Efficient thyroid disorder identification with weighted voting ensemble of super learners by using adaptive synthetic sampling technique

Noor Afshan; Zohaib Mushtaq; Faten S. Alamri; Muhammad Farrukh Qureshi; Nabeel Ahmed Khan; Imran Siddique; Noor Afshan; Zohaib Mushtaq; Faten S. Alamri; Muhammad Farrukh Qureshi; Nabeel Ahmed Khan; Imran Siddique

doi:10.3934/math.20231238

AIMS Mathematics

2023, Volume 8, Issue 10: 24274-24309. doi: 10.3934/math.20231238

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Research article Special Issues

Efficient thyroid disorder identification with weighted voting ensemble of super learners by using adaptive synthetic sampling technique

1.
Department of Software Engineering, Faculty of Computer Science, Lahore Garrison University, Lahore 54000, Pakistan
2.
Department of Electrical Engineering, CET, University of Sargodha, Sargodha 40100, Pakistan
3.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
4.
Department of Electrical Engineering, Riphah International University, Islamabad 44000, Pakistan
5.
Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan

Received: 27 April 2023 Revised: 08 July 2023 Accepted: 01 August 2023 Published: 14 August 2023
MSC : 62J02, 62J99

There are millions of people suffering from thyroid disease all over the world. For thyroid cancer to be effectively treated and managed, a correct diagnosis is necessary. In this article, we suggest an innovative approach for diagnosing thyroid disease that combines an adaptive synthetic sampling method with weighted average voting (WAV) ensemble of two distinct super learners (SLs). Resampling techniques are used in the suggested methodology to correct the class imbalance in the datasets and a group of two SLs made up of various base estimators and meta-estimators is used to increase the accuracy of thyroid cancer identification. To assess the effectiveness of our suggested methodology, we used two publicly accessible datasets: the KEEL thyroid illness (Dataset1) and the hypothyroid dataset (Dataset2) from the UCI repository. The findings of using the adaptive synthetic (ADASYN) sampling technique in both datasets revealed considerable gains in accuracy, precision, recall and F1-score. The WAV ensemble of the two distinct SLs that were deployed exhibited improved performance when compared to prior existing studies on identical datasets and produced higher prediction accuracy than any individual model alone. The suggested methodology has the potential to increase the accuracy of thyroid cancer categorization and could assist with patient diagnosis and treatment. The WAV ensemble strategy computational complexity and the ideal choice of base estimators in SLs continue to be constraints of this study that call for further investigation.

Keywords:

Citation: Noor Afshan, Zohaib Mushtaq, Faten S. Alamri, Muhammad Farrukh Qureshi, Nabeel Ahmed Khan, Imran Siddique. Efficient thyroid disorder identification with weighted voting ensemble of super learners by using adaptive synthetic sampling technique[J]. AIMS Mathematics, 2023, 8(10): 24274-24309. doi: 10.3934/math.20231238

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Abstract

1. Introduction

Differential equations are very useful in many fields of science, including applied sciences and mathematical physics. Partial differential equations (PDEs) are widely employed in engineering to understand the behavior of physical systems through mathematical models. Many efficient approaches for identifying the solutions of nonlinear PDEs have been developed in recent years, such as: Hirotäs bilinear method ^[1], exp $(-\Phi(\xi))$ -expansion method ^[2,3], generalized exponential rational function method ^[4], auxiliary equation method ^[5], first integral method ^[6], homotopy analysis method ^[7,8], tanh-method ^[9], transformed rational function method ^[10], residual power series method ^[11,12], extended direct algebraic method ^[13] and many other powerful mathematical techniques. Nonlinear PDEs are studied extensively because they help to understand the propagation of waves in many areas of mathematical physics, fluid mechanics and electromagnetic theory ^[14,15]. In addition, soliton solutions also have an effective contribution in fields of engineering and nonlinear optics ^{[2,16,18,19,20,21]}.

The traveling wave solutions of nonlinear PDEs equation are essential to explore and interpret various real life physical phenomena. The significance of the traveling wave solutions of nonlinear evolution equations has motivated many researches to investigate exact traveling wave solutions using effective and reliable mathematical techniques. The traveling wave solutions include solitary and other kinds of wave solutions. In particular, solitons are of great significance due to their useful applications in various areas of science and engineering ^[22,23].

This study aims to investigate soliton and other traveling wave solutions of Gardner equation, which is an integrable nonlinear partial differential. The Gardner equation was originally proposed by Clifford Gardner in $1968$ ^[24]. This equation is frequently referred to as combined Korteweg-de Vries-modified Korteweg-de Vries (KdV-mKdV) equation since it can be generalized to Korteweg-de Vries (KdV) equation. Gardner's equation has a wide range of applications in research, including quantum field theory and hydrodynamics ^[25,26,27].

In this work, solitary wave solutions of Gardner equation (GE) are retrieved by utilizing the improved tan $\left(\frac{\Omega(\Upsilon)}{2}\right)$ -expansion method. This technique is a recently developed direct technique which provides a variety of traveling wave solutions for a wide class of nonlinear evolution equations ^{[28,29,30,31]}.

A soliton is an autonomous wave that diffuses while maintaining its shape and velocity. The nonlinear integrable KdV equation can be written, as

$\begin{equation} r_{t}+l rr_{x}+mr_{xxx} = 0, \end{equation}$

(1.1)

where $r(x, t)$ in Eq (1.1) is the appropriate field variable and $l$ , $m$ are real constants, also $x$ is representing the spatial variable and $t$ is indicating the temporal variable. The solitary waves are generated due to nonlinear term $rr_{x}$ and the linear dispersion $r_{xxx}$ . The Gardener equation with constant coefficients ^[32,33] is considered in the form

$\begin{equation} r_{t}-6(r+\delta^{2}r^{2})r_{x}+r_{xxx} = 0, \end{equation}$

(1.2)

where $\delta$ is a nonzero constant. Eq $(1.2)$ is also called the combined KdV-mKdV equation. A higher order nonlinear term was added to the Eq (1.1) to generate the Gardner equation. Like KdV equation, Equation(1.2) is also an integrable equation.

2. Improved tan $\left(\frac{\Omega(\Upsilon)}{2}\right)$ -expansion method

Step 2.1. The nonlinear partial differential equation (NLPDE) for $r(x, t)$ is considered in the form

$\begin{equation} \Omega(r,r_{t},r_{x},r_{tt},r_{xt},r_{xx}, \ldots) = 0. \end{equation}$

(2.1)

By the aid of transformation $\Upsilon = \kappa(x-\varpi t)$ , Eq (2.1) can be converted into an ordinary differential equation, as

$\begin{equation} \Gamma(r,r^{'},-\varpi r^{'},r^{''},\varpi^{2} r^{''} \ldots) = 0, \end{equation}$

(2.2)

where $\varpi$ is to be evaluated later and $\Upsilon$ is a wave variable.

Step 2.2. The solitary wave solution of Eq (2.2) is supposed, as

$\begin{equation} R(\Upsilon) = P(\Phi) = \sum\limits_{j = 0}^{p} G_{j} \bigg[q+\tan \left(\frac{\Omega(\Upsilon)}{2}\right)\bigg]^{j}+\sum\limits_{j = 1}^{p} H_{j} \bigg[q+\tan \left(\frac{\Omega(\Upsilon)}{2}\right)\bigg]^{-j}, \end{equation}$

(2.3)

where $G_{j} (0 \leq j \leq p)$ and $H_{j} (1 \leq j \leq p)$ are constants to be determined later. Also, $G_{p}\neq0, H_{p}\neq0$ and $\Omega = \Omega(\Upsilon)$ satisfy the ordinary differential equation (ODE),

$\begin{equation} \Omega'(\Upsilon) = q_{0} \: \sin(\Omega(\Upsilon)) + q_{1} \:\cos (\Omega(\Upsilon))+ q_{2}. \end{equation}$

(2.4)

Following are the special wave solutions for Eq (2.4).

Family 2.1. For $q_{0}^{2}+q_{1}^{2}-q_{2}^{2} < 0$ and $q_{1}-q_{2}\neq0$ ,

$\begin{eqnarray} \Omega(\Upsilon) = 2\:\arctan \bigg[\frac{q_{0}}{q_{1}-q_{2}}-\frac{\sqrt{q_{2}^{2}-q_{1}^{2}-q_{0}^{2}}}{q_{1}-q_{2}} \tan\bigg(\frac{\sqrt{q_{2}^{2}-q_{1}^{2}-q_{0}^{2}}}{2}(\Upsilon+K)\bigg)\bigg]. \end{eqnarray}$

Family 2.2. For $q_{0}^{2}+q_{1}^{2}-q_{2}^{2} > 0$ and $q_{1}-q_{2}\neq0$ ,

$\begin{eqnarray} \Omega(\Upsilon) = 2 \:\arctan \bigg[\frac{q_{0}}{q_{1}-q_{2}}-\frac{\sqrt{q_{1}^{2}+q_{0}^{2}-q_{2}^{2}}}{q_{1}-q_{2}} \tanh\bigg(\frac{\sqrt{q_{1}^{2}+q_{0}^{2}-q_{2}^{2}}}{2}(\Upsilon+K)\bigg)\bigg]. \end{eqnarray}$

Family 2.3. For $q_{0}^{2}+q_{1}^{2}-q_{2}^{2} > 0$ , $q_{1}\neq0$ and $q_{2} = 0$ ,

$\begin{eqnarray} \Omega(\Upsilon) = 2\: \arctan \bigg[\frac{q_{0}}{q_{1}}-\frac{\sqrt{q_{1}^{2}+q_{0}^{2}}}{q_{1}} \tanh\bigg(\frac{\sqrt{q_{1}^{2}+q_{0}^{2}}}{2}(\Upsilon+K)\bigg)\bigg]. \end{eqnarray}$

Family 2.4. For $q_{0}^{2}+q_{1}^{2}-q_{2}^{2} < 0$ , $q_{2}\neq0$ and $q_{1} = 0$ ,

