An improved dung beetle optimizer based on Padé approximation strategy for global optimization and feature selection

Tianbao Liu; Lingling Yang; Yue Li; Xiwen Qin; Tianbao Liu; Lingling Yang; Yue Li; Xiwen Qin

doi:10.3934/era.2025079

Electronic Research Archive

2025, Volume 33, Issue 3: 1693-1762. doi: 10.3934/era.2025079

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Research article

An improved dung beetle optimizer based on Padé approximation strategy for global optimization and feature selection

School of Mathematics and Statistics, Changchun University of Technology, Changchun 130012, China

Received: 18 December 2024 Revised: 19 February 2025 Accepted: 10 March 2025 Published: 25 March 2025

Feature selection is a crucial data processing method used to reduce dataset dimensionality while preserving key information. In this paper, we proposed a multi-strategy enhanced dung beetle optimization algorithm (mDBO) that integrates multiple strategies to effectively address the feature selection problem. First, a novel population initialization strategy based on a hybrid tent-sine map and random opposition-based learning was proposed to generate initial population. This strategy yielded a more uniform distribution of the initial population, significantly improving the quality of the population distribution within the search space. Second, a new differential evolution mutation strategy with a periodic retrospective adaptive mutation factor was proposed. This strategy effectively improved the algorithm's ability to jump out of the local optimal and explore potential candidate solutions. Third, based on Padé approximation technology and the novel adaptive evolutionary boundary constraint method, an innovative approximation strategy was proposed. The strategy was integrated into the framework of the dung beetle optimizer, significantly improving the solution accuracy and population quality of the algorithm. Finally, the binary version of the mDBO algorithm (bmDBO) was applied to feature selection tasks. Experiments entailing CEC2017 benchmark functions and 17 datasets showed that both mDBO and bmDBO outperformed other algorithms. The mDBO method outperformed other algorithms in 11 of the 29 benchmark functions, ranked second in 8 functions, and achieved an average rank of 1.62 in the Friedman ranking, securing the overall first place; the bmDBO method outperformed in 12 of 17 datasets, achieving an average ranking of 1.35 in the Friedman ranking, securing the first position.

Keywords:

feature selection,
dung beetle optimization algorithm,
differential evolution,
Padé approximation,
adaptive evolutionary boundary constraint handling

Citation: Tianbao Liu, Lingling Yang, Yue Li, Xiwen Qin. An improved dung beetle optimizer based on Padé approximation strategy for global optimization and feature selection[J]. Electronic Research Archive, 2025, 33(3): 1693-1762. doi: 10.3934/era.2025079

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Abstract

1. Introduction

In various fields of science, nonlinear evolution equations practically model many natural, biological and engineering processes. For example, PDEs are very popular and are used in physics to study traveling wave solutions. They have played a crucial role in illustrating the nature of nonlinear problems. PDEs are collected to control the diffusion of chemical reactions. In biology, they play a fundamental role in describing various phenomena, such as population growth. In addition, natural phenomena such as fluid dynamics, plasma physics, optics and optical fibers, electromagnetism, quantum mechanics, ocean waves, and others are studied using PDEs. The qualitative and quantitative characteristics of these phenomena can be identified from the behaviors and shapes of their solutions. Therefore, finding the analytic solutions to such phenomena is a fundamental topic in mathematics. Scientists have developed sparse fundamental approaches to find analytic solutions for nonlinear PDEs. Among these techniques, I present integration methods from ^[1] and ^[2], the modified F-expansion and Generalized Algebraic methods, respectively. Bekir and Unsal ^[3] proposed the first integral method to find the analytical solution of nonlinear equations. Kumar, Seadawy and Joardar ^[4] used the improved Kudryashov technique to extract fractional differential equations. Adomian ^[5] proposed the Adomian decomposition technique to find the solution of frontier problems of physics. ^[6] uses an exploratory method to find explicit solutions of non-linear PDEs. Many different methods of solving equations arising from natural phenomena and some of their analytic solutions, such as dark and light solitons, non-local rogue waves, an occasional wave and mixed soliton solutions, are exhibited and can be found in ^{[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]}.

The Novikov-Veselov (NV) system ^[41,42] is given by

$\begin{equation} \begin{aligned} \Psi_t& +\alpha \Gamma_{xxy}+\beta \Phi_{xyy}+\gamma \Gamma_y \Phi+\gamma \left(\dfrac{\Psi^2}{2}\right)_y+\lambda\, \Gamma\, \Phi_x+\lambda\, \left(\dfrac{\Psi^2}{2}\right)_x = 0, \\ \Gamma_y& = \Psi_x, \\ \Psi_y& = \Phi_x, \\ \end{aligned} \end{equation}$

(1.1)

where $\alpha, \, \beta, \, \gamma$ and $\lambda$ are constants. Barman ^[42] declared that Eq (1.1) is involved to represent tidal and tsunami waves, electro-magnetic waves in communication cables and magneto-sound and ion waves in plasma. In ^[42], the generalized Kudryashov method was utilized to have traveling wave solutions for Eq (1.1). According to Croke ^[43], the Novikov-Veselov system is generalized from the KdV equations which were examined by Novikov and Veselov. Croke ^[43] used several approaches, (the extended mapping, the Hirota and the extended tanh-function approaches) in the proposed system to achieve numerous soliton solutions, such as breathers, and constrained analytic solutions. Boiti, Leon, and Manna ^[44] applied the inverse dispersion technique to solve (1.1) for a particular type of initial value. Numerical solutions and a study of the stability of solutions for the proposed equation were presented by Kazeykina and Klein ^[45]. The Nizhnik-Novikov-Veselov system for two dimensions was also solved using the Kansa technique to find the numerical results ^[46]. To the best of my knowledge, the stability and error analysis of the numerical scheme presented here has not yet been discussed for system (1.1). Therefore, this has motivated me enormously to do so. The primary purpose is to obtain multiple analytic solutions to system (1.1) by using both the modified F-expansion and Generalized Algebraic methods. In connection with the numerical solution, the method of finite differences is utilized to achieve numerical results for the studied system. I graphically and analytically compare the traveling wave solutions and numerical results. Undoubtedly, the presented results strongly contribute to describing physical problems in practice.

The outline of this article is provided in this paragraph. Section 2 summarizes the employed methods. All the analytic solutions are extracted in Section 3. The shooting and BVP results for the proposed system are presented in Section 4. In addition, I examine the numerical solution of the system (1.1) in Section 5. Sections 6 and 7 study the stability and error analysis of the numerical scheme, respectively. Section 8 presents the results and discussion.

2. Summary of proposed methods

Considering the development equation with physical fields $\Psi(x, y, t)$ , $\Phi(x, y, t)$ and $\Gamma(x, y, t)$ in the variables $x,$ $y$ and $t$ is given in the following form:

$\begin{equation} Q_1(\Psi, \Psi_t, \Psi_x, \Psi_y, \Gamma, \Gamma_y, \Gamma_{xxy}, \Phi, \Phi_{xyy}, \Phi_{x}, \dots) = 0. \end{equation}$

(2.1)

Step 1. We extract the traveling-wave solutions of System (1.1) that are formed as follows:

$\begin{equation} \begin{aligned} \Phi(x, y, t) = &\phi(\eta) , &&\qquad&\eta = x+y-wt, \\ \Psi(x, y, t) = &\psi(\eta), \\ \Gamma(x, y, t) = &\Theta(\eta), \end{aligned} \end{equation}$

(2.2)

where $w$ is the wave speed.

Step 2. The nonlinear evolution (2.1) is reduced to the following ODE:

$\begin{equation} Q_2(\psi, \psi_\eta, \Theta, \Theta_\eta, \Theta_{\eta\eta\eta}, \phi, \phi_{\eta\eta\eta}, \phi_{\eta}, \dots) = 0, \end{equation}$

(2.3)

where $Q_2$ is a polynomial in $\psi(\eta), \; \phi(\eta),$ $\Theta(\eta)$ and their total derivatives.

2.1. The modified F-expansion method

According to the modified F-expansion method, the solutions of (2.3) are given by the form

$\begin{equation} \psi(\eta) = \rho_0+\sum\limits_{k = 1}^{N} \left( \rho_{k} F(\eta)^{k}+\dfrac{ q_{k}}{F(\eta)^{k}} \right) , \end{equation}$

(2.4)

and $F(\eta)$ is a solution of the following differential equation:

$\begin{equation} F^{\prime}(\eta) = \mu_0+\mu_1 F(\eta)+\mu_2 F(\eta)^{2}, \end{equation}$

(2.5)

where $\mu_0$ , $\mu_1$ , $\mu_2$ , are given in ^[1], and $\rho_k$ , $q_k$ are to be determined later.

Table 1. The relations among

$\mu_0$ ,

$\mu_1$ ,

$\mu_2$ and the function

$F(\eta)$ .

