar{\mathbf{t}};\frac{1}{\mathbf{n}}\Big) = \Phi(\varkappa, \bar{\mathbf{t}}). \end{eqnarray}$

(3.7)

Consequently, by amplifying $\bar{\mathbf{q}}$ from $0$ to $1/\mathbf{n},$ the solution $\varphi(\varkappa, \bar{\mathbf{t}}; \bar{\mathbf{q}})$ tends from $\Phi_{0}(\varkappa, \bar{\mathbf{t}})$ to the result $\Phi(\varkappa, \bar{\mathbf{t}}).$ Develop the mapping $\varphi(\varkappa, \bar{\mathbf{t}}; \bar{\mathbf{q}})$ in sequence form with the aid of Taylor theorem, one can achieve close to $\bar{\mathbf{q}}$ as

$\begin{eqnarray} \varphi(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}}) = \Phi_{0}(\varkappa, \bar{\mathbf{t}})+\sum\limits_{\mathbf{r} = 1}^{\infty}\Phi_{\mathbf{r}}(\varkappa, \bar{\mathbf{t}})\bar{\mathbf{q}}^{\mathbf{r}}, \end{eqnarray}$

(3.8)

where

$\begin{eqnarray} \Phi_{0}(\varkappa, \bar{\mathbf{t}}) = \frac{1}{\mathbf{r}!}\frac{\partial^{\mathbf{r}}\varphi(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})}{\partial \bar{\mathbf{q}}^{\mathbf{r}}}\Big\vert_{\bar{\mathbf{q}} = 0}. \end{eqnarray}$

(3.9)

With the aid of supplementary linear operator, $\Phi_{0}(\varkappa, \bar{\mathbf{t}}), \; \mathbf{n}, \; \hbar,$ then the (3.9) approaches at $\bar{\mathbf{q}} = 1/\mathbf{n}$ and it yields a solution of (3.3), so we have

$\begin{eqnarray} \Phi(\varkappa, \bar{\mathbf{t}}) = \Phi_{0}(\varkappa, \bar{\mathbf{t}})+\sum\limits_{\mathbf{r} = 0}^{\infty}\Phi_{\mathbf{r}}(\varkappa, \bar{\mathbf{t}})\Big(\frac{1}{\mathbf{n}}\Big)^{\mathbf{r}}. \end{eqnarray}$

(3.10)

The $\mathbf{r}$ -times differentiation of (3.7) considering $\bar{\mathbf{q}}$ and dividing by $\mathbf{r}!,$ later on substituting $\bar{\mathbf{q}} = 0,$ we have

$\begin{eqnarray} \mathcal{A}\big[\Phi_{\mathbf{r}}(\varkappa, \bar{\mathbf{t}})-\mathcal{K}_{\mathbf{r}}\Phi_{\mathbf{r}-1}(\varkappa, \bar{\mathbf{t}})\big] = \hbar\mathcal{R}_{\mathbf{r}}(\overrightarrow{\Phi}_{\mathbf{r}-1}), \end{eqnarray}$

(3.11)

where the vectors are presented as follows:

$\begin{eqnarray} \overrightarrow{\Phi}_{\mathbf{r}} = \big[\Phi_{0}(\varkappa, \bar{\mathbf{t}})+\Phi_{1}(\varkappa, \bar{\mathbf{t}})+....+\Phi_{\mathbf{r}}(\varkappa, \bar{\mathbf{t}})\big]. \end{eqnarray}$

(3.12)

Employing the inverse Aboodh transform on (3.12), yields

$\begin{eqnarray} \Phi_{\mathbf{r}}(\varkappa, \bar{\mathbf{t}}) = \mathcal{K}_{\mathbf{r}}\Phi_{\mathbf{r}-1}(\varkappa, \bar{\mathbf{t}})+\hbar\mathcal{A}^{-1}\big[\mathcal{R}_{\mathbf{r}}(\overrightarrow{\Phi}_{\mathbf{r}-1})\big], \end{eqnarray}$

(3.13)

where

$\mathcal{R}_{\mathbf{r}}(\overrightarrow{\Phi}_{\mathbf{r}-1}) = \mathcal{A}\big[\Phi_{\mathbf{r}-1}(\varkappa, \bar{\mathbf{t}})\big]-\Big(1-\frac{\mathcal{K}_{\mathbf{r}}}{\mathbf{n}}\Big)\bigg(\sum\limits_{\kappa = 0}^{\mathbf{n}-1}\frac{\Phi^{(\kappa)}(\varkappa, 0)}{\omega^{2-\delta+\kappa}}+\frac{1}{\omega^{\delta}}\mathcal{A}\big[\mathcal{F}(\varkappa, \bar{\mathbf{t}})\big]\bigg)\\ +\frac{1}{\omega^{\delta}}\mathcal{A}\big[\mathcal{R}({\Phi}_{\mathbf{r}-1}+\mathcal{H}_{\mathbf{r}-1})\big],$

(3.14)

and

$\begin{eqnarray} \kappa_{\mathbf{r}} = \begin{cases} 0, \; \; \; \; \mathbf{r}\leq1\\ 1, \; \; \; \; \mathbf{r} > 1. \end{cases} \end{eqnarray}$

(3.15)

In (3.33), $\mathcal{H}_{\mathbf{r}}$ indicates the homotopy polynomial and is described as

$\begin{eqnarray} \mathcal{H}_{\mathbf{r}} = \Phi_{0}(\varkappa, \bar{\mathbf{t}}) = \frac{1}{\mathbf{r}!}\frac{\partial^{\mathbf{r}}\varphi(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})}{\partial \bar{\mathbf{q}}^{\mathbf{r}}}\Big\vert_{\bar{\mathbf{q}} = 0} \end{eqnarray}$

(3.16)

and

$\varphi(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}}) = \varphi_{0}+\bar{\mathbf{q}}\varphi_{1}+q^{2}\varphi_{2}+...\; .$

Combining (3.14) and (3.16), we have

$\begin{eqnarray} \Phi_{\mathbf{r}}(\varkappa, \bar{\mathbf{t}})&& = (\kappa_{\mathbf{r}}+\hbar)\Phi_{\mathbf{r}-1}(\varkappa, \bar{\mathbf{t}})-\Big(1-\frac{\kappa_{\mathbf{r}}}{\mathbf{n}}\Big)\mathcal{A}^{-1}\bigg(\sum\limits_{\kappa = 0}^{\mathbf{n}-1}\frac{\Phi^{(\kappa)}(\varkappa, 0)}{\omega^{2-\delta+\kappa}}+\frac{1}{\omega^{\delta}}\mathcal{A}\big[\mathcal{F}(\varkappa, \bar{\mathbf{t}})\big]\bigg)\\&&\quad+\hbar\mathcal{A}^{-1}\bigg(\frac{1}{\omega^{\delta}}\mathcal{A}\big[\mathcal{R}({\Phi}_{\mathbf{r}-1}+\mathcal{H}_{\mathbf{r}-1})\big]\bigg). \end{eqnarray}$

(3.17)

The series solution by $\bar{\mathbf{q}}$ -HATM is expressed as

$\begin{eqnarray} \Phi(\varkappa, \bar{\mathbf{t}}) = \sum\limits_{\mathbf{r} = 0}^{\infty}\Phi_{\mathbf{r}}(\varkappa, \bar{\mathbf{t}}). \end{eqnarray}$

(3.18)

Case II. (ABC fractional derivative operator)

$\begin{eqnarray} \, _{a_{1}}^{ABC}\mathbf{D}_{\bar{\mathbf{t}}}^{\delta}\Phi(\varkappa, \bar{\mathbf{t}})+\mathfrak{R}\Phi(\varkappa, \bar{\mathbf{t}})+\mathfrak{N}\Phi(\varkappa, \bar{\mathbf{t}}) = \mathcal{F}(\varkappa, \bar{\mathbf{t}}), \; \; \mathbf{n}-1 < \delta\leq \mathbf{n}, \end{eqnarray}$

(3.19)

supplemented with the initial conditions (ICs)

$\begin{eqnarray} \Phi(\varkappa, 0) = \Phi_{0}(\varkappa), \end{eqnarray}$

(3.20)

where $\, _{a_{1}}^{ABC}\mathbf{D}_{\bar{\mathbf{t}}}^{\delta}$ indicates the AB fractional derivative in the Caputo sense of $\Phi(\varkappa, \bar{\mathbf{t}}).$

In view of the differentiation rule of Aboodh transform on (3.19), we obtain

$\frac{\omega^{\delta}\mathbb{N}(\delta)}{\delta+(1-\delta) \omega^{\delta}}\mathcal{A}[\Phi(\varkappa, \bar{\mathbf{t}})]-\frac{\delta+(1-\delta) \omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\frac{\Phi(\varkappa, 0)}{\omega^{2}}+\mathcal{A}\big[\mathfrak{R}\Phi(\varkappa, \bar{\mathbf{t}})+\mathfrak{N}\Phi(\varkappa, \bar{\mathbf{t}})\big] \\ = \mathcal{A}\big[\mathcal{F}(\varkappa, \bar{\mathbf{t}})\big].$

(3.21)

After simplification, we have

$\mathcal{A}[\Phi(\varkappa, \bar{\mathbf{t}})]-\frac{1}{\omega^{2}}\Phi(\varkappa, 0)+\frac{\delta+(1-\delta) \omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\big[\mathfrak{R}\Phi(\varkappa, \bar{\mathbf{t}})+\mathfrak{N}\Phi(\varkappa, \bar{\mathbf{t}})\big] \\ = \frac{\delta+(1-\delta) \omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\big[\mathcal{F}(\varkappa, \bar{\mathbf{t}})\big].$

(3.22)

The non-linearity factor can be described as

$\mathfrak{N}\big[\varphi(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})\big] = \mathcal{A}\big[\varphi(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})\big]-\frac{\Phi(\varkappa, 0)}{\omega^{2}}+\frac{\delta+(1-\delta) \omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\big[\mathfrak{R}\varphi(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})\big]+\\ \frac{\delta+(1-\delta) \omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\big[\mathfrak{N}\varphi(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})\big]\\ -\frac{\delta+(1-\delta) \omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\big[\mathcal{F}(\varkappa, \bar{\mathbf{t}})\big],$

(3.23)

where $\bar{\mathbf{q}}\in[0, 1/\mathbf{n}].$ Then, we demonstrate the non-zero auxiliary and embedding factor $\hbar,$ and $\bar{\mathbf{q}},$ respectively, by the classical approach as below

(3.24)

Furthermore, for $\bar{\mathbf{q}} = 0$ and $\bar{\mathbf{q}} = 1/\mathbf{n}$ , then the subsequent assumptions hold true

(3.25)

(3.26)

where

(3.27)

With the aid of supplementary linear operator, $\Phi_{0}(\varkappa, \bar{\mathbf{t}}), \; \mathbf{n}, \; \hbar,$ then the (3.27) approaches at $\bar{\mathbf{q}} = 1/\mathbf{n}$ and it yields a solution of (3.23), so we have

(3.28)

The $\mathbf{r}$ -times differentiation of (3.25) considering $\bar{\mathbf{q}}$ and dividing by $\mathbf{r}!,$ later on substituting $\bar{\mathbf{q}} = 0,$ we have

(3.29)

where the vectors are presented as follows:

(3.30)

Employing the inverse Aboodh transform on (3.30), yields

(3.31)

where

$\begin{eqnarray} \mathcal{R}_{\mathbf{r}}(\overrightarrow{\Phi}_{\mathbf{r}-1})&& = \mathcal{A}\big[\Phi_{\mathbf{r}-1}(\varkappa, \bar{\mathbf{t}})\big]-\Big(1-\frac{\mathcal{K}_{\mathbf{r}}}{\mathbf{n}}\Big)\bigg(\frac{\Phi(\varkappa, 0)}{\omega^{2}}+\frac{\delta+(1-\delta) \omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\big[\mathcal{F}(\varkappa, \bar{\mathbf{t}})\big]\bigg)\\&&\quad+\frac{\delta+(1-\delta) \omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\big[\mathcal{R}({\Phi}_{\mathbf{r}-1}+\mathcal{H}_{\mathbf{r}-1})\big], \end{eqnarray}$

(3.32)

and

$\begin{eqnarray} \kappa_{\mathbf{r}} = \begin{cases} 0, \; \; \; \; \mathbf{r}\leq1\\ 1, \; \; \; \; \mathbf{r} > 1. \end{cases} \end{eqnarray}$

(3.33)

In (3.33), $\mathcal{H}_{\mathbf{r}}$ indicates the homotopy polynomial and is described as

(3.34)

and

$\varphi(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}}) = \varphi_{0}+\bar{\mathbf{q}}\varphi_{1}+q^{2}\varphi_{2}+...\; .$

Combining (3.32) and (3.34), we have

$\begin{eqnarray} \Phi_{\mathbf{r}}(\varkappa, \bar{\mathbf{t}})&& = (\kappa_{\mathbf{r}}+\hbar)\Phi_{\mathbf{r}-1}(\varkappa, \bar{\mathbf{t}})-\Big(1-\frac{\kappa_{\mathbf{r}}}{\mathbf{n}}\Big)\mathcal{A}^{-1}\bigg(\frac{\Phi(\varkappa, 0)}{\omega^{2}}+\frac{\delta+(1-\delta) \omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\big[\mathcal{F}(\varkappa, \bar{\mathbf{t}})\big]\bigg)\\&&\quad+\hbar\mathcal{A}^{-1}\bigg(\frac{\delta+(1-\delta) \omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\big[\mathcal{R}({\Phi}_{\mathbf{r}-1}+\mathcal{H}_{\mathbf{r}-1})\big]\bigg). \end{eqnarray}$

(3.35)