$\begin{eqnarray} \Omega(\Upsilon) = 2\:\arctan \bigg[-\frac{q_{0}}{q_{2}}+\frac{\sqrt{q_{2}^{2}-q_{0}^{2}}}{q_{2}} \tan\bigg(\frac{\sqrt{q_{2}^{2}-q_{0}^{2}}}{2}(\Upsilon+K)\bigg)\bigg]. \end{eqnarray}$

Family 2.5. For $q_{0}^{2}+q_{1}^{2}-q_{2}^{2} > 0$ , $q_{1}-q_{2}\neq0$ and $q_{0} = 0$ ,

$\begin{eqnarray} \Omega(\Upsilon) = 2\:\arctan \bigg[\sqrt {\frac{q_{1}+q_{2}}{q_{1}-q_{2}}} \tanh\bigg(\frac{\sqrt{q_{1}^{2}-q_{2}^{2}}}{2}(\Upsilon+K)\bigg)\bigg]. \end{eqnarray}$

Family 2.6. For $q_{0} = 0$ and $q_{2} = 0$ ,

$\begin{eqnarray} \Omega(\Upsilon) = \arctan \bigg[\frac{e^{2q_{1}(\Upsilon+K)}-1}{e^{2q_{1}(\Upsilon+K)}+1}, \frac{2e^{q_{1}(\Upsilon+K)}}{e^{2q_{1}(\Upsilon+K)}+1}\bigg]. \end{eqnarray}$

Family 2.7. For $q_{1} = 0$ and $q_{2} = 0$ ,

$\begin{eqnarray} \Omega(\Upsilon) = \arctan \bigg[\frac{2e^{q_{0}(\Upsilon+K)}}{e^{2q_{0}(\Upsilon+K)}+1}, \frac{e^{2q_{0}(\Upsilon+K)}-1}{e^{2q_{0}(\Upsilon+K)}+1}\bigg]. \end{eqnarray}$

Family 2.8. For $q_{0}^{2}+q_{1}^{2} = q_{2}^{2}$ ,

$\begin{eqnarray} \Omega(\Upsilon) = -2\:\arctan \bigg[\frac{(q_{1}+q_{2})(q_{0}(\Upsilon+K)+2)}{q_{0}^{2}(\Upsilon+K)}\bigg]. \end{eqnarray}$

Family 2.9. For $q_{0} = q_{1} = q_{2} = il_{0}$ ,

$\begin{eqnarray} \Omega(\Upsilon) = 2\:\arctan \bigg[e^{il_{0}(\Upsilon+K)}-1 \bigg]. \end{eqnarray}$

Family 2.10. For $q_{0} = q_{2} = il_{0}$ and $q_{1} = -il_{0}$ ,

$\begin{eqnarray} \Omega(\Upsilon) = -2\:\arctan \bigg[\frac{e^{il_{0}(\Upsilon+K)}}{-1+e^{il_{0}(\Upsilon+K)}}\bigg]. \end{eqnarray}$

Family 2.11. For $q_{2} = q_{0}$ ,

$\begin{eqnarray} \Omega(\Upsilon) = -2 \:\arctan \bigg[\frac{(q_{0}+q_{1})e^{q_{1}(\Upsilon+K)}-1}{(q_{0}-q_{1})e^{q_{1}(\Upsilon+K)}-1}\bigg]. \end{eqnarray}$

Family 2.12. For $q_{0} = q_{2}$ ,

$\begin{eqnarray} \Omega(\Upsilon) = 2\:\arctan \bigg[\frac{(q_{1}+q_{2})e^{q_{1}(\Upsilon+K)}+1}{(q_{1}-q_{2})e^{q_{1}(\Upsilon+K)}-1}\bigg]. \end{eqnarray}$

Family 2.13. For $q_{2} = -q_{0}$ ,

$\begin{eqnarray} \Omega(\Upsilon) = 2\:\arctan \bigg[\frac{e^{q_{1}(\Upsilon+K)}+q_{1}-q_{0}}{e^{q_{1}(\Upsilon+K)}-q_{1}-q_{0}}\bigg]. \end{eqnarray}$

Family 2.14. For $q_{1} = -q_{2}$ ,

$\begin{eqnarray} \Omega(\Upsilon) = -2\:\arctan \bigg[\frac{q_{0}e^{q_{0}(\Upsilon+K)}}{q_{2}e^{q_{0}(\Upsilon+K)}}\bigg]. \end{eqnarray}$

Family 2.15. For $q_{1} = 0$ , $q_{0} = q_{2}$ ,

$\begin{eqnarray} \Omega(\Upsilon) = -2\:\arctan \bigg[\frac{q_{2}(\Upsilon+K)+2}{q_{2}(\Upsilon+K)}\bigg]. \end{eqnarray}$

Family 2.16. For $q_{0} = 0$ and $q_{1} = q_{2}$ ,

$\begin{eqnarray} \Omega(\Upsilon) = 2\:\arctan \bigg[q_{2}(\Upsilon+K)\bigg]. \end{eqnarray}$

Family 2.17. For $q_{0} = 0$ , $q_{1} = -q_{2}$ ,

$\begin{eqnarray} \Omega(\Upsilon) = -2\:\arctan \bigg[\frac{1}{q_{2}(\Upsilon+K)}\bigg]. \end{eqnarray}$

Family 2.18. For $q_{0} = 0$ and $q_{1} = 0$ ,

$\begin{eqnarray} \Omega(\Upsilon) = q_{2}\Upsilon+K, \end{eqnarray}$

where $q_{0}, \; q_{1}$ , $q_{2}$ and $G_{0}, \; G_{j}, \; H_{j}\; (j = 1, 2, ....p)$ are to be evaluated. Homogeneous balance principle is used to find the value of $p$ by considering highest order derivatives and highest non-linear terms occurring in Eq (2.2). If $p$ is not an integer, then suitable transformation is implemented.

Step 2.3. Once the value of $p$ is obtained, Eq (2.3) is substituted into Eq (2.2). By gathering the coefficients of $\tan\left(\frac{\Omega(\Upsilon)}{2}\right)^{j}$ , $\cot\left(\frac{\Omega(\Upsilon)}{2}\right)^{j}$ $(j = 0, 1, 2, ...)$ and setting each coefficient equal to zero, a set of algebraic equations for $G_{0}, \; G_{j}, \; H_{j}\; (j = 1, 2, .\:.\:.p)$ , $q_{0}, \; q_{1}, \; q_{2}$ and $q$ can be obtained.

Step 2.4. The set of over determined equations are solved and the values of $G_{0}, \; G_{1}, \; H_{1}, ..., \; G_{p}, \; H_{p}, \; \varpi$ and $q$ are substituted in Eq (2.3).

3. Exact soliton solutions of Gardner equation

Consider the integrable nonlinear Gardner equation given by Eq (1.2). Substituting the wave transformation,

$\begin{equation} r(x,t) = R(\Upsilon),\; \Upsilon = \kappa(x-\varpi t), \end{equation}$

(3.1)

into Eq (1.2) yields an ordinary differential equation, as

$\begin{equation} \varpi R+3 R^{2}+2 \delta^{2} R^{3}-\kappa^{2} R^{''} = 0. \end{equation}$

(3.2)

Implementing the homogeneous balance principle the value of positive integer is obtained, as $p = 1$ . The trial solution becomes

$\begin{equation} R(\Upsilon) = G_{0}+G_{1} \bigg [q+\tan \bigg(\frac{\Omega(\Upsilon)}{2}\bigg)\bigg]+H_{1} \bigg [q+\tan \bigg(\frac{\Omega(\Upsilon)}{2}\bigg)\bigg]^{-1}. \end{equation}$

(3.3)

Substituting Eq (3.3) and Eq (2.4) into Eq (3.2), the following set of algebraic equations can be derived for $q_{0}, q_{1}, q_{2}, \kappa, \varpi,$ $G_{0}, G_{1}$ and $H_{1}$ by collecting the terms with the same order of $\tan\left(\frac{\Omega(\Upsilon)}{2}\right)$ and setting every coefficient of all the polynomials equal to zero.

$\begin{array}{l} \bigg(\tan\left(\frac{\Omega(\Upsilon)}{2}\right)\bigg)^{0} = 4\,{\delta }^{2}{H_{{1}}}^{3}-{\kappa}^{2}H_{{1}}{q_{{1}}}^{2}-2\,{\kappa }^{2}H_{{1}}q_{{1}}q_{{2}}-{\kappa}^{2}H_{{1}}{q_{{2}}}^{2},\\ \bigg(\tan\left(\frac{\Omega(\Upsilon)}{2}\right)\bigg)^{1} = 12\,{\delta }^{2}G_{{0}}{H_{{1}}}^{2}-3\,{\kappa}^{2}H_{{1}}q_{{0}}l_{{1} }-3\,{\kappa}^{2}H_{{1}}q_{{0}}q_{{2}}+6\,{H_{{1}}}^{2},\\ \bigg(\tan\left(\frac{\Omega(\Upsilon)}{2}\right)\bigg)^{2} = 12\,{\delta }^{2}{G_{{0}}}^{2}H_{{1}}+12\,{\delta }^{2}G_{{1}}{H_{{1}}}^{2}- 2\,{\kappa}^{2}H_{{1}}{q_{{0}}}^{2}+\\{\kappa}^{2}H_{{1}}{q_{{1}}}^{2}-{ \kappa}^{2}H_{{1}}{q_{{2}}}^{2}+2\,\varpi\,H_{{1}}+12\,G_{{0}}H_{{1}},\\ \bigg(\tan\left(\frac{\Omega(\Upsilon)}{2}\right)\bigg)^{3} = 4\,{\delta }^{2}{G_{{0}}}^{3}+24\,{\delta }^{2}G_{{0}}G_{{1}}H_{{1}}-{\kappa }^{2}G_{{1}}q_{{0}}q_{{1}}-{\kappa}^{2}G_{{1}}q_{{0}}q_{{2}}+{\kappa}^ {2}H_{{1}}q_{{0}}q_{{1}}-{\kappa}^{2}H_{{1}}q_{{0}}q_{{2}}\\ +2\,\varpi\,G_{{0}}+6\,{G_{{0}}}^{2}+12\,G_{{1}}H_{{1}},\\ \bigg(\tan\left(\frac{\Omega(\Upsilon)}{2}\right)\bigg)^{4} = 12\,{\delta }^{2}{G_{{0}}}^{2}G_{{1}}+12\,{\delta }^{2}{G_{{1}}}^{2}H_{{1}}- 2\,{\kappa}^{2}G_{{1}}{q_{{0}}}^{2}+\\{\kappa}^{2}G_{{1}}{q_{{1}}}^{2}-{ \kappa}^{2}G_{{1}}{q_{{2}}}^{2}+2\,\varpi\,G_{{1}}+12\,G_{{0}}G_{{1}},\\ \bigg(\tan\left(\frac{\Omega(\Upsilon)}{2}\right)\bigg)^{5} = 12\,{\delta }^{2}G_{{0}}{G_{{1}}}^{2}+3\,{\kappa}^{2}G_{{1}}q_{{0}}l_{{1} }-3\,{\kappa}^{2}G_{{1}}q_{{0}}q_{{2}}+6\,{G_{{1}}}^{2},\\ \bigg(\tan\left(\frac{\Omega(\Upsilon)}{2}\right)\bigg)^{6} = 4\,{\delta }^{2}{G_{{1}}}^{3}-{\kappa}^{2}G_{{1}}{q_{{1}}}^{2}+2\,{\kappa }^{2}G_{{1}}q_{{1}}q_{{2}}-{\kappa}^{2}G_{{1}}{q_{{2}}}^{2}. \end{array}$