$\mu_0$	$\mu_{1}$	$\mu_{2}$	$F(\eta)$
$\mu_0=0,$	$\mu_1=1,$	$\mu_2=-1,$	$F(\eta)=\dfrac{1}{2}+\dfrac{1}{2} \tanh \left(\dfrac{1}{2} \eta\right).$
$\mu_0=0,$	$\mu_1=-1,$	$\mu_2=1,$	$F(\eta)=\dfrac{1}{2}-\dfrac{1}{2} \operatorname{coth}\left(\dfrac{1}{2} \eta\right).$
$\mu_0=\dfrac{1}{2},$	$\mu_1=0,$	$\mu_2=\dfrac{-1}{2},$	$F(\eta)=\operatorname{coth}(\eta) \pm \operatorname{csch}(\eta),$ $\tanh (\eta) \pm \operatorname{sech}(\eta).$
$\mu_0=1,$	$\mu_1=0,$	$\mu_2=-1,$	$F(\eta)=\tanh (\eta),$ $\operatorname{coth}(\eta).$
$\mu_0=\dfrac{1}{2},$	$\mu_1=0,$	$\mu_2=\dfrac{1}{2},$	$F(\eta)=\sec (\eta)+\tan (\eta),$ $\csc (\eta)-\cot (\eta).$
$\mu_0=\dfrac{1}{2},$	$\mu_1=0,$	$\mu_2=\dfrac{1}{2},$	$F(\eta)=\sec (\eta)-\tan (\eta),$ $\csc (\eta)+\cot (\eta).$
$\mu_0=\pm1,$	$\mu_1=0,$	$\mu_2=\pm1,$	$F(\eta)=\tan (\eta),$ $\cot (\eta).$

| Show Table

DownLoad: CSV

2.2. The generalized algebraic method

According to the generalized direct algebraic method, the solutions of (2.3) are given by

$\begin{equation} \psi(\eta) = \nu_0+\sum\limits_{k = 1}^{N} \left( \nu_{k} G(\eta)^{k} +\dfrac{ r_{k}}{G(\eta) ^{k}} \right) , \end{equation}$

(2.6)

and $G(\eta)$ is a solution of the following differential equation:

$\begin{equation} G^{\prime}(\eta) = \varepsilon \sqrt{\sum\limits_{k = 0}^{4} \delta_{k} G^{k}(\eta)}, \end{equation}$

(2.7)

where $\nu_k,$ and $r_k$ are to be determined, and $N$ is an integer number obtained by the highest degree of the nonlinear terms and the highest order of the derivatives. $\varepsilon$ is user-specified, usually taken with $\varepsilon = \pm1$ , and $\delta_k$ , $k = 0, 1, 2, 3, 4,$ are given in Table 2 ^[2].

Table 2. The relations among

$\delta_k$ ,

$k = 0, 1, 2, 3, 4,$ and the function

$G(\eta)$ .

$\delta_{0}$	$\delta_{1}$	$\delta_{2}$	$\delta_{3}$	$\delta_{4}$	$G(\eta)$
$\delta_{0}=0$ ,	$\delta_{1}=0$ ,	$\delta_{2} > 0$ ,	$\delta_{3}=0$ ,	$\delta_{4} < 0$ ,	$G(\eta)= \varepsilon$ $\sqrt{-\dfrac{\delta_{2}}{\delta_{4}}} \operatorname{sech}\left(\sqrt{\delta_{2}} \eta\right)$ .
$\delta_{0}=\dfrac{\delta_{2}^{2}}{4 c 4}$ ,	$\delta_{1}=0$ ,	$\delta_{2} < 0$ ,	$\delta_{3}=0$ ,	$\delta_{4} > 0$ ,	$G(\eta)=\varepsilon \sqrt{-\dfrac{\delta_{2}}{2 \delta_{4}}} \tanh \left(\sqrt{-\dfrac{\delta_{2}}{2}} \eta\right)$ .
$\delta_{0}=0$ ,	$\delta_{1}=0$ ,	$\delta_{2} < 0$ ,	$\delta_{3}=0$ ,	$\delta_{4} > 0$ ,	$G(\eta)=\varepsilon$ $\sqrt{-\dfrac{\delta_{2}}{\delta_{4}}}$ $\sec$ $\left(\sqrt{-\delta_{2}} \eta\right)$ .
$\delta_{0}=\dfrac{\delta_{2}^{2}}{4 \delta_{4}},$	$\delta_{1}=0$ ,	$\delta_{2} > 0$ ,	$\delta_{3}=0,$	$\delta_{4} > 0,$	$G(\eta)=\varepsilon$ $\sqrt{\dfrac{\delta_{2}}{2 \delta_{4}}} \tan \left(\sqrt{\dfrac{\delta_{2}}{2}} \eta\right)$ .
$\delta_{0}=0,$	$\delta_{1}=0,$	$\delta_{2}=0,$	$\delta_{3}=0,$	$\delta_{4} > 0,$	$G(\eta)=-\dfrac{\varepsilon}{\sqrt{\delta_{4}} \eta}$ .
$\delta_{0}=0,$	$\delta_{1}=0,$	$\delta_{2} > 0,$	$\delta_{3}\neq 0,$	$\delta_{4}=0,$	$G(\eta)=-\dfrac{\delta_{2}}{\delta_{3}}. \operatorname{sech}^{2}\left(\dfrac{\sqrt{\delta_{2}}}{2} \eta\right)$ .

| Show Table

DownLoad: CSV

3. Methodology

Consider the Novikov-Veselov (NV) system

(3.1)

a system of PDEs in the unknown functions $\Psi = \Psi(x, y, t), \; \Phi = \Phi(x, y, t)$ , $\Gamma = \Gamma(x, y, t)$ and their partial derivatives. I plug the transformations

$\begin{equation} \begin{aligned} \Phi(x, y, t) = &\phi(\eta) , &&\qquad&\eta = x+y-wt, \\ \Psi(x, y, t) = &\psi(\eta), \\ \Gamma(x, y, t) = &\Theta(\eta), \end{aligned} \end{equation}$

(3.2)

into Eq (3.1) to reduce it to a system of ODEs given by

$\begin{equation} \begin{aligned} -w\; \psi_\eta& +\alpha \Theta_{\eta\eta\eta}+\beta \phi_{\eta\eta\eta}+\gamma \Theta_\eta \phi+\gamma \left(\dfrac{\psi^2}{2}\right)_\eta+\lambda\, \Theta\, \phi_\eta+\lambda\, \left(\dfrac{\psi^2}{2}\right)_\eta = 0, \\ \Theta_\eta& = \psi_\eta, \\ \psi_\eta& = \phi_\eta.\\ \end{aligned} \end{equation}$

(3.3)

Integrating $\Theta_\eta = \psi_\eta$ and $\phi_\eta = \psi_\eta$ yields

$\begin{equation} \Theta = \psi, \qquad \text{and } \qquad \phi = \psi. \end{equation}$

(3.4)

Substituting (3.4) into the first equation of (3.3) and integrating once with respect to $\eta$ yields

$\begin{equation} \begin{aligned} -w\; \psi& +\left( \alpha +\beta\right) \psi_{\eta\eta}+\left( \gamma+\lambda\right) \psi^2 = 0. \end{aligned} \end{equation}$

(3.5)

Balancing $\psi_{\eta\eta}$ with $\psi^2$ in (3.5) calculates the value of $N = 2.$

3.1. Application of modified F-expansion method

According to the modified F-expansion method with $N = 2$ , the solutions of (3.5) are

$\begin{equation} \psi(\eta) = \rho_0+ \rho_{1} F(\eta)+\dfrac{ q_{1}}{F(\eta)}+ \rho_{2} F(\eta)^{2}+\dfrac{ q_{2}}{F(\eta)^{2}} , \end{equation}$

(3.6)

and $F(\eta)$ is a solution of the following differential equation:

$\begin{equation} F^{\prime}(\eta) = \mu_0+\mu_1 F(\eta)+\mu_2 F(\eta)^{2}, \end{equation}$

(3.7)

where $\mu_0$ , $\mu_1$ , $\mu_2$ are given in Table 1. To explore the analytic solutions to (3.5), I ought to follow the subsequent steps.

$\textbf{Step 1.}$ Placing (3.6) along with (3.7) into Eq (3.5) and gathering the coefficients of $F(\eta)^j$ , $j$ = $-4, -3, -2, -1, 0, 1, 2, 3, 4,$ to zeros gives a system of equations for $\rho_0, \; \rho_k, \; q_k$ , $k = 1, 2.$

$\textbf{Step 2.}$ Solve the resulting system using mathematical software: for example, Mathematica or Maple.

$\textbf{Step 3.}$ Choosing the values of $\mu_0, \; \mu_1$ and $\mu_2$ and the function $F(\eta)$ from and substituting them along with $\rho_0, \; \rho_k, \; q_k$ , $k = 1, 2,$ in (3.6) produces a set of trigonometric function and rational solutions to (3.5).

Applying the above steps, I determine the values of $\rho_0, \; \rho_1, \; \rho_2, \; q_1, \; q_2$ and $w$ as follows:

$(1).$ When $\mu_0 = 0,$ $\mu_1 = 1$ and $\mu_2 = -1$ , I have two cases.