The series solution by $\bar{\mathbf{q}}$ -HATM is expressed as

(3.36)

Theorem 3.1. (Convergence of the series solutions) Suppose $\Phi_{\mathbf{n}}(\varkappa, \bar{\mathbf{t}})$ and $\Phi(\varkappa, \bar{\mathbf{t}})$ are defined in the Banach space $(\mathfrak{B}[0, \mathbf{T}], \|.\|).$ The series solution defined in $(3.18)$ converges to the solution of the $(3.1)$ , if $\lambda\in(0, 1).$

Proof. Consider the sequence $\{S_{\mathbf{n}}\}$ , which is the partial sum of the (3.18), and we have to prove $\{S_{\mathbf{n}}\}$ is the Cauchy sequence in $(\mathfrak{B}[0, \mathbf{T}], \|.\|).$ Now consider

$\begin{eqnarray*} \big\|S_{\mathbf{n}+1}(\varkappa, \bar{\mathbf{t}})-S_{\mathbf{n}}(\varkappa, \bar{\mathbf{t}})\big\|&& = \big\|\Phi_{\mathbf{n}+1}(\varkappa, \bar{\mathbf{t}})\big\|\nonumber\\&&\leq\lambda\big\|\Phi_{\mathbf{n}}(\varkappa, \bar{\mathbf{t}})\big\| \nonumber\\&&\leq\lambda^{2}\big\|\Phi_{\mathbf{n}-1}(\varkappa, \bar{\mathbf{t}})\big\|\leq...\leq\lambda^{\mathbf{n}}\big\|\Phi_{0}(\varkappa, \bar{\mathbf{t}})\big\|. \end{eqnarray*}$

Now, for every $\mathbf{n}, \mathbf{m}\in\mathbb{N}\; (\mathbf{m}\leq\mathbf{n})$

$\begin{eqnarray*} \big\|S_{\mathbf{n}}-S_{\mathbf{m}}\big\|&& = \big\|(S_{\mathbf{n}}-S_{\mathbf{n}-1})+(S_{\mathbf{n}-1}-S_{\mathbf{n}-2})+...+(S_{\mathbf{m}+1}-S_{\mathbf{m}})\big\|\nonumber\\&&\leq\big\|S_{\mathbf{n}}-S_{\mathbf{n}-1}\big\|+\big\|S_{\mathbf{n}-1}-S_{\mathbf{n}-2}\big\|+...+\big\|S_{\mathbf{m}+1}-S_{\mathbf{m}}\big\|\nonumber\\&&\leq(\lambda^{\mathbf{n}}+\lambda^{\mathbf{n}-1}+...\lambda^{\mathbf{m}+1})\big\|\Phi_{0}\big\|\nonumber\\&&\leq\lambda^{\mathbf{m}+1}(\lambda^{\mathbf{n}-\mathbf{m}-1}+\lambda^{\mathbf{n}-\mathbf{m}-2}+...+\lambda+1)\big\|\Phi_{0}\big\|\nonumber\\&&\leq\lambda^{\mathbf{m}+1}\Big(\frac{1-\lambda^{\mathbf{n}-\mathbf{m}}}{1-\lambda}\Big)\big\|\Phi_{0}\big\|. \end{eqnarray*}$

Since $0 < \lambda < 1,$ therefore $\big\|S_{\mathbf{n}}-S_{\mathbf{m}}\big\| = 0.$ Thus $\{S\}_{\mathbf{n}}$ is the Cauchy sequence. This completes the proof.

4. Test examples

Example 4.1. Surmise that the fractional-order WBKEs is presented as follows:

$\begin{eqnarray} &&\mathbf{D}_{\bar{\mathbf{t}}}^{\delta}\Phi(\varkappa, \bar{\mathbf{t}})+\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa} = 0, \; \; 0 < \delta\leq1, \\&& \mathbf{D}_{\bar{\mathbf{t}}}^{\delta}\Psi(\varkappa, \bar{\mathbf{t}})+\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+3\frac{\partial^{3}\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{3}}-\frac{\partial^{2}\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}} = 0, \end{eqnarray}$

(4.1)

supplemented with ICs

$\begin{eqnarray} &&\Phi(\varkappa, 0) = \frac{1}{2}-8\tanh(-2\varkappa), \\&&\Psi(\varkappa, 0) = 16-16\tanh^{2}(-2\varkappa). \end{eqnarray}$

(4.2)

Proof. Primarily, we demonstrate how to solve (4.1) in two different scenarios.

Case I: Initially, we employ the Caputo fractional derivative operator considered with the Aboodh transform and $\bar{q}$ -HATM.

Employing the Aboodh transform on (4.1), we have

$\begin{eqnarray} &&\mathcal{A}\big[\mathbf{D}_{\bar{\mathbf{t}}}^{\delta}\Phi(\varkappa, \bar{\mathbf{t}})\big] = -\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg], \\&& \mathcal{A}\big[\mathbf{D}_{\bar{\mathbf{t}}}^{\delta}\Psi(\varkappa, \bar{\mathbf{t}})\big] = -\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+3\frac{\partial^{3}\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{3}}-\frac{\partial^{2}\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg]. \end{eqnarray}$

(4.3)

It follows that

$\omega^{\delta}\mathcal{A}[\Phi(\varkappa, \bar{\mathbf{t}})]-\sum\limits_{\kappa = 0}^{\mathbf{n}-1}\frac{\Phi^{(\kappa)}(\varkappa, 0)}{\omega^{2-\delta+\kappa}} = -\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg], \\ \omega^{\delta}\mathcal{A}[\Psi(\varkappa, \bar{\mathbf{t}})]-\sum\limits_{\kappa = 0}^{\mathbf{n}-1}\frac{\Psi^{(\kappa)}(\varkappa, 0)}{\omega^{2-\delta+\kappa}} = -\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+3\frac{\partial^{3}\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{3}}\\ -\frac{\partial^{2}\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg].$

(4.4)

Therefore, we have

$\mathcal{A}[\Phi(\varkappa, \bar{\mathbf{t}})] = \frac{1}{\omega^{2}}\Phi(\varkappa, 0)-\frac{1}{\omega^{\delta}}\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg], \\ \mathcal{A}[\Psi(\varkappa, \bar{\mathbf{t}})] = \frac{1}{\omega^{2}}\Psi(\varkappa, 0)-\frac{1}{\omega^{\delta}}\mathcal{A}\\ \bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+3\frac{\partial^{3}\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{3}}-\frac{\partial^{2}\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg].$

(4.5)

With the aid of (4.5), the non-linearity can be expressed as

$\begin{eqnarray} &&\mathcal{N}_{1}\big[\varphi_{1}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}}), \varphi_{2}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})\big]\\&& = \mathcal{A}\Bigg[\varphi_{1}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})-\frac{1}{\omega^{2}}\varphi_{1}(\varkappa, 0)+\frac{1}{\omega^{\delta}}\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg]\Bigg], \\&& \mathcal{N}_{2}\big[\varphi_{1}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}}), \varphi_{2}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})\big]\\&& = \mathcal{A}\Bigg[\varphi_{2}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})-\frac{1}{\omega^{2}}\varphi_{2}(\varkappa, 0)+\frac{1}{\omega^{\delta}}\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\\&&\quad+3\frac{\partial^{3}\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{3}}-\frac{\partial^{2}\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg]\Bigg].\\ \end{eqnarray}$

(4.6)

Utilizing the methodology described in Section 3, the $\mathbf{r}$ th order deformation equation is described as

$\begin{eqnarray} \mathcal{A}\big[\Phi_{\mathbf{r}}(\varkappa, \bar{\mathbf{t}})-\mathcal{K}_{\mathbf{r}}\Phi_{\mathbf{r}-1}(\varkappa, \bar{\mathbf{t}})\big] = \hbar\mathcal{R}_{1, \mathbf{r}}\big[\overrightarrow{\Phi}_{\mathbf{r}-1}, \overrightarrow{\Psi}_{\mathbf{r}-1}\big], \end{eqnarray}$

(4.7)

where

$\begin{eqnarray} \mathcal{R}_{1, \mathbf{r}}\big[\overrightarrow{\Phi}_{\mathbf{r}-1}, \overrightarrow{\Psi}_{\mathbf{r}-1}\big]&& = \mathcal{A}\Bigg[\Phi_{\mathbf{r}-1}(\varkappa, \bar{\mathbf{t}})-\Big(1-\frac{\kappa_{\mathbf{r}}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\big(\frac{1}{2}-8\tanh(-2\varkappa)\big)\Big)\\&&\quad+\frac{1}{\omega^{\delta}}\bigg\{\sum\limits_{\jmath = 0}^{\mathbf{r}-1}\Phi_{\jmath}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{\mathbf{r}-1-\jmath}}{\partial \varkappa}+\frac{\partial\Phi_{\mathbf{r}-1}}{\partial \varkappa}+\frac{\partial\Psi_{\mathbf{r}-1-\jmath}}{\partial \varkappa}\bigg\}\Bigg], \\\mathcal{R}_{2, \mathbf{r}}\big[\overrightarrow{\Phi}_{\mathbf{r}-1}, \overrightarrow{\Psi}_{\mathbf{r}-1}\big]&& = \mathcal{A}\Bigg[\Phi_{\mathbf{r}-1}(\varkappa, \bar{\mathbf{t}})-\Big(1-\frac{\kappa_{\mathbf{r}}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\big(16-16\tanh^{2}(-2\varkappa)\big)\Big)\\&&\quad+\frac{1}{\omega^{\delta}}\bigg\{\sum\limits_{\jmath = 0}^{\mathbf{r}-1}\Phi_{\jmath}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi_{\mathbf{r}-1-\jmath}}{\partial \varkappa}+\sum\limits_{\jmath = 0}^{\mathbf{r}-1}\Psi_{\jmath}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{\mathbf{r}-\jmath-1}}{\partial \varkappa}\\&&\quad+3\frac{\partial^{3}\Phi_{\mathbf{r}-1-\jmath}}{\partial \varkappa^{3}}-\frac{\partial^{2}\Psi_{\mathbf{r}-1-\jmath}}{\partial \varkappa^{2}}\bigg\}\Bigg].\\ \end{eqnarray}$

(4.8)

Thanks to the inverse Aboodh transform, we attain

$\begin{eqnarray} &&\Phi_{\mathbf{r}}(\varkappa, \bar{\mathbf{t}}) = \mathcal{K}_{\mathbf{r}}\Phi_{\mathbf{r}-1}(\varkappa, \bar{\mathbf{t}})+\hbar\mathcal{A}^{-1}\mathcal{R}_{1, \mathbf{r}}\big[\overrightarrow{\Phi}_{\mathbf{r}-1}, \overrightarrow{\Psi}_{\mathbf{r}-1}\big], \\&&\Psi_{\mathbf{r}}(\varkappa, \bar{\mathbf{t}}) = \mathcal{K}_{\mathbf{r}}\Psi_{\mathbf{r}-1}(\varkappa, \bar{\mathbf{t}})+\hbar\mathcal{A}^{-1}\mathcal{R}_{2, \mathbf{r}}\big[\overrightarrow{\Phi}_{\mathbf{r}-1}, \overrightarrow{\Psi}_{\mathbf{r}-1}\big]. \end{eqnarray}$

(4.9)

In view of ICs, we have

$\begin{eqnarray} &&\Phi_{0}(\varkappa, \bar{\mathbf{t}}) = \frac{1}{2}-8\tanh(-2\varkappa), \\&&\Psi_{0}(\varkappa, \bar{\mathbf{t}}) = 16-16\tanh^{2}(-2\varkappa). \end{eqnarray}$

(4.10)

In order to obtain $\Phi_{0}(\varkappa, \bar{\mathbf{t}})$ and $\Psi_{0}(\varkappa, \bar{\mathbf{t}}),$ choosing $\mathbf{r} = 1$ in (4.8), then we acquire

$\begin{eqnarray} &&\Phi_{1}(\varkappa, \bar{\mathbf{t}}) = \mathcal{K}_{1}\Phi_{0}(\varkappa, \bar{\mathbf{t}})+\hbar\mathcal{A}^{-1}\mathcal{R}_{1, 1}\big[\overrightarrow{\Phi}_{0}, \overrightarrow{\Psi}_{0}\big], \\&&\Psi_{1}(\varkappa, \bar{\mathbf{t}}) = \mathcal{K}_{1}\Psi_{0}(\varkappa, \bar{\mathbf{t}})+\hbar\mathcal{A}^{-1}\mathcal{R}_{2, 1}\big[\overrightarrow{\Phi}_{0}, \overrightarrow{\Psi}_{0}\big]. \end{eqnarray}$

(4.11)

For $\mathbf{r} = 1,$ then (4.8) diminish to

$\mathcal{R}_{1, 1}\big[\overrightarrow{\Phi}_{0}, \overrightarrow{\Psi}_{0}\big] = \mathcal{A}\big[\Phi_{0}(\varkappa, \bar{\mathbf{t}})\big]-\Big(1-\frac{\kappa_{1}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\Big(\frac{1}{2}-8\tanh(-2\varkappa)\Big)\\\quad+\frac{1}{\omega^{\delta}}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg\}, \\ \mathcal{R}_{2, 1}\big[\overrightarrow{\Phi}_{0}, \overrightarrow{\Psi}_{0}\big] = \mathcal{A}\big[\Psi_{0}(\varkappa, \bar{\mathbf{t}})\big]-\Big(1-\frac{\kappa_{1}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\big(16-16\tanh^{2}(-2\varkappa)\big)\\\quad+\frac{1}{\omega^{\delta}}\mathcal{A}\\ \bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+3\frac{\partial^{3}\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{3}}-\frac{\partial^{2}\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg\}.\\$

(4.12)