Following are the solutions obtained by solving the system of algebraic equation.

Set 3.1. $\kappa = \kappa, \; \varpi = \frac{1}{\delta^{2}}, \; G_{0} = -\frac{1}{2\delta^{2}}, \; G_{{1}} = \pm\frac{1}{4}{\frac {2\, q_{{1}}\kappa\, \delta -\sqrt {4\, {\delta }^{2}{\kappa}^{2}{q_{{1}}}^{2}-1}}{{\delta }^{2}}}, \; H_{{1}} = \pm\frac{1}{4}{\frac {2\, q_{{1}}\kappa\, \delta +\sqrt {4\, {\delta}^{2}{\kappa}^{2}{q_{{1}}}^{2}-1}}{{\delta }^{2}}}, \; q_{{0}} = 0, \\q_{{1}} = q_{{1}}, \; q_{{2}} = \frac{1}{2}{\frac {\sqrt {4\, {\delta }^{2}{\kappa}^{2}{q_{{1}}}^{2}-1}}{\delta \, \kappa}},$

$\begin{equation} R(\Upsilon) = G_{0}+G_{1}\bigg [\tan \bigg(\frac{\Omega(\Upsilon)}{2}\bigg)\bigg]+H_{1} \bigg [\tan \bigg(\frac{\Omega(\Upsilon)}{2}\bigg)\bigg]^{-1}, \end{equation}$

(3.4)

where $q_{0}$ , $q_{1}$ , $q_{2}$ are arbitrary constants.

Using Eq (3.4) and Families 2.2, 2.5 and 2.18 respectively, yields the following solutions:

$\begin{array}{l} R_{1}(\Upsilon) = -\frac{1}{2\delta^{2}}\pm\frac{1}{4}{\frac {2\,q_{{1}}\kappa\,\delta -\sqrt {4\,{\delta }^{2}{\kappa}^{2}{q_{{1}}}^{2}-1}}{{\delta }^{2}}}\bigg[\frac{q_{0}}{q_{1}-q_{2}}+\\\sqrt{\frac{q_{1}^{2}+q_{0}^{2}-q_{2}^{2}}{q_{1}-q_{2}}}\tanh\bigg(\sqrt{\frac{q_{1}^{2}+q_{0}^{2}-q_{2}^{2}}{2}}(\Upsilon+K)\bigg)\bigg]\\ \pm\frac{1}{4}{\frac {2\,q_{{1}}\kappa\,\delta +\sqrt {4\,{\delta}^{2}{\kappa}^{2}{q_{{1}}}^{2}-1}}{{\delta }^{2}}} \bigg[\frac{q_{0}}{q_{1}-q_{2}}+\\\sqrt{\frac{q_{1}^{2}+q_{0}^{2}-q_{2}^{2}}{q_{1}-q_{2}}}\tanh\bigg(\sqrt{\frac{q_{1}^{2}+q_{0}^{2}-q_{2}^{2}}{2}} (\Upsilon+K)\bigg)\bigg]^{-1}, \end{array}$

(3.5)

$\begin{align} R_{2}(\Upsilon)& = -\frac{1}{2\delta^{2}}\pm\frac{1}{4}{\frac {2\,q_{{1}}\kappa\,\delta -\sqrt {4\,{\delta }^{2}{\kappa}^{2}{q_{{1}}}^{2}-1}}{{\delta }^{2}}}\bigg[\sqrt{\frac{q_{1}+q_{2}}{q_{1}-q_{2}}}\tanh \bigg(\frac{\sqrt{q_{1}^{2}-q_{2}^{2}}}{2}(\Upsilon+K)\bigg) \bigg]\\ &\pm\frac{1}{4}{\frac {2\,q_{{1}}\kappa\,\delta +\sqrt {4\,{\delta}^{2}{\kappa}^{2}{q_{{1}}}^{2}-1}}{{\delta }^{2}}} \bigg[\sqrt{\frac{q_{1}+q_{2}}{q_{1}-q_{2}}}\tanh \bigg(\frac{\sqrt{q_{1}^{2}-q_{2}^{2}}}{2}(\Upsilon+K)\bigg) \bigg]^{-1}, \end{align}$

(3.6)

$\begin{align} R_{3}(\Upsilon)& = -\frac{1}{2\delta^{2}}\pm\frac{1}{4}{\frac {2\,q_{{1}}\kappa\,\delta -\sqrt {4\,{\delta }^{2}{\kappa}^{2}{q_{{1}}}^{2}-1}}{{\delta }^{2}}}\bigg[\tan\bigg(\frac{1}{2}\arctan[\Upsilon\,q_{2}+K]\bigg)\bigg]\\ &\pm\frac{1}{4}{\frac {2\,q_{{1}}\kappa\,\delta +\sqrt {4\,{\delta}^{2}{\kappa}^{2}{q_{{1}}}^{2}-1}}{{\delta }^{2}}} \bigg[\tan\bigg(\frac{1}{2}\arctan[\Upsilon\,q_{2}+K]\bigg)\bigg]^{-1}. \end{align}$

(3.7)

Set 3.2. $\kappa = \kappa, \; \varpi = \frac{1}{\delta^{2}}, \; G_{0} = \frac{1}{2}{\frac {-1+\sqrt {1+ \left(-{q_{{1}}}^{2}+{q_{{2}}}^{2} \right) {\delta }^{2}{\kappa}^{2}}}{{\delta }^{2}}}, \; G_{{1}} = 0, \; H_{{1}} = \mp\frac{1}{2}{\frac {\kappa\, \left(q_{{1}}+q_{{2}} \right) }{\delta }},$ $\; q_{{0}} = \mp{\frac {\sqrt {1+ \left(-{q_{{1}}}^{2}+{q_{{2}}}^{2} \right) {\delta }^{2}{\kappa}^{2}}}{\kappa\, \delta }}, \; q_{{1}} = q_{{1}}, \; q_{{2}} = q_{2},$

$\begin{equation} R(\Upsilon) = G_{0}+H_{1} \bigg [\tan \bigg(\frac{\Omega(\Upsilon)}{2}\bigg)\bigg]^{-1}, \end{equation}$

(3.8)

where $q_{0}$ , $q_{1}$ , $q_{2}$ are arbitrary constants.

The following solutions are determined by using Eq (3.8) and Families 2.2, 2.3, 2.6, 2.7 and 2.11–2.14, respectively.

$\begin{align} R_{4}(\Upsilon)& = \frac{1}{2}{\frac {-1+\sqrt {1+ \left( -{q_{{1}}}^{2}+{q_{{2}}}^{2} \right) {\delta }^{2}{\kappa}^{2}}}{{\delta }^{2}}}\\ &\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }} \bigg[\frac{q_{0}}{q_{1}-q_{2}}+\sqrt{\frac{q_{1}^{2}+q_{0}^{2}-q_{2}^{2}}{q_{1}-q_{2}}}\tanh\bigg(\sqrt{\frac{q_{1}^{2}+q_{0}^{2}-q_{2}^{2}}{2}} (\Upsilon+K)\bigg)\bigg]^{-1}, \end{align}$

(3.9)

$\begin{align} R_{5}(\Upsilon)& = \frac{1}{2}{\frac {-1+\sqrt {1+ \left( -{q_{{1}}}^{2}+{q_{{2}}}^{2} \right) {\delta }^{2}{\kappa}^{2}}}{{\delta }^{2}}}\\ &\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }} \bigg[\frac{q_{0}}{q_{1}}+\frac{\sqrt{q_{1}^{2}+q_{0}^{2}}}{q_{1}} \tanh \bigg(\frac{\sqrt{q_{1}^{2}+q_{0}^{2}}}{2}(\Upsilon+K)\bigg)\bigg]^{-1}, \end{align}$

(3.10)

$\begin{align} R_{6}(\Upsilon)& = \frac{1}{2}{\frac {-1+\sqrt {1+ \left( -{q_{{1}}}^{2}+{q_{{2}}}^{2} \right) {\delta }^{2}{\kappa}^{2}}}{{\delta }^{2}}}\\ &\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }} \quad\bigg[\tan \frac{1}{2}\bigg(\arctan\bigg[{\frac {{e}^{2\,q_{1} \left( \Upsilon+K \right) }-1}{{e}^{2\,q_{1} \left( \Upsilon+K\right) }+1}},{\frac {2{e}^{q_{1}\left(\Upsilon+K \right) }}{{e}^{2\,q_{1} \left( \Upsilon+K\right) }+1}}\bigg]\bigg) \bigg]^{-1}, \end{align}$

(3.11)