$\textbf{Case 1.}$

$\begin{equation} \begin{aligned} \rho_0 = &0, && \rho_1 = \dfrac{6 (\alpha +\beta)}{\gamma +\lambda}, &\rho_2 = & -\dfrac{6 (\alpha +\beta)}{\gamma + \lambda}, & q_1 = q_2 = 0, \quad \text{and}\quad w = \alpha+\beta. \end{aligned} \end{equation}$

(3.8)

The solution is given by

$\begin{equation} \begin{aligned} \Psi_1(x, y, t) = \dfrac{ 3(\alpha+\beta)}{2(\gamma+\lambda)} \operatorname{sech}^2\left(\dfrac{1}{2} \left( x+y-(\alpha +\beta) t+x0\right) \right). \end{aligned} \end{equation}$

(3.9)

$\textbf{Case 2.}$

$\rho_0 = -\dfrac{\alpha+\beta}{\gamma+\lambda}, \rho_1 = \dfrac{6 (\alpha +\beta)}{\gamma +\lambda}, \\ \rho_2 = -\dfrac{6 (\alpha +\beta)}{\gamma + \lambda}, q_1 = q_2 = 0, \quad \text{and}\quad w = -\left( \alpha+\beta\right) .$

(3.10)

The solution is given by

$\begin{equation} \begin{aligned} \Psi_2(x, y, t) = -\dfrac{ (\alpha+\beta)}{2(\gamma+\lambda)} \left( 3 \tanh ^2\left(\dfrac{1}{2} (x+y+t (\alpha+\beta))+x0\right)-1 \right) . \end{aligned} \end{equation}$

(3.11)

presents the time evolution of the analytic solutions $(a)$ $\Psi_1$ and $(b)$ $\Psi_2$ with $t = 0, 10, 20$ . The parameter values are $x0 = -20, \ \alpha = 0.50, \beta = 0.6, \ \gamma = -1.5,$ and $\lambda = 1.$ presents the wave behavior by changing a certain parameter value and fixing the values of the others. presents the behavior of $\Psi_1$ when I change the values of (a) $\alpha$ or $\beta$ and (b) $\gamma$ or $\lambda$ . In it can also be seen that the value of $\alpha$ or $\beta$ affects the direction and amplitude of the wave, such that a negative value always makes the wave negative, its amplitude decreases when $\alpha, \, \beta \to 0$ , and its amplitude increases when $\alpha, \, \beta \to \infty.$ In the value of $\gamma$ or $\lambda$ affects the direction and amplitude of the wave, such that a negative value always makes the wave negative, and its amplitude decreases when the value of $\gamma$ or $\lambda$ increases. In and present the wave behavior of $\Psi_2.$

Figure 1. Time evolution of the analytic solutions

$(a)$

$\Psi_1$ and

$(b)$

$\Psi_2$ with

$t = 0, 10, 20$ . The parameter are given by

$x0 = -20, \ \alpha = 0.50, \beta = 0.6, \ \gamma = -1.5,$ and

$\lambda = 1$ .

DownLoad: Full-Size Img PowerPoint

Figure 2. This figure present the wave behavior when changing a certain parameter value and fixing the values of the others.

$(a)$ presents the behavior when I change the value of

$\alpha$ or

$\beta,$ and

$(b)$ presents when I change the value of

$\gamma$ or

$\lambda$ for the solution

$\Psi_1.$

$(c)$ and

$(d)$ are for

$\Psi_2$ .

DownLoad: Full-Size Img PowerPoint

$(2).$ When $\mu_0 = 0,$ $\mu_1 = -1,$ and $\mu_2 = 1$ , I have two cases.

$\textbf{Case 3.}$ The solution is given by

$\begin{equation} \begin{aligned} \Psi_3(x, y, t) = -\dfrac{ 3(\alpha+\beta)}{2(\gamma+\lambda)} \operatorname{csch}^2\left(\dfrac{1}{2} \left( x+y-(\alpha +\beta) t\right) \right). \end{aligned} \end{equation}$

(3.12)

$\textbf{Case 4.}$ The solution is given by

$\begin{equation} \begin{aligned} \Psi_4(x, y, t) = -\dfrac{ (\alpha+\beta)}{2(\gamma+\lambda)} \left( 3 \coth ^2\left(\dfrac{1}{2} (x+y+t (\alpha+\beta))\right)-1 \right) . \end{aligned} \end{equation}$

(3.13)

$(3).$ When $\mu_0 = 1,$ $\mu_1 = 0,$ and $\mu_2 = -1$ , I have

$\textbf{Case 5.}$ The solution is given by

$\begin{equation} \begin{aligned} \Psi_5(x, y, t) = -\dfrac{8 (\alpha+\beta) }{\gamma+\lambda}(\cosh (4 (16 t (\alpha+\beta)+x+y))+2) \operatorname{csch}^2(2 (16 t (\alpha+\beta)+x+y)). \end{aligned} \end{equation}$

(3.14)

$(4).$ When $\mu_0 = \pm 1,$ $\mu_1 = 0,$ and $\mu_2 = \pm 1$ , I have one case.

$\textbf{Case 6.}$ The solution is given by

$\begin{equation} \begin{aligned} \Psi_6(x, y, t) = -\dfrac{24 (\alpha+\beta) }{\gamma+\lambda}\csc ^2(2 (16 t (\alpha+\beta)+x+y)) . \end{aligned} \end{equation}$

(3.15)

3.2. Application of the generalized algebraic method

According to the generalized algebraic method, the solutions of (3.5) are given by the form

$\begin{equation} \psi(\eta) = \nu_0+ \nu_{1} G(\eta)+ \nu_{2} G(\eta)^{2}+\dfrac{ r_{1}}{G(\eta)}+\dfrac{ r_{2}}{G(\eta)^{2}}, \end{equation}$

(3.16)

where $\nu_k$ , $r_k$ are to be determined later. $G(\eta)$ is a solution of the following differential equation:

$\begin{equation} G^{\prime}(\eta) = \varepsilon \sqrt{\sum\limits_{k = 0}^{4} \delta_{k} G^{k}(\eta)}, \end{equation}$

(3.17)

where $\delta_k$ , $k = 0, 1, 3, 4$ , are given in . In all the cases mentioned above and the subsequent solutions, I used the mathematical software Mathematica to find the values of the constants $\nu_0, \, \nu_1, \, \nu_2, \, r_1, \, r_2$ and $w$ . Thus, the analytic solutions to (3.5) using the generalized algebraic method will be presented here with different values of the constants $\delta_k$ , $k = 0, 1, 3, 4$ .

$(5).$ When $\delta_0 = \dfrac{\delta^2_2}{4 \delta_4},$ $\delta_1 = \delta_3 = 0,$ $\delta_2 < 0$ , $\delta_4 > 0$ , and $\varepsilon = \pm1,$

$\begin{equation} \begin{aligned} \nu_0 = &\frac{\pm\sqrt{4 \delta_2^2 \varepsilon^4 \left(\alpha+\beta\right)^2 \left(\gamma+\lambda\right)^2-3 \delta_2 \varepsilon^4 \left(\alpha+\beta\right)^2 \left(\gamma+\lambda\right)^2}-2 \delta_2 \varepsilon^2 \left(\alpha+\beta\right) \left(\gamma+\lambda\right)}{\left(\gamma+\lambda\right)^2}, \\ \nu_2 = &-\frac{6 \delta_4 \varepsilon^2 \left(\alpha+\beta\right)}{\left(\gamma+\lambda\right)}, \\ \nu_1 = &r_1 = r_2 = 0, \quad\varepsilon = \pm1.\\ w = &\pm\frac{2 \sqrt{\left(4 \delta_2^2-3 \delta_2\right) \varepsilon^4 \left(\alpha+\beta\right)^2 \left(\gamma+\lambda\right)^2}}{\left(\gamma+\lambda\right)}. \end{aligned} \end{equation}$

(3.18)

$\textbf{Case 7.}$ The solution is given by

$\Psi_7(x, y, t) = -\frac{1}{{(\gamma+\lambda)^2}}\\ \left( -3 \delta_2 \varepsilon^4 (\alpha+\beta) (\gamma+\lambda) \tanh ^2\left(\frac{\sqrt{-\delta_2} \left(\frac{2 t \sqrt{\delta_2 (4 \delta_2-3) \varepsilon^4 (\alpha+\beta)^2 (\gamma+\lambda)^2}}{\gamma+\lambda}+x+y\right)}{\sqrt{2}}\right)\right.\\ \left.+2 \delta_2 \varepsilon^2 (\alpha+\beta) (\gamma+\lambda)+\sqrt{\delta_2 (4 \delta_2-3) \varepsilon^4 (\alpha+\beta)^2 (\gamma+\lambda)^2}\right) .$

(3.19)

shows the time evolution of the analytic solutions. shows $\Psi_7$ with $t$ = $0:2:6$ . The parameter values are $\delta_2 = -1$ , $\delta_4 = 1,$ $\epsilon = -1,$ $\alpha = 0.50,$ $\beta = 0.6,$ $\gamma = -1.5,$ $\lambda = 1.8$ and $x0 = -10.$ shows $\Psi_8$ with $t = 0:2:8$ . The parameter values are $\delta_2 = 1$ , $\delta_4 = -1,$ $\epsilon = -1,$ $\alpha = 0.50,$ $\beta = 0.6,$ $\gamma = -1.5,$ $\lambda = 1.8$ and $x0 = -10.$ – present the $3D$ time evolution of the analytic solutions $\Psi_2$ (left) and the numerical solutions (right) obtained employing the scheme 5.1 with $t = 5, 15, 25$ , $M_x = 1600,$ $N_y = 100,$ $x = 0\to60$ and $y = 0\to1.$

Figure 3. Time evolution of the analytic solutions.

$(a)$

$\Psi_7$ with

$t = 0:2:6$ . The parameter values are

$\delta_2 = -1$ ,

$\delta_4 = 1,$

$\epsilon = -1,$

$\alpha = 0.50,$

$\beta = 0.6,$

$\gamma = -1.5,$

$\lambda = 1.8$ and

$x0 = -10.$

$(b)$

$\Psi_8$ with

$t = 0:2:8$ . The parameter values are

$\delta_2 = 1$ ,

$\delta_4 = -1,$

$\epsilon = -1,$

$\alpha = 0.50,$

$\beta = 0.6,$

$\gamma = -1.5,$

$\lambda = 1.8$ and

$x0 = -10$ .