Plugging (3.33) and (4.12) in (4.11), then we have

$\Phi_{1}(\varkappa, \bar{\mathbf{t}}) = \hbar\mathcal{A}^{-1}\bigg[\frac{1}{\omega^{2}}\Big(\frac{1}{2}-8\tanh(-2\varkappa)\Big)-\Big(1-\frac{0}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\Big(\frac{1}{2}-8\tanh(-2\varkappa)\Big)\\ \quad+\frac{1}{\omega^{\delta}}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg\}\bigg], \\ \Psi_{1}(\varkappa, \bar{\mathbf{t}}) = \hbar\mathcal{A}^{-1}\bigg[\frac{1}{\omega^{2}}\big(16-16\tanh^{2}(-2\varkappa)\big)-\Big(1-\frac{0}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\big(16-16\tanh^{2}(-2\varkappa)\big)\\ \quad+\frac{1}{\omega^{\delta}}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+3\frac{\partial^{3}\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{3}}-\frac{\partial^{2}\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg\}\bigg].\\$

(4.13)

Thus, we have

$\begin{eqnarray} \Phi_{1}(\varkappa, \bar{\mathbf{t}}) = -8\hbar\sec h^{2}(-2\varkappa)\frac{\bar{\mathbf{t}}^{\delta}}{\Gamma(\delta+1)}, \; \; \; \; \Psi_{1}(\varkappa, \bar{\mathbf{t}}) = -32\hbar\sec h^{2}(-2\varkappa)\tanh(-2\varkappa)\frac{\bar{\mathbf{t}}^{\delta}}{\Gamma(\delta+1)}. \end{eqnarray}$

(4.14)

Analogously, for $\mathbf{r} = 2,$ then (4.11) and (4.12) yields

$\begin{eqnarray} \Phi_{2}(\varkappa, \bar{\mathbf{t}})&& = \mathbf{n}\Phi_{1}(\varkappa, \bar{\mathbf{t}})+\hbar\mathcal{A}^{-1}\bigg[\mathcal{A}[\Phi_{1}(\varkappa, \bar{\mathbf{t}})]\big]-\Big(1-\frac{\mathbf{n}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\Big(\frac{1}{2}-8\tanh(-2\varkappa)\Big)\\&&\quad+\frac{1}{\omega^{\delta}}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Phi_{1}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Phi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg\}\bigg], \\ \Psi_{2}(\varkappa, \bar{\mathbf{t}})&& = \mathbf{n}\Psi_{1}(\varkappa, \bar{\mathbf{t}})+\hbar\mathcal{A}^{-1}\bigg[\mathcal{A}[\Psi_{1}(\varkappa, \bar{\mathbf{t}})]\big]-\Big(1-\frac{\mathbf{n}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\big(16-16\tanh(-2\varkappa)\big)\\&&\quad+\frac{1}{\omega^{\delta}}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Phi_{1}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa} +\Psi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\\&&\quad+ \Psi_{1}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa} +3\frac{\partial^{3}\Phi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{3}}-\frac{\partial^{2}\Psi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg\}\bigg]. \end{eqnarray}$

(4.15)

After simplification, the foregoing procedure reduces as mentioned

$\begin{eqnarray} \Phi_{2}(\varkappa, \bar{\mathbf{t}})&& = -8(\mathbf{n}+\hbar)\hbar sech^{2}(-2\varkappa)\frac{\bar{\mathbf{t}}^{\delta}}{\Gamma(\delta+1)}-16\hbar^{2}\sec h^{2}(-2\varkappa)\\&&\quad\times\Big(4sech^{2}(-2\varkappa)-8\tanh^{2}(-2\varkappa)+3\tanh(-2\varkappa)\Big)\frac{\bar{\mathbf{t}}^{2\delta}}{\Gamma(2\delta+1)}, \\ \Psi_{2}(\varkappa, \bar{\mathbf{t}})&& = -32(\mathbf{n}+\hbar)\hbar sech^{2}(-2\varkappa)\tanh(-2\varkappa)\frac{\bar{\mathbf{t}}^{\delta}}{\Gamma(\delta+1)}-32\hbar^{2}\sec h^{2}(-2\varkappa)\\&&\quad\times\Big(40\sec h^{2}(-2\varkappa)\tanh(-2\varkappa)+96\tanh^{2}(-2\varkappa)-2\tanh^{2}(-2\varkappa)\\&&\quad-32\tanh^{3}(-2\varkappa)-25\sec h^{2}(-2\varkappa)\Big)\frac{\bar{\mathbf{t}}^{2\delta}}{\Gamma(2\delta+1)}. \end{eqnarray}$

(4.16)

Similarly, we evaluate the remaining term. Then, we have

$\begin{eqnarray} \Phi(\varkappa, \bar{\mathbf{t}})&& = \Phi_{0}(\varkappa, \bar{\mathbf{t}})+\sum\limits_{\mathbf{r} = 1}^{\infty}\Phi_{\mathbf{r}}(\varkappa, \bar{\mathbf{t}})\Big(\frac{1}{\mathbf{n}}\Big)^{\mathbf{r}}, \\\Psi(\varkappa, \bar{\mathbf{t}})&& = \Psi_{0}(\varkappa, \bar{\mathbf{t}})+\sum\limits_{\mathbf{r} = 1}^{\infty}\Psi_{\mathbf{r}}(\varkappa, \bar{\mathbf{t}})\Big(\frac{1}{\mathbf{n}}\Big)^{\mathbf{r}}. \end{eqnarray}$

(4.17)

Case II: Secondly, we employ the ABC fractional derivative operator considered with the Aboodh transform and $\bar{q}$ -HATM.

Employing the Aboodh transform on (4.1), we have

$\begin{eqnarray} &&\frac{\omega^{\delta}\mathbb{N}(\delta)}{\delta+(1-\delta)\omega^{\delta}}\mathcal{A}[\Phi(\varkappa, \bar{\mathbf{t}})]-\frac{\Phi(\varkappa, 0)}{\omega^{2}} = -\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg], \\&& \frac{\omega^{\delta}\mathbb{N}(\delta)}{\delta+(1-\delta)\omega^{\delta}}\mathcal{A}[\Psi(\varkappa, \bar{\mathbf{t}})]-\frac{\Psi(\varkappa, 0)}{\omega^{2}} = -\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\\&&\quad+3\frac{\partial^{3}\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{3}}-\frac{\partial^{2}\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg].\\ \end{eqnarray}$

(4.18)

Therefore, we have

$\begin{eqnarray} \mathcal{A}[\Phi(\varkappa, \bar{\mathbf{t}})]&& = \frac{1}{\omega^{2}}\Phi(\varkappa, 0)-\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg], \\ \mathcal{A}[\Psi(\varkappa, \bar{\mathbf{t}})]&& = \frac{1}{\omega^{2}}\Psi(\varkappa, 0)-\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\\&&\quad+3\frac{\partial^{3}\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{3}}-\frac{\partial^{2}\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg].\\ \end{eqnarray}$

(4.19)

With the aid of (4.19), the non-linearity can be expressed as

$\mathcal{N}_{1}\big[\varphi_{1}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}}), \varphi_{2}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})\big]\\ = \mathcal{A}\Bigg[\varphi_{1}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})-\frac{1}{\omega^{2}}\varphi_{1}(\varkappa, 0)+\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa} \\ +\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg]\Bigg], \\ \mathcal{N}_{2}\big[\varphi_{1}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}}), \varphi_{2}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})\big]\\ = \mathcal{A}\Bigg[\varphi_{2}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})-\frac{1}{\omega^{2}}\varphi_{2}(\varkappa, 0)+\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\\ \quad+3\frac{\partial^{3}\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{3}}-\frac{\partial^{2}\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg]\Bigg].\\$

(4.20)

Utilizing the methodology described in Section 3, the $\mathbf{r}$ th order deformation equation is described as

$\begin{eqnarray} \mathcal{A}\big[\Phi_{\mathbf{r}}(\varkappa, \bar{\mathbf{t}})-\mathcal{K}_{\mathbf{r}}\Phi_{\mathbf{r}-1}(\varkappa, \bar{\mathbf{t}})\big] = \hbar\mathcal{R}_{1, \mathbf{r}}\big[\overrightarrow{\Phi}_{\mathbf{r}-1}, \overrightarrow{\Psi}_{\mathbf{r}-1}\big], \end{eqnarray}$

(4.21)

where

$\begin{eqnarray} \mathcal{R}_{1, \mathbf{r}}\big[\overrightarrow{\Phi}_{\mathbf{r}-1}, \overrightarrow{\Psi}_{\mathbf{r}-1}\big]&& = \mathcal{A}\Bigg[\Phi_{\mathbf{r}-1}(\varkappa, \bar{\mathbf{t}})-\Big(1-\frac{\kappa_{\mathbf{r}}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\big(\frac{1}{2}-8\tanh(-2\varkappa)\big)\Big)\\&&\quad+\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\bigg\{\sum\limits_{\jmath = 0}^{\mathbf{r}-1}\Phi_{\jmath}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{\mathbf{r}-1-\jmath}}{\partial \varkappa}+\frac{\partial\Phi_{\mathbf{r}-1}}{\partial \varkappa}+\frac{\partial\Psi_{\mathbf{r}-1-\jmath}}{\partial \varkappa}\bigg\}\Bigg], \\\mathcal{R}_{2, \mathbf{r}}\big[\overrightarrow{\Phi}_{\mathbf{r}-1}, \overrightarrow{\Psi}_{\mathbf{r}-1}\big]&& = \mathcal{A}\Bigg[\Phi_{\mathbf{r}-1}(\varkappa, \bar{\mathbf{t}})-\Big(1-\frac{\kappa_{\mathbf{r}}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\big(16-16\tanh^{2}(-2\varkappa)\big)\Big)\\&&\quad+\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\bigg\{\sum\limits_{\jmath = 0}^{\mathbf{r}-1}\Phi_{\jmath}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi_{\mathbf{r}-1-\jmath}}{\partial \varkappa}+\sum\limits_{\jmath = 0}^{\mathbf{r}-1}\Psi_{\jmath}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{\mathbf{r}-\jmath-1}}{\partial \varkappa}\\&&\quad+3\frac{\partial^{3}\Phi_{\mathbf{r}-1-\jmath}}{\partial \varkappa^{3}}-\frac{\partial^{2}\Psi_{\mathbf{r}-1-\jmath}}{\partial \varkappa^{2}}\bigg\}\Bigg].\\ \end{eqnarray}$

(4.22)

Thanks to the inverse Aboodh transform, we attain

(4.23)

In view of ICs, we have

$\begin{eqnarray} &&\Phi_{0}(\varkappa, \bar{\mathbf{t}}) = \frac{1}{2}-8\tanh(-2\varkappa), \\&&\Psi_{0}(\varkappa, \bar{\mathbf{t}}) = 16-16\tanh^{2}(-2\varkappa). \end{eqnarray}$

(4.24)

In order to obtain $\Phi_{0}(\varkappa, \bar{\mathbf{t}})$ and $\Psi_{0}(\varkappa, \bar{\mathbf{t}}),$ choosing $\mathbf{r} = 1$ in (4.23), then we acquire

(4.25)

For $\mathbf{r} = 1,$ then (4.22) diminish to

$\begin{eqnarray} \mathcal{R}_{1, 1}\big[\overrightarrow{\Phi}_{0}, \overrightarrow{\Psi}_{0}\big]&& = \mathcal{A}\big[\Phi_{0}(\varkappa, \bar{\mathbf{t}})\big]-\Big(1-\frac{\kappa_{1}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\Big(\frac{1}{2}-8\tanh(-2\varkappa)\Big)\\&&\quad+\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg\}, \\ \mathcal{R}_{2, 1}\big[\overrightarrow{\Phi}_{0}, \overrightarrow{\Psi}_{0}\big]&& = \mathcal{A}\big[\Psi_{0}(\varkappa, \bar{\mathbf{t}})\big]-\Big(1-\frac{\kappa_{1}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\big(16-16\tanh^{2}(-2\varkappa)\big)\\&&\quad+\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\\&&\quad+3\frac{\partial^{3}\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{3}}-\frac{\partial^{2}\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg\}. \end{eqnarray}$

(4.26)

Plugging (3.33) and (4.26) in (4.25), then we have

$\begin{eqnarray} \Phi_{1}(\varkappa, \bar{\mathbf{t}})&& = \hbar\mathcal{A}^{-1}\bigg[\frac{1}{\omega^{2}}\Big(\frac{1}{2}-8\tanh(-2\varkappa)\Big)-\Big(1-\frac{0}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\Big(\frac{1}{2}-8\tanh(-2\varkappa)\Big)\\&&\quad+\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg\}\bigg], \\ \Psi_{1}(\varkappa, \bar{\mathbf{t}})&& = \hbar\mathcal{A}^{-1}\bigg[\frac{1}{\omega^{2}}\big(16-16\tanh^{2}(-2\varkappa)\big)-\Big(1-\frac{0}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\big(16-16\tanh^{2}(-2\varkappa)\big)\\&&\quad+\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\\&&\quad+3\frac{\partial^{3}\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{3}}-\frac{\partial^{2}\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg\}\bigg]. \end{eqnarray}$

(4.27)

Thus, we have

$\begin{eqnarray} \Phi_{1}(\varkappa, \bar{\mathbf{t}})&& = \frac{-8\hbar\sec h^{2}(-2\varkappa)}{\mathbb{N}(\delta)}\bigg\{\frac{\delta \bar{\mathbf{t}}^{\delta}}{\Gamma(\delta+1)}+(1-\delta)\bigg\}, \\\Psi_{1}(\varkappa, \bar{\mathbf{t}})&& = -\frac{32\hbar\sec h^{2}(-2\varkappa)\tanh(-2\varkappa)}{\mathbb{N}(\delta)}\bigg\{\frac{\delta \bar{\mathbf{t}}^{\delta}}{\Gamma(\delta+1)}+(1-\delta)\bigg\}, \end{eqnarray}$