$\begin{align} R_{7}(\Upsilon)& = \frac{1}{2}{\frac {-1+\sqrt {1+ \left( -{q_{{1}}}^{2}+{q_{{2}}}^{2} \right) {\delta }^{2}{\kappa}^{2}}}{{\delta }^{2}}}\\ &\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }}\quad\bigg[\tan\bigg(\frac{1}{2}\arctan\bigg[{\frac {{2e}^{q_{0} \left( \Upsilon+K \right) }}{{e}^{2\,q_{0} \left( \Upsilon+K \right) }+1}}, {\frac {{e}^{2q_{0} \left( \Upsilon+K \right) }-1}{{e}^{2\,q_{0} \left( \Upsilon+K \right) }+1}}\bigg]\bigg)\bigg]^{-1}, \end{align}$

(3.12)

$\begin{align} R_{8}(\Upsilon)& = \frac{1}{2}{\frac {-1+\sqrt {1+ \left( -{q_{{1}}}^{2}+{q_{{2}}}^{2} \right) {\delta }^{2}{\kappa}^{2}}}{{\delta }^{2}}}\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }}\bigg[-{\frac { \left( q_{0}+q_{1} \right) {e}^{q_{1} \left( \Upsilon+K \right) }-1}{\left( q_{0}-q_{1} \right) {e}^{q_{1} \left( \Upsilon+K \right) }-1}}\bigg]^{-1}, \end{align}$

(3.13)

$\begin{align} R_{9}(\Upsilon)& = \frac{1}{2}{\frac {-1+\sqrt {1+ \left( -{q_{{1}}}^{2}+{q_{{2}}}^{2} \right) {\delta }^{2}{\kappa}^{2}}}{{\delta }^{2}}}\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }}\bigg[{\frac { \left( q_{1}+q_{2} \right) {e}^{q_{1} \left( \Upsilon+K \right) }+1}{ \left(q_{1}-q_{2} \right) {e}^{q_{1} \left( \Upsilon+K \right) }-1}}\bigg]^{-1}, \end{align}$

(3.14)

$\begin{align} R_{10}(\Upsilon)& = \frac{1}{2}{\frac {-1+\sqrt {1+ \left( -{q_{{1}}}^{2}+{q_{{2}}}^{2} \right) {\delta }^{2}{\kappa}^{2}}}{{\delta }^{2}}}\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }}\bigg[{\frac {{e}^{q_{1} \left( \Upsilon+K \right) }+q_{1}-q_{0}}{{e}^{q_{1} \left( \Upsilon+K \right) }-q_{1}-q_{0}}}\bigg]^{-1}, \end{align}$

(3.15)

$\begin{align} R_{11}(\Upsilon)& = \frac{1}{2}{\frac {-1+\sqrt {1+ \left( -{q_{{1}}}^{2}+{q_{{2}}}^{2} \right) {\delta }^{2}{\kappa}^{2}}}{{\delta }^{2}}}\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }}\bigg[-{\frac {q_{0}{e}^{q_{0} \left( \Upsilon+K \right) }}{q_{2}{e}^{q_{0} \left( \Upsilon+K \right) }-1}}\bigg]^{-1}. \end{align}$

(3.16)

Set 3.3. $\kappa = \kappa, \; \varpi = \frac{1}{\delta^{2}}, \; G_{0} = \frac{1}{2}{\frac {-1+\sqrt {1+ \left(-{q_{{1}}}^{2}+{q_{{2}}}^{2} \right) {\delta }^{2}{\kappa}^{2}}}{{\delta }^{2}}}, \; G_{{1}} = \mp \frac{1}{2}{\frac {\kappa\, \left(q_{{1}}-q_{{2}} \right) }{\delta }}, \; H_{{1}} = 0, \;$ $q_{{0}} = \mp{\frac {\sqrt {1+ \left(-{q_{{1}}}^{2}+{q_{{2}}}^{2} \right) {\delta }^{2}{\kappa}^{2}}}{\kappa\, \delta }}, \; q_{{1}} = q_{{1}}, \; q_{{2}} = q_{2},$

$\begin{equation} R(\Upsilon) = G_{0}+G_{1} \bigg [\tan \bigg(\frac{\Omega(\Upsilon)}{2}\bigg)\bigg], \end{equation}$

(3.17)

where $q_{0}$ , $q_{1}$ , $q_{2}$ are arbitrary constants.

Using Eq (3.17) and Families 2.2, 2.3, 2.6, 2.7, 2.11–2.14, respectively, yields the following solution:

$\begin{align} R_{12}(\Upsilon)& = \frac{1}{2}{\frac {-1+\sqrt {1+ \left( -{q_{{1}}}^{2}+{q_{{2}}}^{2} \right) {\delta }^{2}{\kappa}^{2}}}{{\delta }^{2}}}\\ &\mp \frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }} \bigg[\frac{q_{0}}{q_{1}-q_{2}}+\sqrt{\frac{q_{1}^{2}+q_{0}^{2}-q_{2}^{2}}{q_{1}-q_{2}}}\tanh\bigg(\sqrt{\frac{q_{1}^{2}+q_{0}^{2}-q_{2}^{2}}{2}} (\Upsilon+K)\bigg)\bigg], \end{align}$

(3.18)

$\begin{align} R_{13}(\Upsilon)& = \frac{1}{2}{\frac {-1+\sqrt {1+ \left( -{q_{{1}}}^{2}+{q_{{2}}}^{2} \right) {\delta }^{2}{\kappa}^{2}}}{{\delta }^{2}}}\\ &\mp \frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }} \bigg[\frac{q_{0}}{q_{1}}+\frac{\sqrt{q_{1}^{2}+q_{0}^{2}}}{q_{1}}\tanh \bigg(\frac{\sqrt{q_{1}^{2}+q_{0}^{2}}}{2}(\Upsilon+K)\bigg)\bigg], \end{align}$

(3.19)

$\begin{align} R_{14}(\Upsilon)& = \frac{1}{2}{\frac {-1+\sqrt {1+ \left( -{q_{{1}}}^{2}+{q_{{2}}}^{2} \right) {\delta }^{2}{\kappa}^{2}}}{{\delta }^{2}}}\\ &\mp \frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\bigg[\tan \frac{1}{2}\bigg(\arctan\bigg[{\frac {{e}^{2\,q_{1} \left( \Upsilon+K \right) }-1}{{e}^{2\,q_{1} \left( \Upsilon+K\right) }+1}},{\frac {2{e}^{q_{1}\left(\Upsilon+K \right) }}{{e}^{2\,q_{1} \left( \Upsilon+K\right) }+1}}\bigg]\bigg) \bigg], \end{align}$

(3.20)

$\begin{align} R_{15}(\Upsilon)& = \frac{1}{2}{\frac {-1+\sqrt {1+ \left( -{q_{{1}}}^{2}+{q_{{2}}}^{2} \right) {\delta }^{2}{\kappa}^{2}}}{{\delta }^{2}}}\\ &\mp \frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\bigg[\tan\bigg(\frac{1}{2}\arctan\bigg[{\frac {{2e}^{q_{0} \left( \Upsilon+K \right) }}{{e}^{2\,q_{0} \left( \Upsilon+K \right) }+1}}, {\frac {{e}^{2q_{0} \left( \Upsilon+K \right) }-1}{{e}^{2\,q_{0} \left( \Upsilon+K \right) }+1}}\bigg]\bigg)\bigg], \end{align}$

(3.21)

$\begin{align} R_{16}(\Upsilon)& = \frac{1}{2}{\frac {-1+\sqrt {1+ \left( -{q_{{1}}}^{2}+{q_{{2}}}^{2} \right) {\delta }^{2}{\kappa}^{2}}}{{\delta }^{2}}}\mp \frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\bigg[-{\frac { \left( q_{0}+q_{1} \right) {e}^{q_{1} \left( \Upsilon+K \right) }-1}{\left( q_{0}-q_{1} \right) {e}^{q_{1} \left( \Upsilon+K \right) }-1}}\bigg], \end{align}$

(3.22)

$\begin{align} R_{17}(\Upsilon)& = \frac{1}{2}{\frac {-1+\sqrt {1+ \left( -{q_{{1}}}^{2}+{q_{{2}}}^{2} \right) {\delta }^{2}{\kappa}^{2}}}{{\delta }^{2}}}\mp \frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\bigg[{\frac { \left( q_{1}+q_{2} \right) {e}^{q_{1} \left( \Upsilon+K \right) }+1}{ \left(q_{1}-q_{2} \right) {e}^{q_{1} \left( \Upsilon+K \right) }-1}}\bigg], \end{align}$

(3.23)

$\begin{align} R_{18}(\Upsilon)& = \frac{1}{2}{\frac {-1+\sqrt {1+ \left( -{q_{{1}}}^{2}+{q_{{2}}}^{2} \right) {\delta }^{2}{\kappa}^{2}}}{{\delta }^{2}}}\mp \frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\bigg[{\frac {{e}^{q_{1} \left( \Upsilon+K \right) }+q_{1}-q_{0}}{{e}^{q_{1} \left( \Upsilon+K \right) }-q_{1}-q_{0}}}\bigg], \end{align}$

(3.24)

$\begin{align} R_{19}(\Upsilon)& = \frac{1}{2}{\frac {-1+\sqrt {1+ \left( -{q_{{1}}}^{2}+{q_{{2}}}^{2} \right) {\delta }^{2}{\kappa}^{2}}}{{\delta }^{2}}}\mp \frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\bigg[-{\frac {q_{0}{e}^{q_{0} \left( \Upsilon+K \right) }}{q_{2}{e}^{q_{0} \left( \Upsilon+K \right) }-1}}\bigg]. \end{align}$

(3.25)

Set 3.4. $\kappa = \kappa, \; \varpi = {\frac {1+ \left({q_{{1}}}^{2}-{q_{{2}}}^{2} \right) {\kappa}^{2}{\delta }^{2}}{{\delta }^{2}}}, \; G_{0} = -\frac{1}{\delta^{2}}, \; G_{{1}} = \mp\frac{1}{2}{\frac {\kappa\, \left(q_{{1}}-q_{{2}} \right) }{\delta }}, \; H_{{1}} = \mp\frac{1}{2}{\frac {\kappa\, \left(q_{{1}}+q_{{2}} \right) }{\delta }}, \; q_{{0}} = \mp\frac{1}{\kappa \delta}, \\q_{{1}} = q_{{1}}, \; q_{{2}} = q_{2},$

$\begin{equation} R(\Upsilon) = G_{0}+G_{1} \bigg [\tan \bigg(\frac{\Omega(\Upsilon)}{2}\bigg)\bigg]+H_{1} \bigg [\tan \bigg(\frac{\Omega(\Upsilon)}{2}\bigg)\bigg]^{-1}, \end{equation}$

(3.26)

where $q_{0}$ , $q_{1}$ , $q_{2}$ are arbitrary constants.