DownLoad: Full-Size Img PowerPoint

Figure 4. 3D graphs presenting the analytic (left) and the numerical (right) solutions of

$\Psi_2(x, y, t)$ at

$t = 5.$ The figures present the strength of agreement between analytic and numerical solutions.

DownLoad: Full-Size Img PowerPoint

Figure 5. 3D graphs presenting the analytic (left) and the numerical (right) solutions of

$\Psi_2(x, y, t)$ at

$t = 15.$ The figures present the strength of agreement between analytic and numerical solutions.

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Figure 6. 3D graphs presenting the analytic (left) and the numerical (right) solutions of

$\Psi_2(x, y, t)$ at

$t = 25.$ The figures present the strength of agreement between analytic and numerical solutions.

DownLoad: Full-Size Img PowerPoint

$(6).$ When $\delta_0 = 0,$ $\delta_1 = \delta_3 = 0,$ $\delta_2 > 0$ , $\delta_4 < 0$ , and $\varepsilon = \pm1,$

$\begin{equation} \begin{aligned} \nu_2 = &-\frac{6 \delta_4 \varepsilon^2 (\alpha+\beta)}{\gamma+\lambda}, \quad \nu_1 = \nu_0 = r_1 = r_2 = 0, \quad w = &4 \delta_2 \varepsilon^2 (\alpha+\beta). \end{aligned} \end{equation}$

(3.20)

$\textbf{Case 8.}$ The solution is given by

$\begin{equation} \begin{aligned} \Psi_8(x, y, t) = &\frac{6 \delta_2\epsilon^4 (\alpha+\beta) \operatorname{sech}^2\left(\sqrt{\delta_2} \left(-4 \delta_2\epsilon^2 t (\alpha+\beta)+x+y\right)\right)}{\gamma+\lambda} \end{aligned} \end{equation}$

(3.21)

$(7).$ When $\delta_0 = 0,$ $\delta_1 = \delta_4 = 0,$ $\delta_3\neq0,$ $\delta_2 > 0$ , $\varepsilon = \pm1$

$\begin{equation} \begin{aligned} \text{Set 1.}\, \nu_1 = & -\tfrac{3 \delta_3 \varepsilon^2 \left(\alpha+\beta\right)}{2 \left(\gamma+\lambda\right)}, \, \nu_0 = \nu_2 = r_1 = r_2 = 0, \, w = \delta_2 \varepsilon^2 \left(\alpha+\beta\right). \\ &\\ \text{Set 2.}\, \nu_0 = & -\tfrac{\delta_2 \varepsilon^2 \left(\alpha+\beta\right)}{\left(\gamma+\lambda\right)}, \, \nu_1 = -\tfrac{3 \delta_3 \varepsilon^2 \left(\alpha+\beta\right)}{2 \left(\gamma+\lambda\right)}, \quad\nu_2 = r_1 = r_2 = 0, \, w = -\delta_2 \varepsilon^2 \left(\alpha+\beta\right). \end{aligned} \end{equation}$

(3.22)

$\textbf{Case 9.}$ The solution is given by

$\begin{equation} \begin{aligned} \Psi_9(x, y, t) = \frac{3 \delta_2 (\alpha+\beta) \operatorname{sech}^2\left(\frac{1}{2} \sqrt{\delta_2} (\delta_2 t (-(\alpha+\beta))+x+y)\right)}{2 (\gamma+\lambda)}. \end{aligned} \end{equation}$

(3.23)

$\textbf{Case 10.}$ The solution is given by

$\begin{equation} \begin{aligned} \Psi_{10}(x, y, t) = \frac{3 \delta_2 (\alpha+\beta) \operatorname{sech}^2\left(\frac{1}{2} \sqrt{\delta_2} (\delta_2 t (\alpha+\beta)+x+y)\right)}{2 (\gamma+\lambda)}-\frac{\delta_2 (\alpha+\beta)}{\gamma+\lambda}. \end{aligned} \end{equation}$

(3.24)

4. Numerical solution

In this section I extract numerical solutions to the resulting ODE system (3.5) using several numerical methods. The purpose of this procedure is to guarantee the accuracy of the analytic solutions. I picked one of the analytic solutions above to be a sample, (3.11). The nonlinear shooting and BVP methods, at $t = 0,$ are used by taking the value of $\psi$ at the right endpoint of the domain $\eta = 0$ with guessing the initial value for $\psi_\eta.$ The new target is to obtain the second boundary condition of $\psi$ at the left endpoint of a particular domain. Once the numerical result is obtained, I compare it with the analytic solution (3.11). The MATLAB solver ODE15s and FSOLVE ^[47] are used to get the numerical solution. The resulting ODE (3.5) is discretized as

$\begin{equation} f(\psi) = 0, \qquad f(\psi_i) = -w\; \psi_i +\dfrac{\alpha +\beta}{\Delta_\eta} \left( \psi_{i+1}-2\psi_i+\psi_{i-1}\right) +\dfrac{ \gamma+\lambda}{2\Delta_\eta} \left( \psi^2_{i+1}-\psi_{i-1}^2\right), \end{equation}$

(4.1)

for the BVP method and

$\begin{equation} \psi_{\eta\eta} = \dfrac{1}{\alpha+\beta}\left( w\psi-\left( \gamma+\lambda\right) \psi^2\right), \end{equation}$

(4.2)

for the shooting method. Figure 7 presents the comparison between the numerical solutions obtained using the above numerical methods and the analytic solution. Figure 7 shows that the solutions are identical to the analytic solution.

Figure 7. Comparing the numerical solutions resulting from the shooting and BVP methods with the analytic solution (3.11) at

$t = 0.$ The parameter values are taken as

$\alpha = 0.50$ ,

$\beta = 0.6$ ,

$\gamma = -1.5, \lambda = 1.8$ , with

$N = 600$ .

DownLoad: Full-Size Img PowerPoint

Thus, it is possible to verify the correctness of the analytic solution. I also accept the obtained numerical solution as an initial condition for the numerical scheme in the next section.

5. Numerical scheme on a fixed mesh

In this section, I use the finite-difference method to obtain the numerical results of system (1.1) over the domain $[a, \, b]\times [c, \, d].$ Here, $a, \; b, \; c$ and $d$ represent the endpoints of the rectangular domain in the $x$ and $y$ directions, respectively, and $T_f$ is a certain time. The domain $[a, \, b]\times [c, \, d]$ is split into $(M_x+1)\times (N_y+1)$ mesh points:

$\begin{equation*} \label{} \begin{aligned} x_m = &a+m\, \Delta_x, &&& m = 0, \, 1, \, 2, \, \dots, \, M_x, \\ y_n& = c+n\, \Delta_y, &&& n = 0, \, 1, \, 2, \, \dots, \, N_y, \end{aligned} \end{equation*}$

where $\Delta_x$ and $\Delta_y$ are the step-sizes of the $x$ and $y$ domains, respectively. The system (1.1) is converted to an ODE system by discretizing the space derivatives while keeping the time derivative continuous. Completing this yields

$\begin{equation} \begin{aligned} &\Psi_t|_{m, n}^k + \dfrac{\alpha}{2\Delta_y \Delta_x^2} \delta_x^2\left( \Gamma^{k+1}_{m, n+1}-\Gamma^{k+1}_{m, n-1}\right) +\dfrac{\beta}{2\Delta_x\Delta_y^2} \delta_y^2\left( \Phi^{k+1}_{m+1, n}-\Phi^{k+1}_{m-1, n}\right)\\ &\qquad \quad- \frac{ \gamma}{4\Delta_y} \left( \left( \Phi^{k+1}_{m, n+1}+\Phi^{k+1}_{m, n} \right) \Gamma^{k+1}_{m, n+1}- \left( \Phi^{k+1}_{m, n}+\Phi^{k+1}_{m, n-1} \right)\Gamma^{k+1}_{m, n-1}\right)\\ &\qquad \quad- \frac{ \lambda}{4\Delta_x} \left( \left( \Gamma^{k+1}_{m+1, n}+\Gamma^{k+1}_{m, n} \right) \Phi^{k+1}_{m+1, n}- \left( \Gamma^{k+1}_{m, n}+\Gamma^{k+1}_{m-1, n} \right)\Phi^{k+1}_{m-1, n}\right)\\ &\qquad \quad+\frac{ \gamma}{4\Delta_y} \left( \left( \Psi^{k+1}_{m, n+1}\right)^2- \left( \Psi^{k+1}_{m, n-1}\right)^2\right) +\frac{ \lambda}{4\Delta_x} \left( \left( \Psi^{k+1}_{m+1, n}\right)^2- \left( \Psi^{k+1}_{m-1, n}\right)^2\right) = 0, \\ &\frac{1}{2\Delta_y}(\Gamma_{m, n+1}^{k+1}-\Gamma_{m, n-1}^{k+1}) = \frac{1}{2\Delta_x}(\Psi_{m+1, n}^{k+1}-\Psi_{m-1, n}^{k+1}), \\ &\frac{1}{2\Delta_y}(\Psi_{m, n+1}^{k+1}-\Psi_{m, n-1}^{k+1}) = \frac{1}{2\Delta_x}(\Phi_{m+1, n}^{k+1}-\Phi_{m-1, n}^{k+1}), \end{aligned} \end{equation}$

(5.1)

where

$\begin{aligned} \delta_x^2\Gamma_{m, n}^{k+1} = &\left(\Gamma^{k+1}_{m+1, n}-2 \Gamma^{k+1}_{m, n}+ \Gamma^{k+1}_{m-1, n}\right), \\ \delta_y^2\Phi_{m, n}^{k+1} = &\left(\Phi^{k+1}_{m, n+1}-2 \Phi^{k+1}_{m, n}+ \Phi^{k+1}_{m, n-1}\right), \end{aligned}$

subject to the boundary conditions:

$\begin{equation} \begin{array}{l} \Psi_{x}(a, \, y, \, t) = \Psi_{x}(b, \, y, \, t) = 0, \, \, \, \, \, \, \, \, \forall y\in [c, \, d], \\ \\ \Psi_{y}(x, \, c, \, t) = \Psi_{y}(x, \, d, \, t) = 0, \, \, \, \, \, \forall x\in [a, \, b]. \end{array} \end{equation}$

(5.2)

Equation (5.2) permits us to use fictitious points in estimating the space derivatives at the domain's endpoints. The initial conditions are generated by

$\begin{equation} \begin{aligned} \Psi_2(x, y, 0) = -\dfrac{ (\alpha+\beta)}{2(\gamma+\lambda)} \left( 3 \tanh ^2\left(\dfrac{1}{2} (x+y+x0\right)-1 \right) , \end{aligned} \end{equation}$

(5.3)

where $\alpha, \, \beta, \, \gamma$ and $\lambda$ are user-defined parameters. In all the numerical results shown in this section, the parameter values are fixed as $\alpha = 0.50, \, \beta = 0.6, \, \gamma = -1.50, \, \lambda = 1.80, \, x0 = -45.0$ , $y = 0\to1,$ $x = 0\to60$ and $t = 0\to 25.$ The above system is solved by using an ODE solver in FORTRAN called the DDASPK solver ^[48]. This solver used a backward differentiation formula. Since I do not have the initial conditions for the space derivatives, I approximate the Jacobian matrix of the linearized system by using LU-Factorization. The obtained numerical results are acceptable. This can be observed from the Figures 8 and 9.

Figure 8. Time change for the numerical results while holding

$y = 0.5$ and

$M_x = 1600$ at

$t = 0:5:25.$ The wave at

$t = 25$ illustrates that the numerical and the analytic solutions are quite identical.

DownLoad: Full-Size Img PowerPoint

Figure 9. The convergence histories of the scheme with the fixation of both

$y = 0.5$ and

$M_x = 1600$ at

$t = 5$ .

DownLoad: Full-Size Img PowerPoint

6. Stability of the numerical scheme

The von Neumann analysis is used to examine the stability of the scheme (5.1). The von Neumann analysis is occasionally called Fourier analysis and is utilized exclusively when the scheme is linear. Hence, I suppose that the linear version is given by

$\begin{equation} \begin{aligned} \Psi_t& +\alpha \Gamma_{xxy}+\beta \Phi_{xyy}+s_0\, \Gamma_y+s_1\, \Psi_y+s_2\, \Phi_x+s_3\, \Psi_x = 0, \\ \Gamma_y& = \Psi_x, \\ \Psi_y& = \Phi_x, \\ \end{aligned} \end{equation}$

(6.1)

where $s_0 = \gamma \Phi$ , $s_1 = \gamma \Psi$ , $s_2 = \lambda \Gamma$ , $s_3 = \lambda \Psi$ are constants. Since $\Gamma_y = \Psi_x,$ and $\Psi_y = \Phi_x,$ the first equation of (6.1) is given by

$\begin{equation} \Psi_t +\alpha \Psi_{xxx}+\beta \Psi_{yyy}+ s_0\, \Psi_y +s_1\, \Psi_y+s_2\, \Psi_x+s_3\, \Psi_x = 0, \end{equation}$

(6.2)

where $\alpha, \, \beta, \, \gamma, \, \lambda, \, s_0, \, s_1, \, s_2, \, s_3, \, l_4$ are constants. I set directly

$\begin{equation} \begin{aligned} \Psi_{m, n}^k = &\mu^k\, \exp(\iota\, \pi\, \xi_0 \, n\, \Delta_x)\exp(\iota\, \pi\, \xi_1 \, m\, \Delta_y), \quad\\ \end{aligned} \end{equation}$

(6.3)

and also I can have

$\begin{aligned} \Psi_{m, n}^{k+1} = &\mu \, \Psi_{m, n}^k, \quad \Psi_{m+1, n}^{k} = \exp(\iota\, \pi\, \xi_0 \, \Delta_x) \, \Psi_{m, n}^k, \quad \Psi_{m, n+1}^{k} = \exp(\iota\, \pi\, \xi_1 \, \Delta_y) \, \Psi_{m, n}^k, \\ & \qquad\qquad\, \, \, \Psi_{m-1, n}^{k} = \exp(-\iota\, \pi\, \xi_0 \, \Delta_x) \, \Psi_{m, n}^k, \quad \Psi_{m, n-1}^{k} = \exp(-\iota\, \pi\, \xi_1 \, \Delta_y) \, \Psi_{m, n}^k , \\ m = &1, \, 2, \dots, \, N_x-1, \quad \quad n = 1, 2, \dots, \, N_y-1. \end{aligned}$

Substituting (6.3) into (6.2) and doing some operations, I have

$\begin{array}{l} 1 = \mu\left( 1-\iota\Delta_t\left( \frac{\sin(\xi_0\pi \Delta_x)\, }{\Delta_x} \left(\frac{4\, \alpha}{\Delta_x^2}\, \sin^2(\tfrac{\xi_0\pi\, \Delta_x}{2})-s_2-s_3\right)+\frac{ \sin(\xi_1\pi \Delta_y)\, }{\Delta_y} \left(\frac{4\, \beta}{\Delta_y^2}\, \sin^2(\tfrac{\xi_1\pi\, \Delta_y}{2})-s_0-s_1\right)\right)\right). \end{array}$

Hence,

$\begin{equation} \begin{array}{l} \mu = \dfrac{1}{ 1-a\iota}, \end{array} \end{equation}$

(6.4)

where

$a = \Delta_t\left( \frac{\sin(\xi_0\pi \Delta_x)\, }{\Delta_x} \left(\frac{4\, \alpha}{\Delta_x^2}\, \sin^2(\tfrac{\xi_0\pi\, \Delta_x}{2})-s_2-s_3\right)+\frac{ \sin(\xi_1\pi \Delta_y)\, }{\Delta_y} \left(\frac{4\, \beta}{\Delta_y^2}\, \sin^2(\tfrac{\xi_1\pi\, \Delta_y}{2})-s_0-s_1\right)\right).$

Thus,

$\begin{equation} \begin{array}{l} |\mu|^2 = \frac{1}{1+a^2}\leq 1.\\ \, \end{array} \end{equation}$

(6.5)

The stability condition of the von Neumann analysis is fulfilled. Consequently, from Eq (6.5), the scheme is unconditionally stable.

7. Error analysis

To examine the accuracy of the numerical scheme (5.1), I study the truncation error utilizing Taylor expansions. Suppose that the error is

$\begin{equation} \begin{array}{l} e^{k+1}_{m, n} = \Psi_{m, n}^{k+1}-\Psi(x_m, y_n, t_{k+1}), \end{array} \end{equation}$

(7.1)

where $\Psi(x_m, y_n, t_{k+1})$ and $\Psi_{m, n}^{k+1}$ are the analytic solution and an approximate solution, respectively. Substituting (7.1) into (5.1) gives

$\begin{equation*} \label{} \begin{aligned} \frac{e^{k+1}_{j, k}-e^{k}_{j, k}}{\Delta_t} = & T^{k+1}_{m, n}- \left( \alpha \frac{1}{2\Delta_x^3 } \delta_x^2\left( e^{k+1}_{m+1, n}-e^{k+1}_{m-1, n}\right) +\beta \frac{1}{2\Delta_y^3 } \delta_y^2\left( e^{k+1}_{m, n+1}-e^{k+1}_{m, n-1}\right)\right.\\ &\left.+ \frac{ s_2+s_3}{2\Delta_x} \left( e^{k+1}_{m+1, n}-e^{k+1}_{m-1, n}\right)+ \frac{ s_0+s_1}{2\Delta_y} \left( e^{k+1}_{m, n+1}-e^{k+1}_{m, n-1}\right)\right), \end{aligned} \end{equation*}$

where

$T^{k+1}_{m, n} = \frac{\alpha}{2\Delta_x^3 } \delta_x^2\left( \Psi(x_{m+1}, y_n, t_{k+1})-\Psi(x_{m-1}, y_n, t_{k+1})\right) \\ +\frac{\beta }{2\Delta_y^3 } \delta_y^2\left( \Psi(x_{m}, y_{n+1}, t_{k+1})-\Psi(x_{m}, y_{n-1}, t_{k+1})\right)\\ + \frac{ s_2+s_3}{2\Delta_x} \left( \Psi(x_{m+1}, y_n, t_{k+1})-\Psi(x_{m-1}, y_n, t_{k+1})\right)+\\ \frac{ s_0+s_1}{2\Delta_y} \left( \Psi(x_{m}, y_{n+1}, t_{k+1})-\Psi(x_{m}, y_{n-1}, t_{k+1})\right).$

Hence,

$\begin{equation*} \label{} \begin{aligned} T^{k+1}_{m, n}\leq& \frac{\Delta_t}{2}\frac{\partial^2\Psi(x_{m}, y_{n}, \xi_{k+1})}{\partial t^2} -\frac{\Delta_x^2}{2}\frac{\partial^5\Psi(\zeta_{m}, y_{n}, t_{k+1})}{\partial x^5} -\frac{\Delta_y^2}{2}\frac{\partial^5\Psi(x_{m}, \eta_{n}, t_{k+1})}{\partial x^5}\\ & -\frac{\Delta_y^2}{6}\frac{\partial^3\Psi(x_{m}, \eta_{n}, t_{k+1})}{\partial x^3}-\frac{\Delta_x^2}{6}\frac{\partial^3\Psi(\zeta_{m}, y_{n}, t_{k+1})}{\partial x^3}. \end{aligned} \end{equation*}$

Accordingly, the truncation error of the numerical scheme is

$\begin{equation*} \label{} \begin{array}{l} T^{k+1}_{m, n} = O(\Delta_t, \Delta_x^2, \Delta_y^2). \end{array} \end{equation*}$

8. Results and discussion

I have prosperously employed several analytical methods to extract the traveling wave solutions to the two-dimensional Novikov-Veselov system, confirming the solutions with numerical results obtained using the numerical scheme (5.1). The major highlights of the results are shown in Table 3 and –, which allow immediate comparison of the analytic solutions with the numerical results. Through these, I can notice that the solutions are identical to a large extent, and the error approaches zero whenever the value of $\Delta_x, \Delta_y\to 0$ . The numerical schemes are unconditionally stable for fixing the parameter values $\alpha = 0.50, \, \beta = 0.6, \, \gamma = -1.50, \, \lambda = 1.80, \, x0 = -45.0$ , $y = 0\to1,$ $x = 0\to60$ and $t = 0\to 25.$

Table 3. The relative error with

$L_2$ norm and CPU at

$t = 20.$ .