(4.28)

Analogously, for $\mathbf{r} = 2,$ then (4.25) and (4.26) yields

$\Phi_{2}(\varkappa, \bar{\mathbf{t}}) = \mathbf{n}\Phi_{1}(\varkappa, \bar{\mathbf{t}})+\hbar\mathcal{A}^{-1}\bigg[\mathcal{A}[\Phi_{1}(\varkappa, \bar{\mathbf{t}})]\big]-\Big(1-\frac{\mathbf{n}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\Big(\frac{1}{2}-8\tanh(-2\varkappa)\Big)\\ \quad+\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Phi_{1}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Phi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg\}\bigg], \\ \Psi_{2}(\varkappa, \bar{\mathbf{t}}) = \mathbf{n}\Psi_{1}(\varkappa, \bar{\mathbf{t}})+\hbar\mathcal{A}^{-1}\bigg[\mathcal{A}[\Psi_{1}(\varkappa, \bar{\mathbf{t}})]\big]-\Big(1-\frac{\mathbf{n}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\big(16-16\tanh(-2\varkappa)\big)\\ \quad+\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Phi_{1}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa} +\Psi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\\ \quad+ \Psi_{1}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa} +3\frac{\partial^{3}\Phi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{3}}-\frac{\partial^{2}\Psi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg\}\bigg].$

(4.29)

After simplification, the foregoing procedure reduces as mentioned

$\Phi_{2}(\varkappa, \bar{\mathbf{t}}) = -\frac{8(\mathbf{n}+\hbar)\hbar sech^{2}(-2\varkappa)}{\mathbb{N}(\delta)}\bigg\{\frac{\delta \bar{\mathbf{t}}^{\delta}}{\Gamma(\delta+1)}+(1-\delta)\bigg\}\\ \quad-\frac{16\hbar^{2}}{\mathbb{N}^{2}(\delta)}\sec h^{2}(-2\varkappa)\Big(4sech^{2}(-2\varkappa)-8\tanh^{2}(-2\varkappa)+3\tanh(-2\varkappa)\Big)\\ \quad\times\bigg\{\frac{\delta^{2}\bar{\mathbf{t}}^{2\delta}}{\Gamma(2\delta+1)}+2\delta(1-\delta)\frac{\delta \bar{\mathbf{t}}^{\delta}}{\Gamma(\delta+1)}+(1-\delta)^{2}\bigg\}, \\ \Psi_{2}(\varkappa, \bar{\mathbf{t}}) = -\frac{32(\mathbf{n}+\hbar)\hbar}{\mathbb{N}(\delta)} sech^{2}(-2\varkappa)\tanh(-2\varkappa)\bigg\{\frac{\delta \bar{\mathbf{t}}^{\delta}}{\Gamma(\delta+1)}+(1-\delta)\bigg\}-\frac{32\hbar^{2}}{\mathbb{N}^{2}(\delta)}\sec h^{2}(-2\varkappa)\\ \quad\times\Big(40\sec h^{2}(-2\varkappa)\tanh(-2\varkappa)+96\tanh^{2}(-2\varkappa)-2\tanh^{2}(-2\varkappa)\\ \quad-32\tanh^{3}(-2\varkappa)-25\sec h^{2}(-2\varkappa)\Big)\bigg\{\frac{\delta^{2}\bar{\mathbf{t}}^{2\delta}}{\Gamma(2\delta+1)}+2\delta(1-\delta)\frac{\delta \bar{\mathbf{t}}^{\delta}}{\Gamma(\delta+1)}+(1-\delta)^{2}\bigg\}.$

(4.30)

Similarly, we evaluate the remaining term. Then, we have

(4.31)

For $\delta = 1,$ then the integer-order solution of (4.1) is

$\begin{eqnarray} \Phi(\varkappa, \bar{\mathbf{t}})&& = \frac{1}{2}-8\tanh \Big(-2\big(\varkappa-\frac{\bar{\mathbf{t}}}{2}\big)\Big), \\ \Psi(\varkappa, \bar{\mathbf{t}})&& = 16-16\tanh^{2} \Big(-2\big(\varkappa-\frac{\bar{\mathbf{t}}}{2}\big)\Big), \\ \end{eqnarray}$

(4.32)

Example 4.2. Surmise that the fractional-order WBKEs is presented as follows:

$\begin{eqnarray} &&\mathbf{D}_{\bar{\mathbf{t}}}^{\delta}\Phi(\varkappa, \bar{\mathbf{t}})+\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{1}{2}\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa} = 0, \; \; 0 < \delta\leq1\\&& \mathbf{D}_{\bar{\mathbf{t}}}^{\delta}\Psi(\varkappa, \bar{\mathbf{t}})+\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}-\frac{1}{2}\frac{\partial^{2}\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}} = 0, \end{eqnarray}$

(4.33)

supplemented with ICs

$\begin{eqnarray} &&\Phi(\varkappa, 0) = \zeta-\kappa\coth\big(\kappa(\varkappa+\theta)\big), \\&&\Psi(\varkappa, 0) = -\kappa^{2}cosec h^{2}\big(\kappa(\varkappa+\theta)\big). \end{eqnarray}$

(4.34)

Proof. Primarily, we demonstrate how to solve (4.33) in two different scenarios.

Case I: Initially, we employ the Caputo fractional derivative operator considered with the Aboodh transform and $\bar{q}$ -HATM.

Employing the Aboodh transform on (4.33), we have

$\begin{eqnarray} &&\mathcal{A}\big[\mathbf{D}_{\bar{\mathbf{t}}}^{\delta}\Phi(\varkappa, \bar{\mathbf{t}})\big] = -\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{1}{2}\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg], \\&& \mathcal{A}\big[\mathbf{D}_{\bar{\mathbf{t}}}^{\delta}\Psi(\varkappa, \bar{\mathbf{t}})\big] = -\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}-\frac{1}{2}\frac{\partial^{2}\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg]. \end{eqnarray}$

(4.35)

It follows that

$\begin{eqnarray} &&\omega^{\delta}\mathcal{A}[\Phi(\varkappa, \bar{\mathbf{t}})]-\sum\limits_{\kappa = 0}^{\mathbf{n}-1}\frac{\Phi^{(\kappa)}(\varkappa, 0)}{\omega^{2-\delta+\kappa}} = -\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{1}{2}\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg], \\&& \omega^{\delta}\mathcal{A}[\Psi(\varkappa, \bar{\mathbf{t}})]-\sum\limits_{\kappa = 0}^{\mathbf{n}-1}\frac{\Psi^{(\kappa)}(\varkappa, 0)}{\omega^{2-\delta+\kappa}} = -\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}-\frac{1}{2}\frac{\partial^{2}\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg].\\ \end{eqnarray}$

(4.36)

Therefore, we have

$\begin{eqnarray} &&\mathcal{A}[\Phi(\varkappa, \bar{\mathbf{t}})] = \frac{1}{\omega^{2}}\Phi(\varkappa, 0)-\frac{1}{\omega^{\delta}}\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{1}{2}\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg], \\&& \mathcal{A}[\Psi(\varkappa, \bar{\mathbf{t}})] = \frac{1}{\omega^{2}}\Psi(\varkappa, 0)-\frac{1}{\omega^{\delta}}\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}-\frac{1}{2}\frac{\partial^{2}\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg].\\ \end{eqnarray}$

(4.37)

With the aid of (4.37), the non-linearity can be expressed as

$\mathcal{N}_{1}\big[\varphi_{1}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}}), \varphi_{2}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})\big]\\ = \mathcal{A}\Bigg[\varphi_{1}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})-\frac{1}{\omega^{2}}\varphi_{1}(\varkappa, 0)+\frac{1}{\omega^{\delta}}\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{1}{2}\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg]\Bigg], \\ \mathcal{N}_{2}\big[\varphi_{1}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}}), \varphi_{2}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})\big]\\ = \mathcal{A}\Bigg[\varphi_{2}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})-\frac{1}{\omega^{2}}\varphi_{2}(\varkappa, 0)+\frac{1}{\omega^{\delta}}\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}-\frac{1}{2}\frac{\partial^{2}\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg]\Bigg].$

(4.38)

Utilizing the methodology described in Section 3, the $\mathbf{r}$ th order deformation equation is described as

$\begin{eqnarray} \mathcal{A}\big[\Phi_{\mathbf{r}}(\varkappa, \bar{\mathbf{t}})-\mathcal{K}_{\mathbf{r}}\Phi_{\mathbf{r}-1}(\varkappa, \bar{\mathbf{t}})\big] = \hbar\mathcal{R}_{1, \mathbf{r}}\big[\overrightarrow{\Phi}_{\mathbf{r}-1}, \overrightarrow{\Psi}_{\mathbf{r}-1}\big], \end{eqnarray}$

(4.39)

where

$\begin{eqnarray} \mathcal{R}_{1, \mathbf{r}}\big[\overrightarrow{\Phi}_{\mathbf{r}-1}, \overrightarrow{\Psi}_{\mathbf{r}-1}\big]&& = \mathcal{A}\Bigg[\Phi_{\mathbf{r}-1}(\varkappa, \bar{\mathbf{t}})-\Big(1-\frac{\kappa_{\mathbf{r}}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\big(\frac{1}{2}-8\tanh(-2\varkappa)\big)\Big)\\&&\quad+\frac{1}{\omega^{\delta}}\bigg\{\sum\limits_{\jmath = 0}^{\mathbf{r}-1}\Phi_{\jmath}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{\mathbf{r}-1-\jmath}}{\partial \varkappa}+\frac{1}{2}\frac{\partial\Phi_{\mathbf{r}-1}}{\partial \varkappa}+\frac{\partial\Psi_{\mathbf{r}-1-\jmath}}{\partial \varkappa}\bigg\}\Bigg], \\\mathcal{R}_{2, \mathbf{r}}\big[\overrightarrow{\Phi}_{\mathbf{r}-1}, \overrightarrow{\Psi}_{\mathbf{r}-1}\big]&& = \mathcal{A}\Bigg[\Phi_{\mathbf{r}-1}(\varkappa, \bar{\mathbf{t}})-\Big(1-\frac{\kappa_{\mathbf{r}}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\big(16-16\tanh^{2}(-2\varkappa)\big)\Big)\\&&\quad+\frac{1}{\omega^{\delta}}\bigg\{\sum\limits_{\jmath = 0}^{\mathbf{r}-1}\Phi_{\jmath}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi_{\mathbf{r}-1-\jmath}}{\partial \varkappa}+\sum\limits_{\jmath = 0}^{\mathbf{r}-1}\Psi_{\jmath}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{\mathbf{r}-\jmath-1}}{\partial \varkappa}\\&&\quad-\frac{1}{2}\frac{\partial^{2}\Psi_{\mathbf{r}-1}}{\partial \varkappa^{2}}\bigg\}\Bigg].\\ \end{eqnarray}$

(4.40)

Thanks to the inverse Aboodh transform, we attain

(4.41)

In view of ICs, we have

$\begin{eqnarray} &&\Phi_{0}(\varkappa, \bar{\mathbf{t}}) = \zeta-\kappa\coth\big(\kappa(\varkappa+\theta)\big), \\&&\Psi_{0}(\varkappa, \bar{\mathbf{t}}) = -\kappa^{2}cosec h^{2}\big(\kappa(\varkappa+\theta)\big). \end{eqnarray}$

(4.42)

In order to obtain $\Phi_{0}(\varkappa, \bar{\mathbf{t}})$ and $\Psi_{0}(\varkappa, \bar{\mathbf{t}}),$ choosing $\mathbf{r} = 1$ in (4.40), then we acquire

(4.43)

For $\mathbf{r} = 1,$ then (4.40) diminish to

$\begin{eqnarray} \mathcal{R}_{1, 1}\big[\overrightarrow{\Phi}_{0}, \overrightarrow{\Psi}_{0}\big]&& = \mathcal{A}\big[\Phi_{0}(\varkappa, \bar{\mathbf{t}})\big]-\Big(1-\frac{\kappa_{1}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\Big(\zeta-\kappa\coth\big(\kappa(\varkappa+\theta)\big)\Big)\\&&\quad+\frac{1}{\omega^{\delta}}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{1}{2}\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg\}, \\ \mathcal{R}_{2, 1}\big[\overrightarrow{\Phi}_{0}, \overrightarrow{\Psi}_{0}\big]&& = \mathcal{A}\big[\Psi_{0}(\varkappa, \bar{\mathbf{t}})\big]-\Big(1-\frac{\kappa_{1}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\big(-\kappa^{2}cosec h^{2}\big(\kappa(\varkappa+\theta)\big)\big)\\&&\quad+\frac{1}{\omega^{\delta}}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}-\frac{1}{2}\frac{\partial^{2}\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg\}.\\ \end{eqnarray}$

(4.44)

Plugging (3.33) and (4.44) in (4.43), then we have

$\begin{eqnarray} \Phi_{1}(\varkappa, \bar{\mathbf{t}})&& = \hbar\mathcal{A}^{-1}\bigg[\frac{1}{\omega^{2}}\Big(\zeta-\kappa\coth\big(\kappa(\varkappa+\theta)\big)\Big)-\Big(1-\frac{0}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\Big(\zeta-\kappa\coth\big(\kappa(\varkappa+\theta)\big)\Big)\\&&\quad+\frac{1}{\omega^{\delta}}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{1}{2}\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg\}\bigg], \\ \Psi_{1}(\varkappa, \bar{\mathbf{t}})&& = \hbar\mathcal{A}^{-1}\bigg[\frac{1}{\omega^{2}}\big(-\kappa^{2}cosec h^{2}\big(\kappa(\varkappa+\theta)\big)\big)-\Big(1-\frac{0}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\big(-\kappa^{2}cosec h^{2}\big(\kappa(\varkappa+\theta)\big)\big)\\&&\quad+\frac{1}{\omega^{\delta}}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}-\frac{1}{2}\frac{\partial^{2}\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg\}\bigg].\\ \end{eqnarray}$