Using Eq (3.26) and Families 2.1–2.4, 2.8–2.10, 2.13–2.15, respectively, gives the following solutions:

$\begin{array}{l} R_{20}(\Upsilon) = -\frac{1}{\delta^{2}}\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\\\bigg[{\frac {q_{0}}{q_{1}-q_{2}}}-\sqrt {{\frac {-{q_{0}}^{2} -{q_{1}}^{2}+{q_{2}}^{2}}{q_{1}-q_{2}}}}\tan \left( \sqrt {\frac{-{q_{0}}^{2}-{q_{1}}^{2}+{q_{2}}^{2}}{2}}(\Upsilon+K) \right) \bigg]\\ \mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }}\\\bigg[{\frac {q_{0}}{q_{1}-q_{2}}}-\sqrt {{\frac {-{q_{0}}^{2} -{q_{1}}^{2}+{q_{2}}^{2}}{q_{1}-q_{2}}}}\tan \left( \sqrt {\frac{-{q_{0}}^{2}-{q_{1}}^{2}+{q_{2}}^{2}}{2}}(\Upsilon+K) \right) \bigg]^{-1}, \end{array}$

(3.27)

$\begin{array}{l} R_{21}(\Upsilon) = -\frac{1}{\delta^{2}}\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\\ \bigg[\frac{q_{0}}{q_{1}-q_{2}}+\sqrt{\frac{q_{1}^{2}+q_{0}^{2}-q_{2}^{2}}{q_{1}-q_{2}}}\tanh\bigg(\sqrt{\frac{q_{1}^{2}+q_{0}^{2}-q_{2}^{2}}{2}} (\Upsilon+K)\bigg)\bigg]\\ \mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }}\\ \bigg[\frac{q_{0}}{q_{1}-q_{2}}+\sqrt{\frac{q_{1}^{2}+q_{0}^{2}-q_{2}^{2}}{q_{1}-q_{2}}}\tanh\bigg(\sqrt{\frac{q_{1}^{2}+q_{0}^{2}-q_{2}^{2}}{2}} (\Upsilon+K)\bigg)\bigg]^{-1}, \end{array}$

(3.28)

$\begin{align} R_{22}(\Upsilon)& = -\frac{1}{\delta^{2}}\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\bigg[\frac{q_{0}}{q_{1}}+\frac{\sqrt{q_{1}^{2}+q_{0}^{2}}}{q_{1}}\tanh \bigg(\frac{\sqrt{q_{1}^{2}+q_{0}^{2}}}{2}(\Upsilon+K)\bigg)\bigg]\\ &\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }}\bigg[\frac{q_{0}}{q_{1}}+\frac{\sqrt{q_{1}^{2}+q_{0}^{2}}}{q_{1}}\tanh \bigg(\frac{\sqrt{q_{1}^{2}+q_{0}^{2}}}{2}(\Upsilon+K)\bigg)\bigg]^{-1}, \end{align}$

(3.29)

$\begin{align} R_{23}(\Upsilon)& = -\frac{1}{\delta^{2}}\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\bigg[-\frac{q_{0}}{q_{2}}+\frac{\sqrt{q_{2}^{2}-q_{0}^{2}}}{q_{2}}\tan \bigg(\frac{\sqrt{q_{2}^{2}-q_{0}^{2}}}{2}(\Upsilon+K)\bigg)\bigg]\\ &\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }}\bigg[-\frac{q_{0}}{q_{2}}+\frac{\sqrt{q_{2}^{2}-q_{0}^{2}}}{q_{2}}\tan \bigg(\frac{\sqrt{q_{2}^{2}-q_{0}^{2}}}{2}(\Upsilon+K)\bigg)\bigg]^{-1}, \end{align}$

(3.30)

$\begin{align} R_{24}(\Upsilon)& = -\frac{1}{\delta^{2}}\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\bigg[-\frac{(q_{1}+q_{2})(q_{0}(\Upsilon+K)+2)}{q_{0}^{2}(\Upsilon+K)}\bigg]\\ &\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }}\bigg[-\frac{(q_{1}+q_{2})(q_{0}(\Upsilon+K)+2)}{q_{0}^{2}(\Upsilon+K)}\bigg]^{-1}, \end{align}$

(3.31)

$\begin{align} R_{25}(\Upsilon)& = -\frac{1}{\delta^{2}}\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\bigg[{e}^{\vartheta\,q_{0} \left( \Upsilon+K \right) }-1\bigg]\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }}\bigg[{e}^{\vartheta\,q_{0} \left( \Upsilon+K \right) }-1\bigg]^{-1}, \end{align}$

(3.32)

$\begin{array}{l} R_{26}(\Upsilon) = -\frac{1}{\delta^{2}}\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\bigg[-{\frac {{e}^{\vartheta\,q_{0} \left( \Upsilon+K \right) }}{-1+{e}^{\vartheta\,q_{0} \left( \Upsilon+K \right) }}}\bigg]\\\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }}\bigg[-{\frac {{e}^{\vartheta\,q_{0} \left( \Upsilon+K \right) }}{-1+{e}^{\vartheta\,q_{0} \left( \Upsilon+K \right) }}}\bigg]^{-1}, \end{array}$

(3.33)

$\begin{array}{l} R_{27}(\Upsilon) = -\frac{1}{\delta^{2}}\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\bigg[{\frac {{e}^{q_{1} \left( \Upsilon+K \right) }+q_{1}-q_{0}}{{e}^{q_{1} \left( \Upsilon+K \right) }-q_{1} -q_{0}}}\bigg]\\\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }}\bigg[{\frac {{e}^{q_{1} \left( \Upsilon+K \right) }+q_{1}-q_{0}}{{e}^{q_{1} \left( \Upsilon+K \right) }-q_{1}-q_{0}}}\bigg]^{-1}, \end{array}$

(3.34)

$\begin{array}{l} R_{28}(\Upsilon) = -\frac{1}{\delta^{2}}\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\bigg[-{\frac {q_{0}{e}^{q_{0} \left( \Upsilon+K \right) }}{q_{2}{e}^{q_{0} \left( \Upsilon+K \right) }-1}}\bigg]\\\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }}\bigg[-{\frac {q_{0}{e}^{q_{0} \left( \Upsilon+K \right) }}{q_{2}{e}^{q_{0} \left( \Upsilon+K \right) }-1}}\bigg]^{-1}, \end{array}$

(3.35)

$\begin{array}{l} R_{29}(\Upsilon) = -\frac{1}{\delta^{2}}\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\bigg[-{\frac {q_{2} \left( \Upsilon+K \right) +2}{q_{2} \left( \Upsilon+K \right) }}\bigg]\\\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }}\bigg[-{\frac {q_{2} \left( \Upsilon+K \right) +2}{q_{2} \left( \Upsilon+K \right) }}\bigg]^{-1}. \end{array}$

(3.36)

Set 3.5. $\kappa = \kappa, \; \varpi = -2\, {\frac { \left(1+ \left({q_{{1}}}^{2}-{q_{{2}}}^{2} \right) {\kappa}^{2}{\delta }^{2} \right) \left(-\frac{1}{2}+ \left({q_{{1}}} ^{2}-{q_{{2}}}^{2} \right) {\kappa}^{2}{\delta }^{2} \right) }{{\delta }^{2} }}, \; G_{0} = {q_{{1}}}^{2}{\kappa}^{2}-{q_{{2}}}^{2}{\kappa}^{2}, \; G_{{1}} = \mp\frac{1}{2}{\frac {\kappa\, \left(q_{{1}}-q_{{2}} \right) }{\delta }}, \\ H_{{1}} = \mp\frac{1}{2}{\frac {\kappa\, \left(q_{{1}}+q_{{2}} \right) }{\delta }}, \; q_{{0}} = \mp{\frac {1+ \left(2\, {q_{{1}}}^{2}-2\, {q_{{2}}}^{2} \right) { \delta }^{2}{\kappa}^{2}}{\kappa\, \delta }}, \; q_{{1}} = q_{{1}}, \; q_{{2}} = q_{2},$

$\begin{equation} R(\Upsilon) = G_{0}+G_{1} \bigg [\tan \bigg(\frac{\Omega(\Upsilon)}{2}\bigg)\bigg]+H_{1} \bigg [\tan \bigg(\frac{\Omega(\Upsilon)}{2}\bigg)\bigg]^{-1}, \end{equation}$

(3.37)

where $q_{0}$ , $q_{1}$ , $q_{2}$ are arbitrary constants.