$\Delta_x$	The Relative Error	CPU
$0.6000 \qquad\qquad$	$5.600\times 10^{-3}\qquad$	$0.063\times10^{3}\, \text{m} \qquad$
$0.3000$	$2.100\times 10^{-3}$	$0.1524\times10^{3} \, s$
$0.1500$	$6.700\times 10^{-4}$	$0.3564\times10^{3}\, s$
$0.0750$	$2.100\times 10^{-4}$	$0.8892\times10^{3}\, s$
$0.0375$	$6.610\times 10^{-5}$	$1.7424\times10^{3}\, s$
$0.0187$	$2.310\times 10^{-5}$	$4.0230\times10^{3}\, s$

| Show Table

DownLoad: CSV

Figure 10. The convergence histories measured utilizing the relative error with

$l_2$ norm as a function of

$\Delta_x$ (see Table 3). Here, I picked a certain value of the variable

$y = 0.5$ at

$t = 20$ and

$x = 0\to60$ .

DownLoad: Full-Size Img PowerPoint

presents the time evolution of the analytic solutions $(a)$ $\Psi_1$ and $(b)$ $\Psi_2$ with $t = 0, 10, 20$ . The parameter values are $x0 = -20, \ \alpha = 0.50, \ \beta = 0.6, \ \gamma = -1.5,$ and $\lambda = 1.$ presents the wave behavior by changing a certain parameter value and fixing the values of the others. presents the behavior of $\Psi_1$ when I change the values of (a) $\alpha$ or $\beta$ and (b) $\gamma$ or $\lambda$ . In it can also be seen that the value of $\alpha$ or $\beta$ affects the direction and amplitude of the wave, such that a negative value always makes the wave negative, its amplitude decreases when $\alpha, \, \beta \to 0$ , and its amplitude increases when $\alpha, \, \beta \to \infty.$ In the value of $\gamma$ or $\lambda$ affects the direction and amplitude of the wave, such that a negative value always makes the wave negative, and its amplitude decreases when the value of $\gamma$ or $\lambda$ increases. In present the wave behavior of $\Psi_2.$ shows the time evolution of the analytic solutions. shows $\Psi_7$ with $t = 0:2:6$ . The parameter values are $\delta_2 = -1$ , $\delta_4 = 1,$ $\epsilon = -1,$ $\alpha = 0.50,$ $\beta = 0.6,$ $\gamma = -1.5,$ $\lambda = 1.8$ and $x0 = -10.$ shows $\Psi_8$ with $t = 0:2:8$ . The parameter values are $\delta_2 = 1$ , $\delta_4 = -1,$ $\epsilon = -1,$ $\alpha = 0.50,$ $\beta = 0.6,$ $\gamma = -1.5,$ $\lambda = 1.8$ and $x0 = -10.$ – present the $3D$ time evolution of the analytic solutions $\Psi_2$ (left) and the numerical solutions (right) obtained employing the scheme 5.1 with $t = 5, 15, 25$ , $M_x = 1600,$ $N_y = 100,$ $x = 0\to60$ and $y = 0\to1.$ These figures provide us with an adequate answer that the numerical and analytic solutions are quite identical. Barman et al. ^[42] accepted several traveling wave solutions for (1.1) as hyperbolic functions. The authors employed other parameters to develop new forms for the accepted solution. They proposed that Eq (1.1) describes tidal and tsunami waves, electromagnetic waves in transmission cables and magneto-sound and ion waves in plasma. In comparison, I have found numerous solutions also as hyperbolic functions. Furthermore, I obtained the numerical solutions to enhance the assurance that the solutions presented here are correct and accurate.

9. Conclusions

I have successfully utilized the generalized algebraic and modified F-expansion methods to acquire the soliton solutions for the two-dimensional Novikov-Veselov system, verifying these solutions with numerical results obtained by employing the numerical scheme (5.1). The major highlights of the results shown in Figures 8– and , which allow immediate comparison of the analytic solutions with the numerical results. Through these, I can notice that the solutions are identical to a large extent, and the error approaches zero whenever the value of $\Delta_x, \Delta_y\to 0$ . The numerical schemes are unconditionally stable for fixing the parameter values $\alpha = 0.50, \, \beta = 0.6, \, \gamma = -1.50, \, \lambda = 1.80,$ $x0 = -45.0$ , $y = 0\to1,$ $x = 0\to60$ and $t = 0\to 25.$ The Jacobi elliptic functions have effectively deteriorated to hyperbolic functions. The applied numerical schemes have provided reliable numerical solutions when using a small value of $\Delta_x, \Delta_y\to0$ .

Ultimately, I can deduce that the methods used are valuable and applicable to extract soliton solutions for other nonlinear evolutionary systems found in chemistry, engineering, physics and other sciences.

Conflict of interest

The author declares that he has no potential conflict of interest in this article.