(4.45)

Thus, we have

$\begin{eqnarray} &&\Phi_{1}(\varkappa, \bar{\mathbf{t}}) = -\zeta\hbar\kappa^{2}cosec h^{2}\big(\kappa(\varkappa+\theta)\big)\frac{\bar{\mathbf{t}}^{\delta}}{\Gamma(\delta+1)}, \\&&\Psi_{1}(\varkappa, \bar{\mathbf{t}}) = -\zeta\hbar\kappa^{2}cosec h^{2}\big(\kappa(\varkappa+\theta)\big)\coth\big(\kappa(\varkappa+\theta)\big)\frac{\bar{\mathbf{t}}^{\delta}}{\Gamma(\delta+1)}, \end{eqnarray}$

(4.46)

Analogously, for $\mathbf{r} = 2,$ then (4.43) and (4.44) yields

$\Phi_{2}(\varkappa, \bar{\mathbf{t}}) = \mathbf{n}\Phi_{1}(\varkappa, \bar{\mathbf{t}})+\hbar\mathcal{A}^{-1}\bigg[\mathcal{A}[\Phi_{1}(\varkappa, \bar{\mathbf{t}})]\big]-\Big(1-\frac{\mathbf{n}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\Big(\zeta-\kappa\coth\big(\kappa(\varkappa+\theta)\big)\Big)\\ +\frac{1}{\omega^{\delta}}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{1}{2}\Phi_{1}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Phi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg\}\bigg], \\ \Psi_{2}(\varkappa, \bar{\mathbf{t}}) = \mathbf{n}\Psi_{1}(\varkappa, \bar{\mathbf{t}})+\hbar\mathcal{A}^{-1}\bigg[\mathcal{A}[\Psi_{1}(\varkappa, \bar{\mathbf{t}})]\big]-\Big(1-\frac{\mathbf{n}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\big(16-16\tanh(-2\varkappa)\big)\\ +\frac{1}{\omega^{\delta}}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Phi_{1}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa} +\Psi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\\ + \Psi_{1}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa} -\frac{1}{2}\frac{\partial^{2}\Psi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg\}\bigg].$

(4.47)

After simplification, the foregoing procedure reduces as mentioned

$\Phi_{2}(\varkappa, \bar{\mathbf{t}}) = -\zeta(\mathbf{n}+\hbar)\hbar \kappa^{2}cosech^{2}\big(\kappa(\varkappa+\theta)\big)\frac{\bar{\mathbf{t}}^{\delta}}{\Gamma(\delta+1)}+\zeta\hbar^{2}\kappa^{4}cosech^{2}\big(\kappa(\varkappa+\theta)\big)\\ \times\bigg\{\frac{2\zeta\kappa \Gamma(2\delta+1)\bar{\mathbf{t}}^{3\delta}}{\Gamma(3\delta+1)\Gamma^{2}(\delta+1)}-\frac{\bar{\mathbf{t}}^{2\delta}\big(3\coth^{2}\big(\big(\kappa(\varkappa+\theta)\big)-1\big)\big)}{\Gamma(2\delta+1)}\bigg\}, \\ \Psi_{2}(\varkappa, \bar{\mathbf{t}}) = -\zeta(\mathbf{n}+\hbar)\hbar \kappa^{2}cosech^{2}\big(\kappa(\varkappa+\theta)\big)\coth\big(\kappa(\varkappa+\theta)\big)\frac{\bar{\mathbf{t}}^{\delta}}{\Gamma(\delta+1)}+\frac{2\zeta\kappa^{5}\hbar^{2}cosech^{2}\big(\kappa(\varkappa+\theta)\big)}{\Gamma(\delta+1)}\\ \times\bigg\{\frac{\bar{\mathbf{t}}^{3\delta}\big(\zeta\kappa cosech^{2}\big(3coth^{2}(\big(\kappa(\varkappa+\theta)\big)-1)\big)+2\zeta\kappa cosech^{2}\coth^{2}\big(\kappa(\varkappa+\theta)\big)\big)}{\Gamma(\delta+1)\Gamma(3\delta+1)}\\ -\frac{2\bar{\mathbf{t}}^{\delta}\zeta\coth(3cosech^{2}(\big(\kappa(\varkappa+\theta)\big)-1))}{\Gamma(2\delta+1)}\bigg\}.$

(4.48)

Similarly, we evaluate the remaining term. Then, we have

(4.49)

Case II: Secondly, we employ the ABC fractional derivative operator considered with the Aboodh transform and $\bar{q}$ -HATM.

Employing the Aboodh transform on (4.33), we have

$\frac{\omega^{\delta}\mathbb{N}(\delta)}{\delta+(1-\delta)\omega^{\delta}}\mathcal{A}[\Phi(\varkappa, \bar{\mathbf{t}})]-\frac{\Phi(\varkappa, 0)}{\omega^{2}} = -\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{1}{2}\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg], \\ \frac{\omega^{\delta}\mathbb{N}(\delta)}{\delta+(1-\delta)\omega^{\delta}}\mathcal{A}[\Psi(\varkappa, \bar{\mathbf{t}})]-\frac{\Psi(\varkappa, 0)}{\omega^{2}} =\\ -\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}-\frac{1}{2}\frac{\partial^{2}\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg].$

(4.50)

Therefore, we have

$\mathcal{A}[\Phi(\varkappa, \bar{\mathbf{t}})] = \frac{1}{\omega^{2}}\Phi(\varkappa, 0)-\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{1}{2}\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg], \\ \mathcal{A}[\Psi(\varkappa, \bar{\mathbf{t}})] = \frac{1}{\omega^{2}}\Psi(\varkappa, 0)-\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\\ \mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}-\frac{1}{2}\frac{\partial^{2}\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg].$

(4.51)

With the aid of (4.51), the non-linearity can be expressed as

$\mathcal{N}_{1}\big[\varphi_{1}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}}), \varphi_{2}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})\big]\\ = \mathcal{A}\Bigg[\varphi_{1}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})-\frac{1}{\omega^{2}}\varphi_{1}(\varkappa, 0)+\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{1}{2}\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg]\Bigg], \\ \mathcal{N}_{2}\big[\varphi_{1}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}}), \varphi_{2}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})\big]\\ = \mathcal{A}\Bigg[\varphi_{2}(\varkappa, \bar{\mathbf{t}};\bar{\mathbf{q}})-\frac{1}{\omega^{2}}\varphi_{2}(\varkappa, 0)+\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\\ \mathcal{A}\bigg[\Phi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}-\frac{1}{2}\frac{\partial^{2}\Psi(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg]\Bigg].\\$

(4.52)

Utilizing the methodology described in Section 3, the $\mathbf{r}$ th order deformation equation is described as

$\begin{eqnarray} \mathcal{A}\big[\Phi_{\mathbf{r}}(\varkappa, \bar{\mathbf{t}})-\mathcal{K}_{\mathbf{r}}\Phi_{\mathbf{r}-1}(\varkappa, \bar{\mathbf{t}})\big] = \hbar\mathcal{R}_{1, \mathbf{r}}\big[\overrightarrow{\Phi}_{\mathbf{r}-1}, \overrightarrow{\Psi}_{\mathbf{r}-1}\big], \end{eqnarray}$

(4.53)

where

$\mathcal{R}_{1, \mathbf{r}}\big[\overrightarrow{\Phi}_{\mathbf{r}-1}, \overrightarrow{\Psi}_{\mathbf{r}-1}\big] = \mathcal{A}\Bigg[\Phi_{\mathbf{r}-1}(\varkappa, \bar{\mathbf{t}})-\Big(1-\frac{\kappa_{\mathbf{r}}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\big(\zeta-\kappa\coth\big(\kappa(\varkappa+\theta)\big)\big)\Big)\\ \quad+\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\bigg\{\sum\limits_{\jmath = 0}^{\mathbf{r}-1}\Phi_{\jmath}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{\mathbf{r}-1-\jmath}}{\partial \varkappa}+\frac{1}{2}\frac{\partial\Phi_{\mathbf{r}-1}}{\partial \varkappa}+\frac{\partial\Psi_{\mathbf{r}-1-\jmath}}{\partial \varkappa}\bigg\}\Bigg], \\\mathcal{R}_{2, \mathbf{r}}\big[\overrightarrow{\Phi}_{\mathbf{r}-1}, \overrightarrow{\Psi}_{\mathbf{r}-1}\big] = \mathcal{A}\Bigg[\Phi_{\mathbf{r}-1}(\varkappa, \bar{\mathbf{t}})-\Big(1-\frac{\kappa_{\mathbf{r}}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\big(-\kappa^{2}cosec h^{2}\big(\kappa(\varkappa+\theta)\big)\big)\Big)\\ \quad+\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\bigg\{\sum\limits_{\jmath = 0}^{\mathbf{r}-1}\Phi_{\jmath}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi_{\mathbf{r}-1-\jmath}}{\partial \varkappa}+\sum\limits_{\jmath = 0}^{\mathbf{r}-1}\Psi_{\jmath}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{\mathbf{r}-\jmath-1}}{\partial \varkappa}-\frac{1}{2}\frac{\partial^{2}\Psi_{\mathbf{r}-1}}{\partial \varkappa^{2}}\bigg\}\Bigg].$

(4.54)

Thanks to the inverse Aboodh transform, we attain

(4.55)

In view of ICs, we have

(4.56)

In order to obtain $\Phi_{0}(\varkappa, \bar{\mathbf{t}})$ and $\Psi_{0}(\varkappa, \bar{\mathbf{t}}),$ choosing $\mathbf{r} = 1$ in (4.55), then we acquire

(4.57)

For $\mathbf{r} = 1,$ then (4.54) diminish to

$\mathcal{R}_{1, 1}\big[\overrightarrow{\Phi}_{0}, \overrightarrow{\Psi}_{0}\big] = \mathcal{A}\big[\Phi_{0}(\varkappa, \bar{\mathbf{t}})\big]-\Big(1-\frac{\kappa_{1}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\Big(\zeta-\kappa\coth\big(\kappa(\varkappa+\theta)\big)\Big)\\ +\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{1}{2}\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg\}, \\ \mathcal{R}_{2, 1}\big[\overrightarrow{\Phi}_{0}, \overrightarrow{\Psi}_{0}\big] = \mathcal{A}\big[\Psi_{0}(\varkappa, \bar{\mathbf{t}})\big]-\Big(1-\frac{\kappa_{1}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\big(-\kappa^{2}cosec h^{2}\big(\kappa(\varkappa+\theta)\big)\big)\\ +\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}-\frac{1}{2}\frac{\partial^{2}\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg\}.$

(4.58)

Plugging (3.33) and (4.58) in (4.57), then we have

$\begin{eqnarray} \Phi_{1}(\varkappa, \bar{\mathbf{t}})&& = \hbar\mathcal{A}^{-1}\bigg[\frac{1}{\omega^{2}}\Big(\zeta-\kappa\coth\big(\kappa(\varkappa+\theta)\big)\Big)-\Big(1-\frac{0}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\Big(\zeta-\kappa\coth\big(\kappa(\varkappa+\theta)\big)\Big)\\&&\quad+\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{1}{2}\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg\}\bigg], \\ \Psi_{1}(\varkappa, \bar{\mathbf{t}})&& = \hbar\mathcal{A}^{-1}\bigg[\frac{1}{\omega^{2}}\big(-\kappa^{2}cosec h^{2}\big(\kappa(\varkappa+\theta)\big)\big)-\Big(1-\frac{0}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\big(-\kappa^{2}cosec h^{2}\big(\kappa(\varkappa+\theta)\big)\big)\\&&\quad+\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Psi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}-\frac{1}{2}\frac{\partial^{2}\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg\}\bigg]. \end{eqnarray}$

(4.59)

Thus, we have

$\begin{eqnarray} \Phi_{1}(\varkappa, \bar{\mathbf{t}})&& = -\frac{\zeta\hbar\kappa^{2}}{\mathbb{N}(\delta)}cosec h^{2}\big(\kappa(\varkappa+\theta)\big)\bigg\{\frac{\delta \bar{\mathbf{t}}^{\delta}}{\Gamma(\delta+1)}+(1-\delta)\bigg\}, \\\Psi_{1}(\varkappa, \bar{\mathbf{t}})&& = -\frac{\zeta\hbar\kappa^{2}}{\mathbb{N}(\delta)}cosec h^{2}\big(\kappa(\varkappa+\theta)\big)\coth\big(\kappa(\varkappa+\theta)\big)\bigg\{\frac{\delta \bar{\mathbf{t}}^{\delta}}{\Gamma(\delta+1)}+(1-\delta)\bigg\}, \end{eqnarray}$

(4.60)