Using Eq (3.37) and Families 2.1–2.3, 2.9, 2.10, 2.13 and 2.14, respectively, yields the following solutions:

$\begin{array}{l} R_{30}(\Upsilon) = {q_{{1}}}^{2}{\kappa}^{2}-{q_{{2}}}^{2}{\kappa}^{2}\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\bigg[{\frac {q_{0}}{q_{1}-q_{2}}}\\ -\sqrt {{\frac {-{q_{0}}^{2}-{q_{1}}^{2}+{q_{2}}^{2}}{q_{1}-q_{2}}}}\tan \left( \sqrt {\frac{-{q_{0}}^{2}-{q_{1}}^{2}+{q_{2}}^{2}}{2}}(\Upsilon+K) \right) \bigg]\\ \mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }}\\\bigg[{\frac {q_{0}}{q_{1}-q_{2}}}-\sqrt {{\frac {-{q_{0}}^{2} -{q_{1}}^{2}+{q_{2}}^{2}}{q_{1}-q_{2}}}}\tan \left( \sqrt {\frac{-{q_{0}}^{2}-{q_{1}}^{2}+{q_{2}}^{2}}{2}}(\Upsilon+K) \right) \bigg]^{-1}, \end{array}$

(3.38)

$\begin{array}{l} R_{31}(\Upsilon) = {q_{{1}}}^{2}{\kappa}^{2}-{q_{{2}}}^{2}{\kappa}^{2}\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\\\bigg[\frac{q_{0}}{q_{1}-q_{2}}+\sqrt{\frac{q_{1}^{2}+q_{0}^{2}-q_{2}^{2}}{q_{1}-q_{2}}}\tanh\bigg(\sqrt{\frac{q_{1}^{2}+q_{0}^{2}-q_{2}^{2}}{2}} (\Upsilon+K)\bigg)\bigg]\\ \mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }}\bigg[\frac{q_{0}}{q_{1}-q_{2}}+\sqrt{\frac{q_{1}^{2}+q_{0}^{2}-q_{2}^{2}}{q_{1}-q_{2}}}\tanh\bigg(\sqrt{\frac{q_{1}^{2}+q_{0}^{2}-q_{2}^{2}}{2}} (\Upsilon+K)\bigg)\bigg]^{-1}, \end{array}$

(3.39)

$\begin{align} R_{32}(\Upsilon)& = {q_{{1}}}^{2}{\kappa}^{2}-{q_{{2}}}^{2}{\kappa}^{2}\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\bigg[\frac{q_{0}}{q_{1}}+\frac{\sqrt{q_{1}^{2}+q_{0}^{2}}}{q_{1}}\tanh \bigg(\frac{\sqrt{q_{1}^{2}+q_{0}^{2}}}{2}(\Upsilon+K)\bigg)\bigg]\\ &\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }}\bigg[\frac{q_{0}}{q_{1}}+\frac{\sqrt{q_{1}^{2}+q_{0}^{2}}}{q_{1}}\tanh \bigg(\frac{\sqrt{q_{1}^{2}+q_{0}^{2}}}{2}(\Upsilon+K)\bigg)\bigg]^{-1}, \end{align}$

(3.40)

$\begin{align} R_{33}(\Upsilon)& = {q_{{1}}}^{2}{\kappa}^{2}-{q_{{2}}}^{2}{\kappa}^{2}\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\bigg[{e}^{\vartheta\,q_{0} \left( \Upsilon+K \right) }-1\bigg]\\ &\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }}\bigg[{e}^{\vartheta\,q_{0} \left( \Upsilon+K \right) }-1\bigg]^{-1}, \end{align}$

(3.41)

$\begin{align} R_{34}(\Upsilon)& = {q_{{1}}}^{2}{\kappa}^{2}-{q_{{2}}}^{2}{\kappa}^{2}\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\bigg[-{\frac {{e}^{\vartheta\,q_{0} \left( \Upsilon+K \right) }}{-1+{e}^{\vartheta\,q_{0} \left( \Upsilon+K \right) }}}\bigg]\\ &\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }}\bigg[-{\frac {{e}^{\vartheta\,q_{0} \left( \Upsilon+K \right) }}{-1+{e}^{\vartheta\,q_{0} \left( \Upsilon+K \right) }}}\bigg]^{-1}, \end{align}$

(3.42)

$\begin{align} R_{35}(\Upsilon)& = {q_{{1}}}^{2}{\kappa}^{2}-{q_{{2}}}^{2}{\kappa}^{2}\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\bigg[{\frac {{e}^{q_{1} \left( \Upsilon+K \right) }+q_{1}-q_{0}}{{e}^{q_{1} \left( \Upsilon+K \right) }-q_{1} -q_{0}}}\bigg]\\ &\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }}\bigg[{\frac {{e}^{q_{1} \left( \Upsilon+K \right) }+q_{1}-q_{0}}{{e}^{q_{1} \left( \Upsilon+K \right) }-q_{1}-q_{0}}}\bigg]^{-1}, \end{align}$

(3.43)

$\begin{align} R_{36}(\Upsilon)& = {q_{{1}}}^{2}{\kappa}^{2}-{q_{{2}}}^{2}{\kappa}^{2}\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}-q_{{2}} \right) }{\delta }}\bigg[-{\frac {q_{0}{e}^{q_{0} \left( \Upsilon+K \right) }}{q_{2}{e}^{q_{0} \left( \Upsilon+K \right) }-1}}\bigg]\\ &\mp\frac{1}{2}{\frac {\kappa\, \left( q_{{1}}+q_{{2}} \right) }{\delta }}\bigg[-{\frac {q_{0}{e}^{q_{0} \left( \Upsilon+K \right) }}{q_{2}{e}^{q_{0} \left( \Upsilon+K \right) }-1}}\bigg]^{-1}, \end{align}$

(3.44)

where $\Upsilon = \kappa(x-\varpi t).$

4. Graphical illustration

Some of the obtained soliton solutions are graphically represented in this section. Kink solitons, dark-bright solitons, bright solitons, singular solitons and periodic wave solutions are retrieved.

The $3$ D-graph and contour plot for the solution $R_{4}(\Upsilon)$ are shown in Figure 1. The solution $R_{4}(\Upsilon)$ is derived from Family $2.2$ of solution Set $2.2$ as defined by Eq $(3.9)$ . The obtained graphs show a kink soliton solution. Kink soliton is a type of solitary wave that ascend or descend from one asymptotic state to another. The contour graph is also included along with surface graph to illustrate the wave structure corresponding to the obtained solution.

Figure 1. This figure demonstrates the 3D graph and corresponding contour of Eq

$(3.9)$ at

$\kappa = 1, ~\delta = 1, ~q_{0} = -2.82, ~q_{1} = -3, ~q_{2} = 4, ~G_{0} = 0.914, ~G_{1} = 0, ~H_{1} = 1, ~K = -0.5, ~\varpi = 1$ .

DownLoad: Full-Size Img PowerPoint

The graphical illustration of $R_{23}(\Upsilon)$ is presented in Figure 2. The solution $R_{23}(\Upsilon)$ is given by Equation $(3.30)$ using the values of Set $2.4$ for Family $2.4$ . The graphs in Figure 2 show a bright soliton. The surface graph shows a localized intensity peak above the continuous wave background which means that there is a temporary increase in the wave amplitude.

Figure 2. This figure demonstrates the 3D graph and corresponding contour of Eq

$(3.30)$ at

$\kappa = 1, ~\delta = 2, ~q_{0} = 0.5, ~q_{1} = 0, ~q_{2} = 0.5, ~G_{0} = -0.25, ~G_{1} = -0.125, ~H_{1} = -0.125, ~K = 1, ~\varpi = 0, ~\vartheta = 1$ .

DownLoad: Full-Size Img PowerPoint

Figure 3 shows the $3$ D plot and the corresponding contour plot of $R_{20}(\Upsilon)$ given by Eq $(3.27)$ . The graph of $R_{12}(\Upsilon)$ given by Eq $(3.18)$ is illustrated in Figure 4. Figure 5 provides the graphical illustration of $R_{30}(\Upsilon)$ given by Eq $(3.38)$ . Figure 6 shows the graph of a dark-bright soliton which is graphical illustration of the solution $R_{3}(\Upsilon)$ expressed by Eq $(3.7)$ . Similarly, – presents the graphical illustrations for the solutions presented by Eq $(3.35)$ , Eq $(3.6)$ and Eq $(3.32)$ , respectively.

Figure 3. This figure demonstrates the 3D graph and corresponding contour of Eq

$(3.27)$ at

$\kappa = 1, ~\delta = 1, ~q_{0} = 1, ~q_{1} = 1, ~q_{2} = -3, ~G_{0} = -1, ~G_{1} = 2, ~H_{1} = 1, K = 1, ~\varpi = -7$ .

DownLoad: Full-Size Img PowerPoint

Figure 4. This figure demonstrates the 3D graph and corresponding contour of Eq

$(3.18)$ at

$\kappa = 1, ~\delta = 1, ~q_{0} = 2.828, ~q_{1} = -3, ~q_{2} = 4, ~G_{0} = 0.914, ~G_{1} = 3.5, ~H_{1} = 0, ~K = 1, ~\varpi = 1$ .

DownLoad: Full-Size Img PowerPoint

Figure 5. This figure demonstrates the 3D graph and corresponding contour of Eq

$(3.38)$ at

$\kappa = 1, ~\delta = 3, ~q_{0} = 0.206, ~q_{1} = 0, ~q_{2} = 0.3, ~G_{0} = -0.09, ~G_{1} = -0.05, ~H_{1} = -0.05, ~K = 1, ~\varpi = 0.055, ~\vartheta = 1$ .

DownLoad: Full-Size Img PowerPoint

Figure 6. This figure demonstrates the 3D graph and corresponding contour of Eq

$(3.7)$ at

$\kappa = 1, ~\delta = 1, ~q_{0} = 0, ~q_{1} = 0, ~q_{2} = 0.5I, ~G_{0} = -0.5, ~G_{1} = -0.25I, ~H_{1} = 0.25I, ~K = 1, ~\varpi = 1$ .

DownLoad: Full-Size Img PowerPoint

Figure 7. This figure demonstrates the 3D graph and corresponding contour of Eq

$(3.35)$ at

$\kappa = 1, \; \delta = 3, \; q_{0} = 0.33, \; q_{1} = 3, \; q_{2} = -3, \; G_{0} = -0.11, \; G_{1} = 1, \; H_{1} = 0, \; K = 1, \; \varpi = 0.11, \; \vartheta = 1$ .

DownLoad: Full-Size Img PowerPoint

Figure 8. This figure demonstrates the 3D graph and corresponding contour of Eq

$(3.6)$ at

$\kappa = 1, ~\delta = 1, ~q_{0} = 0, ~q_{1} = 2.5, ~q_{2} = 2.449, ~G_{0} = -0.5, ~G_{1} = 0.025,$

$~H_{1} = 2.474, ~K = 1, ~\varpi = 1$ .