References

[1]	M. Azizi, U. Aickelin, H. A. Khorshidi, M. Baghalzadeh-Shishehgarkhaneh, Energy valley optimizer: A novel metaheuristic algorithm for global and engineering optimization, Sci. Rep., 13 (2023), 226. https://doi.org/10.1038/s41598-022-27344-y doi: 10.1038/s41598-022-27344-y
[2]	F. A. Hashim, E. H. Houssein, K. Hussain, M. S. Mabrouk, W. Al-Atabany, Honey badger algorithm: New metaheuristic algorithm for solving optimization problems, Math. Comput. Simul., 192 (2022), 84–110. https://doi.org/10.1016/j.matcom.2021.08.013 doi: 10.1016/j.matcom.2021.08.013
[3]	S. Gupta, H. Abderazek, B. S. Yıldız, A. R. Yildiz, S. Mirjalili, S. M. Sait, Comparison of metaheuristic optimization algorithms for solving constrained mechanical design optimization problems, Exp. Syst. Appl., 183 (2021), 115351. https://doi.org/10.1016/j.eswa.2021.115351 doi: 10.1016/j.eswa.2021.115351
[4]	S. Aslan, T. Erkin, An immune plasma algorithm based approach for ucav path planning, J. King Saud Univ. Comput. Inf. Sci., 35 (2023), 56–69. https://doi.org/10.1016/j.jksuci.2022.06.004 doi: 10.1016/j.jksuci.2022.06.004
[5]	L. Abualigah, K. H. Almotairi, M. A. Elaziz, Multilevel thresholding image segmentation using meta-heuristic optimization algorithms: Comparative analysis, open challenges and new trends, Appl. Intell., 53 (2023), 11654–11704. https://doi.org/10.1007/s10489-022-04064-4 doi: 10.1007/s10489-022-04064-4
[6]	M. Chan-Ley, G. Olague, Categorization of digitized artworks by media with brain programming, Appl. Opt., 59 (2020), 4437–4447, 2020. https://doi.org/10.1364/AO.385552 doi: 10.1364/AO.385552
[7]	Y. Li, G. Tian, Y. Yi, Y. Yuan, Improved artificial rabbit optimization and its application in multichannel signal denoising, IEEE Sensors J., 2024 (2024). https://doi.org/10.1109/JSEN.2024.3456290 doi: 10.1109/JSEN.2024.3456290
[8]	Y. Xiao, H. Cui, A. G. Hussien, F. A. Hashim, MSAO: A multi-strategy boosted snow ablation optimizer for global optimization and real-world engineering applications, Adv. Eng. Inf., 61 (2024), 102464. https://doi.org/10.1016/j.aei.2024.102464 doi: 10.1016/j.aei.2024.102464
[9]	X. Fei, J. Wang, S. Ying, Z. Hu, J. Shi, Projective parameter transfer based sparse multiple empirical kernel learning machine for diagnosis of brain disease, Neurocomputing, 413 (2020), 271–283. https://doi.org/10.1016/j.neucom.2020.07.008 doi: 10.1016/j.neucom.2020.07.008
[10]	Z. Ma, X. Li, An improved supervised and attention mechanism-based u-net algorithm for retinal vessel segmentation, Comput. Biol. Med., 168 (2024), 107770. https://doi.org/10.1016/j.compbiomed.2023.107770 doi: 10.1016/j.compbiomed.2023.107770
[11]	Y. Xiao, Y. Guo, H. Cui, Y. Wang, J. Li, Y. Zhang, IHAOAVOA: An improved hybrid aquila optimizer and African vultures optimization algorithm for global optimization problems, Math. Biosci. Eng., 19 (2022), 10963–11017. https://doi.org/10.3934/mbe.2022512 doi: 10.3934/mbe.2022512
[12]	J. Zhang, M. Xiao, L. Gao, Q. Pan, Queuing search algorithm: A novel metaheuristic algorithm for solving engineering optimization problems, Appl. Math. Modell., 63 (2018), 464–490. https://doi.org/10.1016/j.apm.2018.06.036 doi: 10.1016/j.apm.2018.06.036
[13]	U. Kamath, K. De Jong, A. Shehu, Effective automated feature construction and selection for classification of biological sequences, PloS One, 9 (2014), e99982. https://doi.org/10.1371/journal.pone.0099982 doi: 10.1371/journal.pone.0099982
[14]	F. Thabtah, F. Kamalov, S. Hammoud, S. R. Shahamiri, Least loss: A simplified filter method for feature selection, Inf. Sci., 534 (2020), 1–15. https://doi.org/10.1016/j.ins.2020.05.017 doi: 10.1016/j.ins.2020.05.017
[15]	L. Sun, J. Zhang, W. Ding, J. Xu, Feature reduction for imbalanced data classification using similarity-based feature clustering with adaptive weighted k-nearest neighbors, Inf. Sci., 593 (2022), 591–613. https://doi.org/10.1016/j.ins.2022.02.004 doi: 10.1016/j.ins.2022.02.004
[16]	Z. Tao, L. Huiling, W. Wenwen, Y. Xia, GA-SVM based feature selection and parameter optimization in hospitalization expense modeling, Appl. Soft Comput., 75 (2019), 323–332. https://doi.org/10.1016/j.asoc.2018.11.001 doi: 10.1016/j.asoc.2018.11.001
[17]	H. Liu, Z. Zhao, Manipulating data and dimension reduction methods: Feature selection, in Computational Complexity: Theory, Techniques, and Applications, Springer, (2012), 790–1800. https://doi.org/10.1007/978-1-4614-1800-9_115
[18]	M. Rostami, K. Berahmand, E. Nasiri, S. Forouzandeh, Review of swarm intelligence-based feature selection methods, Eng. Appl. Artif. Intell., 100 (2021), 104210. https://doi.org/10.1016/j.engappai.2021.104210 doi: 10.1016/j.engappai.2021.104210
[19]	D. H. Wolpert, W. G. Macready, No free lunch theorems for optimization, IEEE Trans. Evol. Comput., 1 (1997), 67–82. https://doi.org/10.1109/4235.585893 doi: 10.1109/4235.585893
[20]	X. Song, Y. Zhang, D. Gong, X. Sun, Feature selection using bare-bones particle swarm optimization with mutual information, Pattern Recognit., 112 (2021), 107804. https://doi.org/10.1016/j.patcog.2020.107804 doi: 10.1016/j.patcog.2020.107804
[21]	H. Faris, M. M. Mafarja, A. A. Heidari, I. Aljarah, A. M. Al-Zoubi, S. Mirjalili, et al., An efficient binary salp swarm algorithm with crossover scheme for feature selection problems, Knowl. Based Syst., 154 (2018), 43–67. https://doi.org/10.1016/j.knosys.2018.05.009 doi: 10.1016/j.knosys.2018.05.009
[22]	E. Emary, H. M. Zawbaa, A. E. Hassanien, Binary grey wolf optimization approaches for feature selection, Neurocomputing, 172 (2016), 371–381. https://doi.org/10.1016/j.neucom.2015.06.083 doi: 10.1016/j.neucom.2015.06.083
[23]	Y. Zhao, J. Dong, X. Li, H. Chen, S. Li, A binary dandelion algorithm using seeding and chaos population strategies for feature selection, Appl. Soft Comput., 125 (2022), 109166. https://doi.org/10.1016/j.asoc.2022.109166 doi: 10.1016/j.asoc.2022.109166
[24]	M. Mafarja, S. Mirjalili, Whale optimization approaches for wrapper feature selection, Appl. Soft Comput., 62 (2018), 441–453. https://doi.org/10.1016/j.asoc.2017.11.006 doi: 10.1016/j.asoc.2017.11.006
[25]	A. Adamu, M. Abdullahi, S. B. Junaidu, I. H. Hassan, An hybrid particle swarm optimization with crow search algorithm for feature selection, Mach. Learn. Appl., 6 (2021), 100108. https://doi.org/10.1016/j.mlwa.2021.100108 doi: 10.1016/j.mlwa.2021.100108
[26]	Y. Xue, T. Tang, A. X. Liu, Large-scale feedforward neural network optimization by a self-adaptive strategy and parameter based particle swarm optimization, IEEE Access, 7 (2019), 52473–52483. https://doi.org/10.1109/ACCESS.2019.2911530 doi: 10.1109/ACCESS.2019.2911530
[27]	E. Aličković, A. Subasi, Breast cancer diagnosis using GA feature selection and rotation forest, Neural Comput. Appl., 28 (2017), 753–763. https://doi.org/10.1007/s00521-015-2103-9 doi: 10.1007/s00521-015-2103-9
[28]	B. Xue, M. Zhang, W. N. Browne, X. Yao, A survey on evolutionary computation approaches to feature selection, IEEE Trans. Evol. Comput., 20 (2015), 606–626. https://doi.org/10.1109/TEVC.2015.2504420 doi: 10.1109/TEVC.2015.2504420
[29]	N. Khodadadi, E. Khodadadi, Q. Al-Tashi, E. S. M. El-Kenawy, L. Abualigah, S. J. Abdulkadir, et al., BAOA: Binary arithmetic optimization algorithm with K-nearest neighbor classifier for feature selection, IEEE Access, 11 (2023), 94094–94115. https://doi.org/10.1109/ACCESS.2023.3310429 doi: 10.1109/ACCESS.2023.3310429
[30]	W. N. Chen, D. Z. Tan, Q. Yang, T. Gu, J. Zhang, Ant colony optimization for the control of pollutant spreading on social networks, IEEE Trans. Cybern., 50 (2019), 4053–4065. https://doi.org/10.1109/TCYB.2019.2922266 doi: 10.1109/TCYB.2019.2922266
[31]	H. Hichem, M. Elkamel, M. Rafik, M. T. Mesaaoud, C. Ouahiba, A new binary grasshopper optimization algorithm for feature selection problem, J. King Saud University Comput. Inf. Sci., 34 (2022), 316–328. https://doi.org/10.1016/j.jksuci.2019.11.007 doi: 10.1016/j.jksuci.2019.11.007
[32]	A. A. Alhussan, A. A. Abdelhamid, E. S. M. El-Kenawy, A. Ibrahim, M. M. Eid, D. S. Khafaga, et al., A binary waterwheel plant optimization algorithm for feature selection, IEEE Access, 11 (2023), 94227–94251. https://doi.org/10.1109/ACCESS.2023.3312022 doi: 10.1109/ACCESS.2023.3312022
[33]	K. Nag, N. R. Pal, A multiobjective genetic programming-based ensemble for simultaneous feature selection and classification, IEEE Trans. Cybern., 46 (2015), 499–510. https://doi.org/10.1109/TCYB.2015.2404806 doi: 10.1109/TCYB.2015.2404806
[34]	J. Xue, B. Shen, Dung beetle optimizer: A new meta-heuristic algorithm for global optimization, J. Supercomput., 79 (2023), 7305–7336. https://doi.org/10.1007/s11227-022-04959-6 doi: 10.1007/s11227-022-04959-6
[35]	X. Yao, Y. Liu, G. Lin, Evolutionary programming made faster, IEEE Trans. Evol. Comput., 3 (1999), 82–102. https://doi.org/10.1109/4235.771163 doi: 10.1109/4235.771163