Analogously, for $\mathbf{r} = 2,$ then (4.57) and (4.58) yields

$\Phi_{2}(\varkappa, \bar{\mathbf{t}}) = \mathbf{n}\Phi_{1}(\varkappa, \bar{\mathbf{t}})+\hbar\mathcal{A}^{-1}\bigg[\mathcal{A}[\Phi_{1}(\varkappa, \bar{\mathbf{t}})]\big]-\Big(1-\frac{\mathbf{n}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\Big(\zeta-\kappa\coth\big(\kappa(\varkappa+\theta)\big)\Big)\\ +\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Phi_{1}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{1}{2}\frac{\partial\Phi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\frac{\partial\Psi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\bigg\}\bigg], \\ \Psi_{2}(\varkappa, \bar{\mathbf{t}}) = \mathbf{n}\Psi_{1}(\varkappa, \bar{\mathbf{t}})+\hbar\mathcal{A}^{-1}\bigg[\mathcal{A}[\Psi_{1}(\varkappa, \bar{\mathbf{t}})]\big]-\Big(1-\frac{\mathbf{n}}{\mathbf{n}}\Big)\frac{1}{\omega^{2}}\big(16-16\tanh(-2\varkappa)\big)\\ +\frac{\delta+(1-\delta)\omega^{\delta}}{\omega^{\delta}\mathbb{N}(\delta)}\mathcal{A}\bigg\{\Phi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}+\Phi_{1}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Psi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa} +\Psi_{0}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa}\\ + \Psi_{1}(\varkappa, \bar{\mathbf{t}})\frac{\partial\Phi_{0}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa} -\frac{1}{2}\frac{\partial^{2}\Psi_{1}(\varkappa, \bar{\mathbf{t}})}{\partial \varkappa^{2}}\bigg\}\bigg],$

(4.61)

After simplification, the foregoing procedure reduces as mentioned

$\Phi_{2}(\varkappa, \bar{\mathbf{t}}) = -(\mathbf{n}+\hbar)\frac{\zeta\kappa^{2}}{\mathbb{N}(\delta)}cosec h^{2}\big(\kappa(\varkappa+\theta)\big)\bigg\{\frac{\delta \bar{\mathbf{t}}^{\delta}}{\Gamma(\delta+1)}+(1-\delta)\bigg\}\\ -\frac{\zeta\kappa^{4}\hbar^{2}}{\mathbb{N}^{2}(\delta)}cosech^{2}\big(\kappa(\varkappa+\theta)\big)\big(3coth^{2}(\big(\kappa(\varkappa+\theta)\big)-1)\big)\\ \times\bigg\{\frac{\delta^{2}\bar{\mathbf{t}}^{2\delta}}{\Gamma(2\delta+1)}+2\delta(1-\delta)\frac{\delta \bar{\mathbf{t}}^{\delta}}{\Gamma(\delta+1)}+(1-\delta)^{2}\bigg\}+\frac{2\hbar^{2}\zeta^{2}\kappa^{5}}{\mathbb{N}^{3}(\delta)}cosech^{2}\big(\kappa(\varkappa+\theta)\big)\\ \times\bigg\{\frac{\delta^{3}\bar{\mathbf{t}}^{3\delta}}{\Gamma(3\delta+1)}+3\delta^{2}(1-\delta)\frac{ \bar{\mathbf{t}}^{2\delta}}{\Gamma(2\delta+1)}+3\delta(1-\delta)^{2}\frac{ \bar{\mathbf{t}}^{\delta}}{\Gamma(\delta+1)}+(1-\delta)^{3}\bigg\}, \\ \Psi_{2}(\varkappa, \bar{\mathbf{t}}) = -(\mathbf{n}+\hbar)\frac{\zeta\hbar\kappa^{2}}{\mathbb{N}(\delta)}cosec h^{2}\big(\kappa(\varkappa+\theta)\big)\coth\big(\kappa(\varkappa+\theta)\big)\bigg\{\frac{\delta \bar{\mathbf{t}}^{\delta}}{\Gamma(\delta+1)}+(1-\delta)\bigg\}\\ -\frac{2\zeta\hbar^{2}}{\mathbb{N}^{2}(\delta)}coth^{2}\big(3cosech^{2}\big(\kappa(\varkappa+\theta)-1\big)\bigg\{\frac{\delta^{2}\bar{\mathbf{t}}^{2\delta}}{\Gamma(2\delta+1)}+2\delta(1-\delta)\frac{\delta \bar{\mathbf{t}}^{\delta}}{\Gamma(\delta+1)}+(1-\delta)^{2}\bigg\}\\ +\frac{2\hbar^{2}\zeta\kappa^{5}}{\mathbb{N}^{3}(\delta)}\Big(\zeta\kappa cosech^{2}\big(\kappa(\varkappa+\theta)\big)\big(3coth^{2}\big(\kappa(\varkappa+\theta)-1\big)+2\zeta\kappa cosech^{2}\big(\kappa(\varkappa+\theta)\big)\big(coth^{2}\big(\kappa(\varkappa+\theta)\big)\big)\Big)\\ \times\bigg\{\frac{\delta^{3}\bar{\mathbf{t}}^{3\delta}}{\Gamma(3\delta+1)}+3\delta^{2}(1-\delta)\frac{ \bar{\mathbf{t}}^{2\delta}}{\Gamma(2\delta+1)}+3\delta(1-\delta)^{2}\frac{ \bar{\mathbf{t}}^{\delta}}{\Gamma(\delta+1)}+(1-\delta)^{3}\bigg\}.$

(4.62)

Similarly, we evaluate the remaining term. Then, we have

(4.63)

For $\delta = 1,$ then the integer-order solution of (4.33) is

$\begin{eqnarray} \Phi(\varkappa, \bar{\mathbf{t}})&& = \zeta-\kappa\coth\big(\kappa(\varkappa+\theta-\zeta \bar{\mathbf{t}})\big), \\ \Psi(\varkappa, \bar{\mathbf{t}})&& = -\kappa^{2}cosech^{2}\big(\kappa(\varkappa+\theta-\zeta \bar{\mathbf{t}})\big). \end{eqnarray}$

(4.64)

5. Numerical results and their physical interpretation

The fractional WBKE is solved using the $\bar{q}$ -HATM merged with the Aboodh transform. For the most part, MATLAB 21 has been implemented. The mathematical models eqrefkb1 and (4.33) were carried out in an attempt to determine that the prospective approach would result in increased reliability via the Caputo and ABC fractional derivative operators. The projected method, as shown by the effort it will lead, provides amazing precision in respect to the approach described in these articles ^[59,60], which is referenced for both of the instances in Tables 1 and 2. For Example 4.1, and examine the correlation of generated results to precise responses and approximate solutions for both $\Phi(\varkappa, \bar{\mathbf{t}})$ and $\Psi(\varkappa, \bar{\mathbf{t}}),$ respectively. The behaviour of coupled system of (4.1) is presented in as a three-dimensional and two-dimensional plots. illustrates the exact and approximate two-dimensional behaviour of $\Phi(\varkappa, \bar{\mathbf{t}})$ and $\Psi(\varkappa, \bar{\mathbf{t}}),$ respectively. This behaviour shows how the exact and approximate solutions are in close agreement with each other. The behavior of derived findings for the Example 4.1 involving various fractional and integer order is shown in . Also, the $\bar{q}$ -HATM results for different $\hbar$ are shown in Figure 5, which help us customize and manage the convergence area. – show the significance of the asymptotic component in the $q$ -HATM formulation.

Table 1. Comparison analysis in respects of absolute error between ADM ^[59], VIM ^[60] and

$\bar{q}$ -HATM for the approximate findings of

$\Phi(\varkappa, \bar{\mathbf{t}})$ at

$\mathbf{n} = 1, \; \hbar = -1$ and

$\delta = 1$ for Example 4.2.

$(\varkappa, \bar{\mathbf{t}})$	$\big\vert\Phi_{E}-\Phi_{ADM}\big\vert$	$\big\vert\Phi_{E}-\Phi_{VIM}\big\vert$	$\big\vert\Phi_{E}-\Phi_{\bar{q}-HATM(CFD)}\big\vert$	$\big\vert\Phi_{E}-\Phi_{\bar{q}-HATM(ABC)}\big\vert$
(0.1, 0.1)	8.16297 $\times10^{-7}$	6.35269 $\times10^{-5}$	5.89129 $\times10^{-17}$	5.55112 $\times10^{-17}$
(0.1, 0.3)	7.64247 $\times10^{-7}$	1.90854 $\times10^{-4}$	5.89129 $\times10^{-17}$	5.55112 $\times10^{-17}$
(0.1, 0.5)	7.16083 $\times10^{-7}$	3.18549 $\times10^{-4}$	5.89129 $\times10^{-17}$	5.55112 $\times10^{-17}$
(0.2, 0.1)	3.26243 $\times10^{-6}$	6.18430 $\times10^{-5}$	5.89129 $\times10^{-17}$	5.55112 $\times10^{-17}$
(0.2, 0.3)	3.05458 $\times10^{-6}$	1.85945 $\times10^{-4}$	5.89129 $\times10^{-17}$	5.55112 $\times10^{-17}$
(0.2, 0.5)	2.86226 $\times10^{-6}$	3.10352 $\times10^{-4}$	7.798625 $\times10^{-16}$	7.77156 $\times10^{-16}$
(0.3, 0.1)	3.26243 $\times10^{-6}$	6.18430 $\times10^{-5}$	5.89129 $\times10^{-17}$	5.55112 $\times10^{-17}$
(0.3, 0.3)	3.05458 $\times10^{-6}$	1.85945 $\times10^{-4}$	5.89129 $\times10^{-17}$	5.55112 $\times10^{-17}$
(0.3, 0.5)	2.86226 $\times10^{-6}$	3.10352 $\times10^{-4}$	7.798625 $\times10^{-16}$	7.77156 $\times10^{-16}$
(0.4, 0.1)	1.30286 $\times10^{-5}$	5.87746 $\times10^{-5}$	5.89129 $\times10^{-17}$	5.55112 $\times10^{-17}$
(0.4, 0.3)	1.22000 $\times10^{-5}$	1.76574 $\times10^{-4}$	5.89129 $\times10^{-17}$	5.55112 $\times10^{-17}$
(0.4, 0.5)	1.14333 $\times10^{-5}$	2.44707 $\times10^{-4}$	5.89129 $\times10^{-17}$	5.55112 $\times10^{-17}$
(0.5, 0.1)	2.03415 $\times10^{-5}$	5.72867 $\times10^{-5}$	0	0
(0.5, 0.3)	1.90489 $\times10^{-5}$	1.72102 $\times10^{-4}$	1.73301 $\times10^{-16}$	1.11022 $\times10^{-16}$
(0.5, 0.5)	1.78528 $\times10^{-5}$	2.87241 $\times10^{-4}$	6.74434 $\times10^{-17}$	6.10623 $\times10^{-16}$

| Show Table

DownLoad: CSV

Table 2. Comparison analysis in respects of absolute error between ADM ^[59], VIM ^[60] and

$\bar{q}$ -HATM for the approximate findings of

$\Psi(\varkappa, \bar{\mathbf{t}})$ at

$\mathbf{n} = 1, \; \hbar = -1$ and

$\delta = 1$ for Example 4.2.

$(\varkappa, \bar{\mathbf{t}})$	$\big\vert\Phi_{E}-\Phi_{ADM}\big\vert$	$\big\vert\Phi_{E}-\Phi_{VIM}\big\vert$	$\big\vert\Phi_{E}-\Phi_{\bar{q}-HATM(CFD)}\big\vert$	$\big\vert\Phi_{E}-\Phi_{\bar{q}-HATM(ABC)}\big\vert$
(0.1, 0.1)	4.81902 $\times10^{-4}$	1.23033 $\times10^{-4}$	1.453423 $\times10^{-18}$	1.73472 $\times10^{-18}$
(0.1, 0.3)	4.50818 $\times10^{-4}$	1.76000 $\times10^{-4}$	2.90009 $\times10^{-17}$	2.60209 $\times10^{-17}$
(0.1, 0.5)	4.22221 $\times10^{-4}$	2.69597 $\times10^{-4}$	1.98220 $\times10^{-16}$	1.80411 $\times10^{-16}$
(0.2, 0.1)	9.76644 $\times10^{-4}$	2.69597 $\times10^{-4}$	3.94560 $\times10^{-9}$	3.82100 $\times10^{-9}$
(0.2, 0.3)	9.13502 $\times10^{-4}$	2.69597 $\times10^{-4}$	2.34188 $\times10^{-7}$	2.34188 $\times10^{-7}$
(0.2, 0.5)	8.55426 $\times10^{-4}$	2.69597 $\times10^{-4}$	1.78935 $\times10^{-16}$	1.73472 $\times10^{-16}$
(0.3, 0.1)	1.48482 $\times10^{-3}$	2.69597 $\times10^{-4}$	3.89567 $\times10^{-18}$	3.46945 $\times10^{-18}$
(0.3, 0.3)	1.38858 $\times10^{-3}$	2.69597 $\times10^{-4}$	5.89129 $\times10^{-17}$	5.55112 $\times10^{-17}$
(0.3, 0.5)	1.30009 $\times10^{-3}$	2.69597 $\times10^{-4}$	1.69879 $\times10^{-16}$	1.61329 $\times10^{-16}$
(0.4, 0.1)	2.00705 $\times10^{-3}$	2.69597 $\times10^{-4}$	2.91458 $\times10^{-18}$	2.60209 $\times10^{-18}$
(0.4, 0.3)	1.87661 $\times10^{-3}$	2.69597 $\times10^{-4}$	1.98455 $\times10^{-17}$	1.73472 $\times10^{-17}$
(0.4, 0.5)	1.75670 $\times10^{-3}$	2.69597 $\times10^{-4}$	1.59884 $\times10^{-16}$	1.52656 $\times10^{-16}$
(0.5, 0.1)	2.54396 $\times10^{-3}$	2.69597 $\times10^{-4}$	8.98734 $\times10^{-19}$	8.67362 $\times10^{-19}$
(0.5, 0.3)	2.37815 $\times10^{-3}$	2.69597 $\times10^{-4}$	2.98823 $\times10^{-17}$	2.08167 $\times10^{-17}$
(0.5, 0.5)	2.22578 $\times10^{-3}$	2.69597 $\times10^{-4}$	1.48878 $\times10^{-16}$	1.43982 $\times10^{-16}$

| Show Table

DownLoad: CSV

Figure 1. Illustration in three dimensions of the exact and approximate solutions for

$\Phi(\varkappa, \bar{\mathbf{t}})$ of Example 4.1 when

$\delta = 1$ .