DownLoad: Full-Size Img PowerPoint

Figure 9. This figure demonstrates the 3D graph and corresponding contour of Eq

$(3.32)$ at

$\kappa = 1, \; \delta = 1, \; q_{0} = 1, \; q_{1} = 1, \; q_{2} = 1, \; G_{0} = -1, \; G_{1} = 0, \; H_{1} = -1, \; K = 1, \; \varpi = 1, \; \vartheta = 1$ .

DownLoad: Full-Size Img PowerPoint

It can be easily observed that improved tan $\left(\frac{\Omega(\Upsilon)}{2}\right)$ technique is a spectacular technique as compared to many other direct techniques as it gave abundant soliton solutions. Using this technique, kink, singular, bright and dark-bright soliton solutions have been retrieved in this paper. This method is clearly more powerful than many other methods, such as: the tanh-method ^[34], the $\frac{G'}{G}$ expansion method ^[35] and the generalized exponential rational function method, the Jacobi elliptic solution method ^[36], because the improved tan $\left(\frac{\Omega(\Upsilon)}{2}\right)$ method retrieved many more new solutions than the previously mentioned techniques.

5. Conclusions

In this study, the soliton and other solitary wave solutions of the constant-coefficient Gardner equation are investigated using tan $\left(\frac{\Omega(\Upsilon)}{2}\right)$ -expansion method. A variety of precise closed form traveling wave solutions have been constructed including bright solitons, dark-bright solitons, kink solitons and periodic wave solutions. Some of the obtained solutions are illustrated using graphical simulations for suitable choice of parameters. The wave profile corresponding to the obtained solutions is depicted through 3D-surface graphs and corresponding 2D-contour plots. Comparison of the obtained results with those available in the literature depict the efficacy and productivity of the improved tan $\left(\frac{\Omega(\Upsilon)}{2}\right)$ technique. Mathematical computations and simulations were obtained using Maple software. The reported results may be helpful in further explorations of the nonlinear physical problems governed by the Gardner equation in fluid dynamics, plasma physics and other fields. The improved tan $\left(\frac{\Omega(\Upsilon)}{2}\right)$ technique will be useful for the analytic study of a large class of nonlinear PDEs that are widely used in engineering, physics and other sciences.

Conflict of interest

The authors declare no conflicts of interest.