DownLoad: Full-Size Img PowerPoint

Figure 2. Illustration in three dimensions of the exact and approximate solutions for

$\Psi(\varkappa, \bar{\mathbf{t}})$ of Example 4.1 when

$\delta = 1$ .

DownLoad: Full-Size Img PowerPoint

Figure 3. (a) Three-dimensional illustration (b) Two-dimensional illustrations of the approximate solutions for

$\Phi(\varkappa, \bar{\mathbf{t}})$ and

$\Psi(\varkappa, \bar{\mathbf{t}})$ , respectively, of Example 4.1 when

$\delta = 0.8$ .

DownLoad: Full-Size Img PowerPoint

Figure 4. (a) Two-dimensional illustration of the exact and approximate solutions of

$\Phi(\varkappa, \bar{\mathbf{t}})$ of Example 4.1 when

$\delta = 1, \; \; \hbar = -1, \; \mathbf{n} = 1$ and

$\bar{\mathbf{t}} = 0.7.$ (b) (a) Two-dimensional illustration of the exact and approximate solutions of

$\Psi(\varkappa, \bar{\mathbf{t}})$ of Example 4.1 when

$\delta = 1, \; \; \hbar = -1, \; \mathbf{n} = 1$ and

$\bar{\mathbf{t}} = 0.7$ .

DownLoad: Full-Size Img PowerPoint

Figure 5. (a) Two-dimensional illustration of the approximate solutions of

$\Phi(\varkappa, \bar{\mathbf{t}})$ of Example 4.1 having multiple fractional order when

$\hbar = -1, \; \mathbf{n} = 1$ and

$\bar{\mathbf{t}} = 0.7.$ (b) Two-dimensional illustration of the approximate solutions of

$\Psi(\varkappa, \bar{\mathbf{t}})$ of Example 4.1 having multiple fractional order when

$\hbar = -1, \; \mathbf{n} = 1$ and

$\bar{\mathbf{t}} = 0.7$ .

DownLoad: Full-Size Img PowerPoint

Furthermore, Figure 6 depicts the characteristics of produced results in relation to precise values for Example 4.2, with Figure 7 demonstrating the preciseness of the achieved result and approximate solution. Figure 8 shows the performance of the resulting approach for a couple of WBKE model 4.33, in two and three dimensional views. The characteristics of acquired treatment for specific may be seen in Figure 9, which help us govern the converged zone. Subsequently, the multiple fractional orders are depicted in Figure 10, and the integer-order depicts the converging domain for the WBKE. Example 4.1 and Example 4.2 address the correlated texture of the WBKEs, which are presented in Figures 6–10, respectively, to assist in considering the characteristics of coupled equations.

Figure 6. Illustration in three dimensions of the exact and approximate solutions for

$\Phi(\varkappa, \bar{\mathbf{t}})$ of Example 4.2 when

$\delta = 1$ .

DownLoad: Full-Size Img PowerPoint

Figure 7. Illustration in three dimensions of the exact and approximate solutions for

$\Psi(\varkappa, \bar{\mathbf{t}})$ of Example 4.2 when

$\delta = 1$ .

DownLoad: Full-Size Img PowerPoint

Figure 8. Illustration in three dimensions of the exact and approximate solutions for

$\Phi(\varkappa, \bar{\mathbf{t}})$ of Example 4.2 when

$\delta = 1$ .

DownLoad: Full-Size Img PowerPoint

Figure 9. (a) Two-dimensional illustration of the approximate solutions of

$\Phi(\varkappa, \bar{\mathbf{t}})$ of Example 4.2 having multiple fractional order when

$\hbar = -1, \; \mathbf{n} = 1$ and

$\bar{\mathbf{t}} = 0.7.$ (b) Two-dimensional illustration of the approximate solutions of

$\Psi(\varkappa, \bar{\mathbf{t}})$ of Example 4.2 having multiple fractional order when

$\hbar = -1, \; \mathbf{n} = 1$ and

$\bar{\mathbf{t}} = 0.7$ .

DownLoad: Full-Size Img PowerPoint

Figure 10. (a) Two-dimensional illustration of the approximate solutions of

$\Phi(\varkappa, \bar{\mathbf{t}})$ of Example 4.2 having multiple fractional order when

$\hbar = -1, \; \mathbf{n} = 1$ and

$\bar{\mathbf{t}} = 0.7.$ (b) Two-dimensional illustration of the approximate solutions of

$\Psi(\varkappa, \bar{\mathbf{t}})$ of Example 4.2 having multiple fractional order when

$\hbar = -1, \; \mathbf{n} = 1$ and

$\bar{\mathbf{t}} = 0.7$ .

DownLoad: Full-Size Img PowerPoint

6. Conclusions

In this investigation, we employed $\bar{\mathbf{q}}$ -HATM outcomes to construct the solution for a prospective dynamical model, WBKE. More interestingly, we investigated the Mittag-Leffler function, a revolutionary fractional operator that is generated with the help of a singular/non-singular fractional operator. These formulations aid experts in emulating the performance of a variety of real-world systems. The present investigation illuminates the employed nonlinear behaviors in a way that is clearly appropriate for the time instant and the time history, which can be effectively demonstrated using the fractional calculus idea. The graphs depicting the simulated interactions in this work can help practitioners determine several intriguing and important consequences of examining the paired process. They demonstrate the influence of the fractional operator by using an inexpensive approach to solve real-world situations. The current study can aid in the exploration of increasingly complicated and dynamic systems, as well as the acquisition of additional repercussions of particular instances of WBKE. The graphic representations provided are quite helpful in comprehending the networks' intricacy. The computational domain is used to verify the effectiveness and to demonstrate how the result moves from a mathematical formulation to an approximate finding as the series of solution components accumulates. Also, the proposed technique generates a new solution by avoiding any convolutions, transformations, or perturbations. Furthermore, we can state that the present investigation will assist scholars in demonstrating the characteristics of various frameworks.

Acknowledgements

This research was supported by Taif University Research Supporting Project Number (TURSP2020/96), Taif University, Taif, Saudi Arabia.

Conflict of interest

The authors declare that they have no competing interests.

References

[1]	H. I. Abdel-Gawad, M. Tantawy, D. Baleanu, Fractional KdV and Boussenisq-Burger's equations, reduction to PDE and stability approaches, Math. Method. Appl. Sci., 43 (2020), 4125–4135. http://doi.org/10.1002/mma.6178 doi: 10.1002/mma.6178
[2]	H. I. Abdel-Gawad, M. Tantawy, Traveling wave solutions of DNA-Torsional model of fractional order, Appl. Math. Inf. Sci. Lett., 6 (2018), 85–89. http://doi.org/10.18576/amisl/060205 doi: 10.18576/amisl/060205
[3]	N. I. Okposo, P. Veeresha, E. N. Okposo, Solutions for time-fractional coupled nonlinear Schrödinger equations arising in optical solitons, Chinese J. Phy., 77 (2022), 965–984. http://doi.org/10.1016/j.cjph.2021.10.014 doi: 10.1016/j.cjph.2021.10.014
[4]	P. Veeresha, M. Yavuz, C. Baishya, A computational approach for shallow water forced Korteweg-De Vries equation on critical flow over a hole with three fractional operators, IJOCTA, 11 (2021), 52–67. http://doi.org/10.11121/ijocta.2021.1177 doi: 10.11121/ijocta.2021.1177
[5]	P. Veeresha, E. Ilhan, H. M. Baskonus, Fractional approach for analysis of the model describing wind-influenced projectile motion, Phy. Scr., 96 (2021), 075209. http://doi.org/10.1088/1402-4896/abf868 doi: 10.1088/1402-4896/abf868
[6]	P. Veeresha, D. Baleanu, A unifying computational framework for fractional Gross-Pitaevskii equations, Phy. Scr., 96 (2021), 125010. http://doi.org/10.1088/1402-4896/ac28c9 doi: 10.1088/1402-4896/ac28c9
[7]	P. Veeresha, D. G. Prakasha, H. M. Baskonus, New numerical surfaces to the mathematical model of cancer chemotherapy effect in Caputo fractional derivatives, Chaos, 29 (2019), 013119. http://doi.org/10.1063/1.5074099 doi: 10.1063/1.5074099
[8]	C. Baishya, P. Veeresha, Laguerre polynomial-based operational matrix of integration for solving fractional differential equations with non-singular kernel, Proc. R. Soc. A, 477 (2021), 20210438. http://doi.org/10.1098/rspa.2021.0438 doi: 10.1098/rspa.2021.0438
[9]	S. N. Hajiseyedazizi, M. E. Samei, J. Alzabut, Y.-M. Chu, On multi-step methods for singular fractional $q$ -integro-differential equations, Open Math., 19 (2021), 1378–1405. http://doi.org/10.1515/math-2021-0093 doi: 10.1515/math-2021-0093
[10]	F. Jin, Z.-S. Qian, Y.-M. Chu, M. ur Rahman, On nonlinear evolution model for drinking behavior under Caputo-Fabrizio derivative, J. Appl. Anal. Comput., 12 (2022), 790–806. http://doi.org/10.11948/20210357 doi: 10.11948/20210357
[11]	S. Rashid, E. I. Abouelmagd, A. Khalid, F. B. Farooq, Y.-M. Chu, Some recent developments on dynamical $\hbar$ -discrete fractional type inequalities in the frame of nonsingular and nonlocal kernels, Fractals, 30 (2022), 2240110. http://doi.org/10.1142/S0218348X22401107 doi: 10.1142/S0218348X22401107
[12]	F.-Z. Wang, M. N. Khan, I. Ahmad, H. Ahmad, H. Abu-Zinadah, Y.-M. Chu, Numerical solution of traveling waves in chemical kinetics: time-fractional Fishers equations, Fractals, 30 (2022), 2240051. http://doi.org/10.1142/S0218348X22400515 doi: 10.1142/S0218348X22400515
[13]	S. Rashid, E. I. Abouelmagd, S. Sultana, Y.-M. Chu, New developments in weighted $n$ -fold type inequalities via discrete generalized h-proportional fractional operators, Fractals, 30 (2022), 2240056. http://doi.org/10.1142/S0218348X22400564 doi: 10.1142/S0218348X22400564
[14]	Y.-M. Chu, S. Bashir, M. Ramzan, M. Y. Malik, Model-based comparative study of magnetohydrodynamics unsteady hybrid nanofluid flow between two infinite parallel plates with particle shape effects, Math. Method. Appl. Sci., in press. http://doi.org/10.1002/mma.8234
[15]	S. A. Iqbal, M. G. Hafez, Y.-M. Chu, C. Park, Dynamical Analysis of nonautonomous RLC circuit with the absence and presence of Atangana-Baleanu fractional derivative, J. Appl. Anal. Comput., 12 (2022), 770–789. http://doi.org/10.11948/20210324 doi: 10.11948/20210324
[16]	C. Baishya1, S. J. Achar, P. Veeresha, D. G. Prakasha, Dynamics of a fractional epidemiological model with disease infection in both the populations, Chaos, 31 (2021), 043130. http://doi.org/10.1063/5.0028905 doi: 10.1063/5.0028905
[17]	E. Ilhan, P. Veeresha, H. M. Baskonus, Fractional approach for a mathematical model of atmospheric dynamics of CO2 gas with an efficient method, Chaos Soliton. Fract., 152 (2021), 111347. http://doi.org/10.1016/j.chaos.2021.111347 doi: 10.1016/j.chaos.2021.111347
[18]	P. Veeresha, H. M. Baskonus, W. Gao, Strong interacting internal waves in rotating ocean: Novel fractional approach, Axioms, 10 (2021), 123. http://doi.org/10.3390/axioms10020123 doi: 10.3390/axioms10020123
[19]	P. Veeresha, D. G. Prakasha, D. Kumar, An efficient technique for nonlinear time-fractional Klein–Fock–Gordon equation, Appl. Math. Comput., 364 (2020), 124637. http://doi.org/10.1016/j.amc.2019.124637 doi: 10.1016/j.amc.2019.124637
[20]	P. Veeresha, D. G. Prakasha, M. A. Qurashi, D. Baleanu, A reliable technique for fractional modified Boussinesq and approximate long wave equations, Adv. Differ. Equ., 2019 (2019), 253. http://doi.org/10.1186/s13662-019-2185-2 doi: 10.1186/s13662-019-2185-2
[21]	T.-H. Zhao, O. Castillo, H. Jahanshahi, A. Yusuf, M. O. Alassafi, F. E. Alsaadi, A fuzzy-based strategy to suppress the novel coronavirus (2019-NCOV) massive outbreak, Appl. Comput. Math., 20 (2021), 160–176.
[22]	M. Nazeer, F. Hussain, M. Ijaz Khan, Asad-ur-Rehman, E. R. El-Zahar, Y.-M. Chu, Theoretical study of MHD electro-osmotically flow of third-grade fluid in micro channel, Appl. Math. Comput., 420 (2022), 126868. http://doi.org/10.1016/j.amc.2021.126868 doi: 10.1016/j.amc.2021.126868
[23]	Y.-M. Chu, B. M. Shankaralingappa, B. J. Gireesha, F. Alzahrani, M. Ijaz Khan, S. U. Khan, Combined impact of Cattaneo-Christov double diffusion and radiative heat flux on bio-convective flow of Maxwell liquid configured by a stretched nano- material surface, Appl. Math. Comput., 419 (2022), 126883. http://doi.org/10.1016/j.amc.2021.126883 doi: 10.1016/j.amc.2021.126883
[24]	T.-H. Zhao, M. Ijaz Khan, Y.-M. Chu, Artificial neural networking (ANN) analysis for heat and entropy generation in flow of non-Newtonian fluid between two rotating disks, Math. Method. Appl. Sci., in press. http://doi.org/10.1002/mma.7310
[25]	K. Karthikeyan, P. Karthikeyan, H. M. Baskonus, K. Venkatachalam, Y.-M. Chu, Almost sectorial operators on $\Psi$ -Hilfer derivative fractional impulsive integro-differential equations, Math. Method. Appl. Sci., in press. https://doi.org/10.1002/mma.7954
[26]	Y.-M. Chu, U. Nazir, M. Sohail, M. M. Selim, J.-R. Lee, Enhancement in thermal energy and solute particles using hybrid nanoparticles by engaging activation energy and chemical reaction over a parabolic surface via finite element approach, Fractal Fract., 5 (2021), 119. http://doi.org/10.3390/fractalfract5030119 doi: 10.3390/fractalfract5030119
[27]	S. Rashid, S. Sultana, Y. Karaca, A. Khalid, Y.-M. Chu, Some further extensions considering discrete proportional fractional operators, Fractals, 30 (2022), 2240026. http://doi.org/10.1142/S0218348X22400266 doi: 10.1142/S0218348X22400266
[28]	B. S. T. Alkahtani, A. Atangana, Analysis of non-homogenous heat model with new trend of derivative with fractional order, Chaos Soliton. Fract., 89 (2016), 566–571. https://doi.org/10.1016/j.chaos.2016.03.027 doi: 10.1016/j.chaos.2016.03.027
[29]	A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769.
[30]	T.-H. Zhao, M.-K. Wang, G.-J. Hai, Y.-M. Chu, Landen inequalities for Gaussian hypergeometric function, RACSAM, 116, (2022), 53. https://doi.org/10.1007/s13398-021-01197-y
[31]	T.-H. Zhao, M.-K. Wang, Y.-M. Chu, On the bounds of the perimeter of an ellipse, Acta Math. Sci., 42 (2022), 491–501. https://doi.org/10.1007/s10473-022-0204-y doi: 10.1007/s10473-022-0204-y
[32]	T.-H. Zhao, Z.-Y. He, Y.-M. Chu, Sharp bounds for the weighted Hölder mean of the zero-balanced generalized complete elliptic integrals, Comput. Methods Funct. Theory, 21 (2021), 413–426. https://doi.org/10.1007/s40315-020-00352-7 doi: 10.1007/s40315-020-00352-7
[33]	L. Wang, X. Chen, Approximate analytical solutions of time fractional Whitham–Broer–Kaup equations by a residual power series method, Entropy, 17 (2015), 6519–6533. http://doi.org/10.3390/e17096519 doi: 10.3390/e17096519
[34]	S. Rashid, K. T. Kubra, H. Jafari, S. U. Lehre, A semi-analytical approach for fractional order Boussinesq equation in a gradient unconfined aquifers, Math. Method. Appl. Sci., 45 (2022), 1033–1062. http://doi.org/10.1002/mma.7833 doi: 10.1002/mma.7833
[35]	P. Veeresha, D. G. Prakasha, H. M. Baskonus, Novel simulations to the time-fractional Fisher's equation, Math. Sci., 13 (2019), 33–42. http://doi.org/10.1007/s40096-019-0276-6 doi: 10.1007/s40096-019-0276-6
[36]	M. I. El-Bahi, K. Hilal, Lie symmetry analysis, exact solutions, and conservation laws for the generalized time-fractional KdV-Like equation, J. Funct. Space., 2021 (2021), 6628130. http://doi.org/10.1155/2021/6628130 doi: 10.1155/2021/6628130
[37]	S. C. Shiralashetti, S. Kumbinarasaiah, Laguerre wavelets collocation method for the numerical solution of the Benjamina–Bona-Mohany equations, J. Taibah Univ. Sci., 13 (2019), 9–15. http://doi.org/10.1080/16583655.2018.1515324 doi: 10.1080/16583655.2018.1515324
[38]	A. Kadem, D. Baleanu, On fractional coupled Whitham–Broer–Kaup equations, Rom. J. Phys., 56 (2011), 629–635.
[39]	G. B. Whitham, Variational methods and applications to water waves, Proc. R. Soc. Lond. A, 299 (1967), 6–25. http://doi.org/10.1098/rspa.1967.0119 doi: 10.1098/rspa.1967.0119
[40]	L. J. F. Broer, Approximate equations for long water waves, Appl. Sci. Res., 31 (1975), 377–395. http://doi.org/10.1007/BF00418048 doi: 10.1007/BF00418048
[41]	D. J. Kaup, A higher-order water-wave equation and the method for solving it, Progress of Theoretical Physics, 54 (1975), 396–408. http://doi.org/10.1143/PTP.54.396 doi: 10.1143/PTP.54.396
[42]	M. Al-Qurashi, S. Rashid, F. Jarad, M. Tahir, A. M. Alsharif, New computations for the two-mode version of the fractional Zakharov-Kuznetsov model in plasma fluid by means of the Shehu decomposition method, AIMS Mathematics, 7 (2022), 2044–2060. http://doi.org/10.3934/math.2022117 doi: 10.3934/math.2022117
[43]	S. Rashid, F. Jarad, T. M. Jawa, A study of behaviour for fractional order diabetes model via the nonsingular kernel, AIMS Mathematics, 7 (2022), 5072–5092. http://doi.org/10.3934/math.2022282 doi: 10.3934/math.2022282
[44]	S. Rashid, S. Sultana, R. Ashraf, M. K. A. Kaabar, On comparative analysis for the Black-Scholes model in the generalized fractional derivatives sense via Jafari transform, J. Funct. Space., 2021 (2021), 7767848. http://doi.org/10.1155/2021/7767848 doi: 10.1155/2021/7767848
[45]	S. T. Mohyud-Din, A. Ydrm, G. Demirli, Traveling wave solutions of Whitham–Broer–Kaup equations by homotopy perturbation method, J. King Saud Univ. Sci., 22 (2010), 173–176. http://doi.org/10.1016/j.jksus.2010.04.008 doi: 10.1016/j.jksus.2010.04.008
[46]	S. Saha Ray, A novel method for travelling wave solutions of fractional Whitham-Broer-Kaup, fractional modified Boussinesq and fractional approximate long wave equations in shallow water, Math. Method. Appl. Sci., 38 (2015), 1352–1368. http://doi.org/10.1002/mma.3151 doi: 10.1002/mma.3151
[47]	B. Tian, Y. Qiu, Exact and explicit solutions of Whitham–Broer–Kaup equations in shallow water, Pure and Applied Mathematics Journal, 5 (2016), 174–180. http://doi.org/10.11648/j.pamj.20160506.11 doi: 10.11648/j.pamj.20160506.11
[48]	P. Veeresha, D. G. Prakasha, H. M. Baskonus, An efficient technique for coupled fractional Whitham–Broer–Kaup equations describing the propagation of shallow water waves, In: 4th International conference on computational mathematics and engineering sciences (CMES-2019), Cham: Springer, 2019, 49–75. http://doi.org/10.1007/978-3-030-39112-6_4
[49]	J. Singh, D. Kumar, R. Swroop, Numerical solution of time- and space-fractional coupled Burgers' equations via homotopy algorithm, Alex. Eng. J., 55 (2016), 1753–1763. http://doi.org/10.1016/j.aej.2016.03.028 doi: 10.1016/j.aej.2016.03.028
[50]	S. J. Liao, Homotopy analysis method: A new analytical technique for nonlinear problems, Commun. Nonlinear Sci. Numer. Simul., 2 (1997), 95–100. https://doi.org/10.1016/S1007-5704(97)90047-2 doi: 10.1016/S1007-5704(97)90047-2
[51]	S. J. Liao, Homotopy analysis method: a new analytic method for nonlinear problems, Appl. Math. Mech., 19 (1998), 957–962. http://doi.org/10.1007/BF02457955 doi: 10.1007/BF02457955
[52]	K. S. Aboodh, The new integral transform "Aboodh transform", Global Journal of Pure and Applied Mathematics, 9 (2013), 35–43.
[53]	H. Bulut, D. Kumar, J. Singh, R. Swroop, H. M. Baskonus, Analytic study for a fractional model of HIV infection of CD4+T lymphocyte cells, Math. Nat. Sci., 2 (2018), 33–43. http://doi.org/10.22436/mns.02.01.04 doi: 10.22436/mns.02.01.04
[54]	P. Veeresha, D. G. Prakasha, Solution for fractional Zakharov-Kuznetsov equations by using two reliable techniques, Chinese J. Phys., 60 (2019), 313–330. http://doi.org/10.1016/j.cjph.2019.05.009 doi: 10.1016/j.cjph.2019.05.009
[55]	D. G. Prakasha, P. Veeresha, Analysis of lakes pollution model with Mittag-Leffler kernel, J. Ocean Eng. Sci., 5 (2020), 310–322. http://doi.org/10.1016/j.joes.2020.01.004 doi: 10.1016/j.joes.2020.01.004
[56]	M. H. Cherif, D. Ziane, A new numerical technique for solving systems of nonlinear fractional partial differential equations, Int. J. Anal. Appl., 15 (2017), 188–197.
[57]	M. A. Awuya, D. Subasi, Aboodh transform iterative method for solving fractional partial differential equation with Mittag–Leffler kernel, Symmetry, 13 (2021), 2055. http://doi.org/10.3390/sym13112055 doi: 10.3390/sym13112055
[58]	G. M. Mittag-Leffler, Sur La nonvelle Fonction $E_{\alpha}(x)$ , C. R. Acad. Sci. Paris, (Ser. II), 137 (1903), 554–558.
[59]	S. M. El-Sayed, D. Kaya, Exact and numerical travelling wave solutions of Whitham–Broer–Kaup equations, Appl. Math. Comput., 167 (2005), 1339–1349. http://doi.org/10.1016/j.amc.2004.08.012 doi: 10.1016/j.amc.2004.08.012
[60]	M. Rafei, H. Daniali, Application of the variational iteration method to the Whitham-Broer-Kaup equations, Comput. Math. Appl, 54 (2007), 1079–1085. http://doi.org/10.1016/j.camwa.2006.12.054 doi: 10.1016/j.camwa.2006.12.054

This article has been cited by:

1.	Timilehin Kingsley Akinfe, Adedapo Chris Loyinmi, An improved differential transform scheme implementation on the generalized Allen–Cahn equation governing oil pollution dynamics in oceanography, 2022, 6, 26668181, 100416, 10.1016/j.padiff.2022.100416
2.	Saima Rashid, Fahd Jarad, Hajid Alsubaie, Ayman A. Aly, Ahmed Alotaibi, A novel numerical dynamics of fractional derivatives involving singular and nonsingular kernels: designing a stochastic cholera epidemic model, 2023, 8, 2473-6988, 3484, 10.3934/math.2023178
3.	Maysaa Al-Qurashi, Sobia Sultana, Shazia Karim, Saima Rashid, Fahd Jarad, Mohammed Shaaf Alharthi, Identification of numerical solutions of a fractal-fractional divorce epidemic model of nonlinear systems via anti-divorce counseling, 2022, 8, 2473-6988, 5233, 10.3934/math.2023263
4.	Saima Rashid, Bushra Kanwal, Muhammad Attique, Ebenezer Bonyah, Ibrahim Mahariq, An Efficient Technique for Time-Fractional Water Dynamics Arising in Physical Systems Pertaining to Generalized Fractional Derivative Operators, 2022, 2022, 1563-5147, 1, 10.1155/2022/7852507
5.	M. S. Alqurashi, Saima Rashid, Bushra Kanwal, Fahd Jarad, S. K. Elagan, A novel formulation of the fuzzy hybrid transform for dealing nonlinear partial differential equations via fuzzy fractional derivative involving general order, 2022, 7, 2473-6988, 14946, 10.3934/math.2022819
6.	Zharasbek Baishemirov, Abdumauvlen Berdyshev, Dossan Baigereyev, Kulzhamila Boranbek, Efficient Numerical Implementation of the Time-Fractional Stochastic Stokes–Darcy Model, 2024, 8, 2504-3110, 476, 10.3390/fractalfract8080476
7.	Kingsley Timilehin Akinfe, A reliable analytic technique for the modified prototypical Kelvin–Voigt viscoelastic fluid model by means of the hyperbolic tangent function, 2023, 7, 26668181, 100523, 10.1016/j.padiff.2023.100523
8.	P Karunakar, S Chakraverty, TD Rao, K Ramesh, AK Hussein, Fuzzy fractional shallow water wave equations: analysis, convergence of solutions, and comparative study with depth as triangular fuzzy number, 2024, 99, 0031-8949, 125216, 10.1088/1402-4896/ad8afd

Reader Comments

Your name:*

Email:*
© 2022 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)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(2332) PDF downloads(149) Cited by(8)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(10) / Tables(2)

AIMS Mathematics

Fractional-order partial differential equations describing propagation of shallow water waves depending on power and Mittag-Leffler memory

Related Papers:

Abstract

1. Introduction

2. Preliminaries

3. Road map of $\bar{\mathbf{q}}$ -homotopy analysis transform method

4. Test examples

5. Numerical results and their physical interpretation

6. Conclusions

Acknowledgements

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

AIMS Mathematics

Fractional-order partial differential equations describing propagation of shallow water waves depending on power and Mittag-Leffler memory

Related Papers:

Abstract

1. Introduction

2. Preliminaries

3. Road map of ˉq \bar{\mathbf{q}} -homotopy analysis transform method

4. Test examples

5. Numerical results and their physical interpretation

6. Conclusions

Acknowledgements

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

3. Road map of $\bar{\mathbf{q}}$ -homotopy analysis transform method