References

[1]	S. Grodski, T. Brown, S. Sidhu, A. Gill, B. Robinson, D. Learoyd, et al., Increasing incidence of thyroid cancer is due to increased pathologic detection, Surgery, 144 (2008), 1038–1043.
[2]	J. Kim, J. E. Gosnell, S. A. Roman, Geographic influences in the global rise of thyroid cancer, Nat. Rev. Endocrinol., 16 (2020), 17–29.
[3]	H. Sung, J. Ferlay, R. L. Siegel, M. Laversanne, I. Soerjomataram, A. Jemal, Global cancer statistics 2020: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries, CA: Cancer J. Clin., 71 (2021), 209–249. https://doi.org/10.3322/caac.21660 doi: 10.3322/caac.21660
[4]	L. Enewold, K. Zhu, E. Ron, A. J. Marrogi, A. Stojadinovic, G. E. Peoples, et al., Rising thyroid cancer incidence in the United States by demographic and tumor characteristics, 1980–2005, Cancer Epidem. Biomar., 18 (2009), 784–791. https://doi.org/10.1109/JMEMS.2009.2023841 doi: 10.1109/JMEMS.2009.2023841
[5]	L. Davies, H. G. Welch, Current thyroid cancer trends in the United States, JAMA Otolaryngology-Head Neck Surgery, 140 (2014), 317. https://doi.org/10.1016/j.neucom.2014.03.007 doi: 10.1016/j.neucom.2014.03.007
[6]	P. B. Manoj, A. Innisai, D. S. Hameed, A. Khader, M. Gopanraj, N. H. Ihare, Correlation of high-resolution ultrasonography findings of thyroid nodules with ultrasound-guided fine-needle aspiration cytology in detecting malignant nodules: A retrospective study in Malabar region of Kerala, South India, J. Fam. Med. Prim. Care, 8 (2019), 1613.
[7]	H. Tan, Z. Li, N. Li, J. Qian, F. Fan, H. Zhong, et al., Thyroid imaging reporting and data system combined with Bethesda classification in qualitative thyroid nodule diagnosis, Medicine, 98 (2019), 2019.
[8]	A. N. Rajalakshmi, F. Begam, Thyroid Hormones in the Human Body: A review, J. Drug Delivery Ther., 11 (2021), 178–182. https://doi.org/10.22270/jddt.v11i5.5039 doi: 10.22270/jddt.v11i5.5039
[9]	A. K. Lee, P. M. A. Tacanay, P. Siy, D. T. Argamosa, Ectopic papillary thyroid carcinoma presenting as right lateral neck mass, JAFES, 37 (2022), 2022.
[10]	M. I. Larg, D. Apostu, C. Peștean, K. Gabora, I. C. Bădulescu, E. Olariu, et al., Evaluation of malignancy risk in 18F-FDG PET/CT thyroid incidentalomas, Diagnostics, 9 (2019), 92. https://doi.org/10.3390/diagnostics9030092 doi: 10.3390/diagnostics9030092
[11]	M. Hanan, E. Fatma, A. Aly, A. Medhat, Evaluation of Incidental Thyroid Findings Detected by Positron Emission Tomography/Computed Tomography, Medical J. Cairo University, 87 (2019), 819–826. https://doi.org/10.21608/mjcu.2019.52541 doi: 10.21608/mjcu.2019.52541
[12]	S. Quazi, Artificial intelligence and machine learning in precision and genomic medicine, Med. Oncol., 39 (2022), 120.
[13]	K. Preuss, N. Thach, X. Liang, M. Baine, J. Chen, C. Zhang, et al., Using quantitative imaging for personalized medicine in pancreatic cancer: a review of radiomics and deep learning applications, Cancers, 14 (2022), 1654. https://doi.org/10.3390/cancers14071654 doi: 10.3390/cancers14071654
[14]	N. Shusharina, D. Yukhnenko, S. Botman, V. Sapunov, V. Savinov, G. Kamyshov, et al., Modern methods of diagnostics and treatment of neurodegenerative diseases and depression, Diagnostics, 13 (2023), 573. https://doi.org/10.3390/diagnostics13030573 doi: 10.3390/diagnostics13030573
[15]	S. Khalil, U. Nawaz, Zubariah, Z. Mushtaq, S. Arif, M. Z. ur Rehman, et al., Enhancing ductal carcinoma Classification using transfer learning with 3D U-Net models in breast cancer imaging, Appl. Sci., 13 (2023), 4255.
[16]	A. M. Antoniadi, Y. Du, Y. Guendouz, L. Wei, C. Mazo, B. A. Becker, et al., Current challenges and future opportunities for XAI in machine learning-based clinical decision support systems: A systematic review, Appl. Sci., 11 (2021), 5088.
[17]	Z. Mushtaq, M. F. Qureshi, M. J. Abbass, S. M. Q. AlFakih, Effective kernelprincipal component analysis based approach for wisconsin breast cancer diagnosis, Electron. Lett., 59 (2023).
[18]	X. M. Keutgen, H. Li, K. Memeh, J. Conn Busch, J. Williams, L. Lan, D. Sarne, et al., A machine-learning algorithm for distinguishing malignant from benign indeterminate thyroid nodules using ultrasound radiomic features, J. Med. Imaging, 9 (2022), 034501–034501.
[19]	V. V. Vadhiraj, A. Simpkin, J. O'Connell, N. Singh Ospina, S. Maraka, D. T. O'Keeffe, Ultrasound image classification of thyroid nodules using machine learning techniques, Medicina, 57 (2021), 527. https://doi.org/10.3390/medicina57060527 doi: 10.3390/medicina57060527
[20]	M. Bereby-Kahane, R. Dautry, E. Matzner-Lober, F. Cornelis, D. Sebbag-Sfez, V. Place, et al., Prediction of tumor grade and lymphovascular space invasion in endometrial adenocarcinoma with MR imaging-based radiomic analysis, Diagn. Interv. Imag., 101 (2020), 401–411.
[21]	K. E. Fasmer, E. Hodneland, J. A. Dybvik, K. Wagner-Larsen, J. Trovik, A. Salvesen, et al., Whole-volume tumor MRI radiomics for prognostic modeling in endometrial cancer, J. Magn. Reson. Imaging, 53 (2021), 928–937.
[22]	A. Prete, P. Borges de Souza, S. Censi, M. Muzza, N. Nucci, M. Sponziello, Update on fundamental mechanisms of thyroid cancer, Front. Endocrinol., 11 (2020), 102.
[23]	N. Pozdeyev, M. M. Rose, D. W. Bowles, R. E. Schweppe, Molecular therapeutics for anaplastic thyroid cancer, In: Seminars in Cancer Biology, 61 (2020), 23–29. https://doi.org/10.1016/j.semcancer.2020.01.005
[24]	Y. C. Zhu, P. F. Jin, J. Bao, Q. Jiang, X. Wang, Thyroid ultrasound image classification using a convolutional neural network, Ann. Transl. Med., 9 (2021).
[25]	M. R. Kwon, J. H. Shin, H. Park, H. Cho, S. Y. Hahn, K. W. Park, Radiomics study of thyroid ultrasound for predicting BRAF mutation in papillary thyroid carcinoma: Preliminary results, Am. J. Neuroradiol., 41 (2020), 700–705. https://doi.org/10.3174/ajnr.A6505 doi: 10.3174/ajnr.A6505
[26]	Y. Wang, W. Yue, X. Li, S. Liu, L. Guo, H. Xu, et al., Comparison study of radiomics and deep learning-based methods for thyroid nodules classification using ultrasound images, Ieee Access, 8 (2020), 52010–52017.
[27]	D. Chen, J. Hu, M. Zhu, N. Tang, Y. Yang, Y. Feng, Diagnosis of thyroid nodules for ultrasonographic characteristics indicative of malignancy using random forest, BioData Min., 13 (2020), 1–21.
[28]	H. K. Shivastuti, J. Manhas, V. Sharma, Performance evaluation of SVM and random forest for the diagnosis of thyroid disorder, Int. J. Res. Appl. Sci. Eng. Technol., 9 (2021), 945–947.
[29]	H. Abbad Ur Rehman, C. Y. Lin, Z. Mushtaq, Effective K-nearest neighbor algorithms performance analysis of thyroid disease, J. Chin. Inst. Eng., 44 (2021), 77–87. https://doi.org/10.14358/PERS.87.2.77 doi: 10.14358/PERS.87.2.77
[30]	T. Akhtar, S. O. Gilani, Z. Mushtaq, S. Arif, M. Jamil, Y. Ayaz, et al., Effective voting ensemble of homogenous ensembling with multiple attribute-selection approaches for improved identification of thyroid disorder, Electronics, 10 (2021), 3026.
[31]	L. C. Zhu, Y. L. Ye, W. H. Luo, M. Su, H. P. Wei, X. B. Zhang, et al., A model to discriminate malignant from benign thyroid nodules using artificial neural network, PLoS One, 8 (2013), e82211. https://doi.org/10.1371/journal.pone.0082211 doi: 10.1371/journal.pone.0082211
[32]	B. Zhang, J. Tian, S. Pei, Y. Chen, X. He, Y. Dong, et al., Machine learning–assisted system for thyroid nodule diagnosis, Thyroid, 29 (2019), 858–867. https://doi.org/10.1089/thy.2018.0380 doi: 10.1089/thy.2018.0380
[33]	A. K. Singh, A comparative study on disease classification using machine learning algorithms, In Proceedings of 2nd International Conference on Advanced Computing and Software Engineering (ICACSE), 2019.
[34]	E. Sonuç, Thyroid disease classification using machine learning algorithms, In: Journal of Physics: Conference Series, vol. 1963, p. 012140, IOP Publishing, 2021. https://doi.org/10.1088/1742-6596/1963/1/012140
[35]	P. Poudel, A. Illanes, E. J. Ataide, N. Esmaeili, S. Balakrishnan, M. Friebe, Thyroid ultrasound texture classification using autoregressive features in conjunction with machine learning approaches, IEEE Access, 7 (2019), 79354–79365. https://doi.org/10.1109/ACCESS.2019.2923547 doi: 10.1109/ACCESS.2019.2923547
[36]	D. C. Yadav, S. Pal, Thyroid prediction using ensemble data mining techniques, Int. J. Inf. Technol., 14 (2022), 1273–1283.
[37]	S. S. Z. Mousavi, M. M. Zanjireh, M. Oghbaie, Applying computational classification methods to diagnose Congenital Hypothyroidism: A comparative study, Inf. Medicine Unlocked, 18 (2020), 100281.
[38]	D. T. Nguyen, J. K. Kang, T. D. Pham, G. Batchuluun, K. R. Park, Ultrasound image-based diagnosis of malignant thyroid nodule using artificial intelligence, Sensors, 20 (2020), 1822. https://doi.org/10.3390/s20071822 doi: 10.3390/s20071822
[39]	G. Chaubey, D. Bisen, S. Arjaria, V. Yadav, Thyroid disease prediction using machine learning approaches, Natl. Acad. Sci. Lett., 44 (2021), 233–238.
[40]	M. Garcia de Lomana, A. G. Weber, B. Birk, R. Landsiedel, J. Achenbach, K. J. Schleifer, et al., In silico models to predict the perturbation of molecular initiating events related to thyroid hormone homeostasis, Chem. Res. Toxicol., 34 (2020), 396–411.
[41]	K. Shankar, S. K. Lakshmanaprabu, D. Gupta, A. Maseleno, V. H. C. De Albuquerque, Optimal feature-based multi-kernel SVM approach for thyroid disease classification, J. Supercomput., 76 (2020), 1128–1143.
[42]	H. Abbad Ur Rehman, C. Y. Lin, Z. Mushtaq, S. F. Su, Performance analysis of machine learning algorithms for thyroid disease, Arab. J. Sci. Eng., 1–13, 2021.
[43]	R. Das, S. Saraswat, D. Chandel, S. Karan, J. S. Kirar, An AI Driven Approach for Multiclass Hypothyroidism Classification, In: Advanced Network Technologies and Intelligent Computing: First International Conference, ANTIC 2021, Varanasi, India, December 17–18, 2021, Proceedings, pp. 319–327, Springer, 2022.
[44]	M. Hosseinzadeh, O. H. Ahmed, M. Y. Ghafour, F. Safara, H. K. Hama, S. Ali, et al., A multiple multilayer perceptron neural network with an adaptive learning algorithm for thyroid disease diagnosis in the internet of medical things, J. Supercomput., 77 (2021), 3616–3637.
[45]	M. Riajuliislam, K. Z. Rahim, A. Mahmud, Prediction of Thyroid Disease (Hypothyroid) in Early Stage Using Feature Selection and Classification Techniques, In: 2021 International Conference on Information and Communication Technology for Sustainable Development (ICICT4SD), pp. 60–64, IEEE, 2021.
[46]	R. Jha, V. Bhattacharjee, A. Mustafi, Increasing the prediction accuracy for thyroid disease: A step towards better health for society, Wireless Pers. Commun., 122 (2022), 1921–1938. https://doi.org/10.1155/2022/9809932 doi: 10.1155/2022/9809932
[47]	T. Alyas, M. Hamid, K. Alissa, T. Faiz, N. Tabassum, A. Ahmad, Empirical method for thyroid disease classification using a machine learning approach, BioMed Res. Int., 22 (2022).
[48]	S. Sankar, A. Potti, G. N. Chandrika, S. Ramasubbareddy, Thyroid disease prediction using XGBoost algorithms, J. Mob. Multimed, 18 (2022), 1–18.
[49]	I. Ali, Z. Mushtaq, S. Arif, A. Algarni, N. Soliman, W. El-Shafai, Hyperspectral images-based crop classification scheme for agricultural remote sensing, Comput. Syst. Sci. Eng., 46 (2023), 303–319.
[50]	S. Arif, S. Munawar, H. Ali, Driving drowsiness detection using spectral signatures of EEG-based neurophysiology, Front. Physiol., 14 (2023), 1153268.
[51]	S. Arif, M. Arif, S. Munawar, Y. Ayaz, M. J. Khan, N. Naseer, EEG spectral comparison between occipital and prefrontal cortices for early detection of driver drowsiness, In: 2021 International Conference on Artificial Intelligence and Mechatronics Systems (AIMS), pp. 1–6, IEEE, 2021.
[52]	S. Arif, M. J. Khan, N. Naseer, K. S. Hong, H. Sajid, Y. Ayaz, Vector phase analysis approach for sleep stage classification: A functional near-infrared spectroscopy-based passive brain–computer interface, Front. Hum. Neurosci., 15 (2021), 658444.
[53]	T. Akhtar, S. Arif, Z. Mushtaq, S. O. Gilani, M. Jamil, Y. Ayaz, et al., Ensemble-based effective diagnosis of thyroid disorder with various feature selection techniques, In: 2022 2nd International Conference of Smart Systems and Emerging Technologies (SMARTTECH), pp. 14–19, IEEE, 2022.
[54]	K. Chandel, V. Kunwar, S. Sabitha, T. Choudhury, S. Mukherjee, A comparative study on thyroid disease detection using K-nearest neighbor and Naive Bayes classification techniques, CSI Transactions ICT, 4 (2016), 313–319. https://doi.org/10.1111/twec.13285 doi: 10.1111/twec.13285
[55]	R. Pal, T. Anand, S. K. Dubey, Evaluation and performance analysis of classification techniques for thyroid detection, Int. J. Bus. Inf. Syst., 28 (2018), 163–177.
[56]	M. Saktheeswari, T. Balasubramanian, Multi-layer tree liquid state machine recurrent auto encoder for thyroid detection, Multimed. Tools Appl., 80 (2021), 17773–17783. https://doi.org/10.1007/s11042-020-10243-7 doi: 10.1007/s11042-020-10243-7
[57]	A. Tyagi, R. Mehra, A. Saxena, Interactive Thyroid Disease Prediction System Using Machine Learning Technique, In: 2018 Fifth International Conference on Parallel, Distributed and Grid Computing (PDGC), (Solan Himachal Pradesh, India), pp. 689–693, IEEE, Dec. 2018.
[58]	S. Mishra, Y. Tadesse, A. Dash, L. Jena, P. Ranjan, Thyroid Disorder Analysis Using Random Forest Classifier, In: Intelligent and Cloud Computing (D. Mishra, R. Buyya, P. Mohapatra, and S. Patnaik, eds.), Smart Innovation, Systems and Technologies, (Singapore), pp. 385–390, Springer, 2021.
[59]	K. Guleria, S. Sharma, S. Kumar, S. Tiwari, Early prediction of hypothyroidism and multiclass classification using predictive machine learning and deep learning, Measurement: Sensors, 24 (2022), 100482. https://doi.org/10.1016/j.measen.2022.100482 doi: 10.1016/j.measen.2022.100482
[60]	H. Zhang, C. Li, D. Li, Y. Zhang, W. Peng, Fault detection and diagnosis of the air handling unit via an enhanced kernel slow feature analysis approach considering the time-wise and batch-wise dynamics, Energ. Buildings, 253 (2021), 111467. https://doi.org/10.1016/j.enbuild.2021.111467 doi: 10.1016/j.enbuild.2021.111467
[61]	H. Zhang, W. Yang, W. Yi, J. B. Lim, Z. An, C. Li, Imbalanced data based fault diagnosis of the chiller via integrating a new resampling technique with an improved ensemble extreme learning machine, J. Build. Eng., 70 (2023), 106338. https://doi.org/10.1016/j.jobe.2023.106338 doi: 10.1016/j.jobe.2023.106338
[62]	H. Zhang, C. Li, Q. Wei, Y. Zhang, Fault detection and diagnosis of the air handling unit via combining the feature sparse representation based dynamic SFA and the LSTM network, Energ. Buildings, 269 (2022), 112241.

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