[36]	T. Y. Wu, H. Li, S. C. Chu, CPPE: An improved phasmatodea population evolution algorithm with chaotic maps, Mathematics, 11 (2023), 1977. https://doi.org/10.3390/math11091977 doi: 10.3390/math11091977
[37]	P. Qu, Q. Yuan, F. Du, Q. Gao, An improved manta ray foraging optimization algorithm, Sci. Rep., 14 (2024), 10301. https://doi.org/10.1038/s41598-024-59960-1 doi: 10.1038/s41598-024-59960-1
[38]	H. Yu, Y. Wang, H. Jia, L. Abualigah, Modified prairie dog optimization algorithm for global optimization and constrained engineering problems, Math. Biosci. Eng, 20 (2023), 19086–19132. https://doi.org/10.3934/mbe.2023844 doi: 10.3934/mbe.2023844
[39]	C. Liu, L. Li, Y. Qiang, S. Zhang, Predicting construction accidents on sites: An improved atomic search optimization algorithm approach, Eng. Rep., 6 (2024), e12773. https://doi.org/10.1002/eng2.12773 doi: 10.1002/eng2.12773
[40]	A. Bisht, M. Dua, S. Dua, A novel approach to encrypt multiple images using multiple chaotic maps and chaotic discrete fractional random transform, J. Ambient Intell. Humanized Comput., 10 (2019), 3519–3531. https://doi.org/10.1007/s12652-018-1072-0 doi: 10.1007/s12652-018-1072-0
[41]	Y. Zhou, L. Bao, C. L. P. Chen, A new 1D chaotic system for image encryption, Signal Process., 97 (2014), 172–182. https://doi.org/10.1016/j.sigpro.2013.10.034 doi: 10.1016/j.sigpro.2013.10.034
[42]	F. Han, X. Liao, B. Yang, Y. Zhang, A hybrid scheme for self-adaptive double color-image encryption, Multimedia Tools Appl., 77 (2018), 14285–14304. https://doi.org/10.1007/s11042-017-5029-7 doi: 10.1007/s11042-017-5029-7
[43]	H. R. Tizhoosh, Opposition-based learning: A new scheme for machine intelligence, in International Conference on Computational Intelligence for Modelling, Control and Automation and International Conference on Intelligent Agents, Web Technologies and Internet Commerce (CIMCA-IAWTIC'06), 1 (2005), 695–701. https://doi.org/10.1109/CIMCA.2005.1631345
[44]	S. Rahnamayan, H. R. Tizhoosh, M. M. A. Salama, Opposition-based differential evolution, IEEE Trans. Evol. Comput., 12 (2008), 64–79. https://doi.org/10.1109/TEVC.2007.894200 doi: 10.1109/TEVC.2007.894200
[45]	H. Wang, Z. Wu, S. Rahnamayan, Y. Liu, M. Ventresca, Enhancing particle swarm optimization using generalized opposition-based learning, Inf. Sci., 181 (2011), 4699–4714. https://doi.org/10.1016/j.ins.2011.03.016 doi: 10.1016/j.ins.2011.03.016
[46]	M. Abd Elaziz, D. Oliva, S. Xiong, An improved opposition-based sine cosine algorithm for global optimization, Exp. Syst. Appl., 90 (2017), 484–500. https://doi.org/10.1016/j.eswa.2017.07.043 doi: 10.1016/j.eswa.2017.07.043
[47]	W. Long, J. Jiao, X. Liang, S. Cai, M. Xu, A random opposition-based learning grey wolf optimizer, IEEE Access, 7 (2019), 113810–113825. https://doi.org/10.1109/ACCESS.2019.2934994 doi: 10.1109/ACCESS.2019.2934994
[48]	R. Storn, K. Price, Differential evolution–-a simple and efficient heuristic for global optimization over continuous spaces, J. Global Optim., 11 (1997), 341–359. https://doi.org/10.1023/A:1008202821328 doi: 10.1023/A:1008202821328
[49]	A. Nickabadi, M. M. Ebadzadeh, R. Safabakhsh, A novel particle swarm optimization algorithm with adaptive inertia weight, Appl. Soft Comput., 11 (2011), 3658–3670. https://doi.org/10.1016/j.asoc.2011.01.037 doi: 10.1016/j.asoc.2011.01.037
[50]	Y. Honshuku, H. Isakari, A topology optimisation of acoustic devices based on the frequency response estimation with the Padé approximation, Appl. Math. Modell., 110 (2022), 819–840. https://doi.org/10.1016/j.apm.2022.06.020 doi: 10.1016/j.apm.2022.06.020
[51]	H. Vazquez-Leal, B. Benhammouda, U. Filobello-Nino, A. Sarmiento-Reyes, V. M. Jimenez-Fernandez, J. L. Garcia-Gervacio, et al., Direct application of Padé approximant for solving nonlinear differential equations, SpringerPlus, 3 (2014), 1–11. https://doi.org/10.1186/2193-1801-3-563 doi: 10.1186/2193-1801-3-563
[52]	I. V. Andrianov, A. Shatrov, Padé approximants, their properties, and applications to hydrodynamic problems, Symmetry, 13 (2021), 1869. https://doi.org/10.3390/sym13101869 doi: 10.3390/sym13101869
[53]	J. Xu, T. Wang, L. Pei, S. Mao, C. Zhu, Parameter identification of electrolyte decomposition state in lithium-ion batteries based on a reduced pseudo two-dimensional model with Padé approximation, J. Power Sources, 460 (2020), 228093. https://doi.org/10.1016/j.jpowsour.2020.228093 doi: 10.1016/j.jpowsour.2020.228093
[54]	A. H. Gandomi, X. S. Yang, Evolutionary boundary constraint handling scheme, Neural Comput. Appl., 21 (2012), 1449–1462. https://doi.org/10.1007/s00521-012-1069-0 doi: 10.1007/s00521-012-1069-0
[55]	A. H. Gandomi, A. R. Kashani, Evolutionary bound constraint handling for particle swarm optimization, in 2016 4th International Symposium on Computational and Business Intelligence (ISCBI), (2016), 148–152. https://doi.org/10.1109/ISCBI.2016.7743274
[56]	A. H. Gandomi, A. R. Kashani, M. Mousavi, Boundary constraint handling affection on slope stability analysis, in Engineering and Applied Sciences Optimization. Computational Methods in Applied Sciences (eds. N. Lagaros and M. Papadrakakis), Springer, (2015), 341–358. https://doi.org/10.1007/978-3-319-18320-6_18
[57]	A. H. Gandomi, A. R. Kashani, F. Zeighami, Retaining wall optimization using interior search algorithm with different bound constraint handling, Int. J. Numer. Anal. Methods Geomech., 41 (2017), 1304–1331. https://doi.org/10.1002/nag.2678 doi: 10.1002/nag.2678
[58]	A. C. Cinar, A comprehensive comparison of accuracy-based fitness functions of metaheuristics for feature selection, Soft Comput., 27 (2023), 8931–8958. https://doi.org/10.1007/s00500-023-08414-3 doi: 10.1007/s00500-023-08414-3
[59]	I. Al-Shourbaji, N. Helian, Y. Sun, S. Alshathri, M. Abd Elaziz, Boosting ant colony optimization with reptile search algorithm for churn prediction, Mathematics, 10 (2022), 1031. https://doi.org/10.3390/math10071031 doi: 10.3390/math10071031
[60]	E. S. M. El-Kenawy, S. Mirjalili, A. Ibrahim, M. Alrahmawy, M. El-Said, R. M. Zaki, et al., Advanced meta-heuristics, convolutional neural networks, and feature selectors for efficient COVID-19 X-ray chest image classification, IEEE Access, 9 (2021). 36019–36037. https://doi.org/10.1109/ACCESS.2021.3061058
[61]	T. Khosla, O. P. Verma, An adaptive hybrid particle swarm optimizer for constrained optimization problem, in 2021 International Conference in Advances in Power, Signal, and Information Technology (APSIT), IEEE, (2021), 1–7. https://doi.org/10.1109/APSIT52773.2021.9641410
[62]	B. D. Kwakye, Y. Li, H. H. Mohamed, E. Baidoo, T. Q. Asenso, Particle guided metaheuristic algorithm for global optimization and feature selection problem, Exp. Syst. Appl., 248 (2024), 123362. https://doi.org/10.1016/j.eswa.2024.123362 doi: 10.1016/j.eswa.2024.123362
[63]	G. Wu, R. Mallipeddi, P. N. Suganthan, Problem definitions and evaluation criteria for the CEC 2017 competition on constrained real-parameter optimization, National University of Defense Technology, Changsha, Hunan, PR China and Kyungpook National University, Daegu, South Korea and Nanyang Technological University, Singapore, Technical Report, 2017.
[64]	S. Mirjalili, A. Lewis, The whale optimization algorithm, Adv. Eng. Software, 95 (2016), 51–67. https://doi.org/10.1016/j.advengsoft.2016.01.008 doi: 10.1016/j.advengsoft.2016.01.008
[65]	S. Mirjalili, SCA: A sine cosine algorithm for solving optimization problems, Knowl. Based Syst., 96 (2016), 120–133. https://doi.org/10.1016/j.knosys.2015.12.022 doi: 10.1016/j.knosys.2015.12.022
[66]	J. Kennedy, R. Eberhart, Particle swarm optimization, in Proceedings of ICNN'95-International Conference on Neural Networks, 4 (1995), 1942–1948. https://doi.org/10.1109/ICNN.1995.488968
[67]	S. Mirjalili, S. M. Mirjalili, A. Lewis, Grey wolf optimizer, Adv. Eng. Software, 69 (2014), 46–61. https://doi.org/10.1016/j.advengsoft.2013.12.007 doi: 10.1016/j.advengsoft.2013.12.007
[68]	A. A. Heidari, S. Mirjalili, H. Faris, I. Aljarah, M. Mafarja, H. Chen, Harris Hawks optimization: Algorithm and applications, Future Gener. Comput. Syst., 97 (2019), 849–872. https://doi.org/10.1016/j.future.2019.02.028 doi: 10.1016/j.future.2019.02.028
[69]	L. Abualigah, A. Diabat, S. Mirjalili, M Abd Elaziz, A. H. Gandomi, The arithmetic optimization algorithm, Comput. Methods Appl. Mech. Eng., 376 (2021), 113609. https://doi.org/10.1016/j.cma.2020.113609 doi: 10.1016/j.cma.2020.113609
[70]	S. Arora, P. Anand, Binary butterfly optimization approaches for feature selection, Exp. Syst. Appl., 116 (2019), 147–160. https://doi.org/10.1016/j.eswa.2018.08.051 doi: 10.1016/j.eswa.2018.08.051
[71]	R. Tanabe, A. S. Fukunaga, Improving the search performance of SHADE using linear population size reduction, in 2014 IEEE Congress on Evolutionary Computation (CEC), (2014), 1658–1665. https://doi.org/10.1109/CEC.2014.6900380
[72]	A. W. Mohamed, A. A. Hadi, A. M. Fattouh, K. M. Jambi, LSHADE with semi-parameter adaptation hybrid with CMA-ES for solving CEC 2017 benchmark problems, in 2017 IEEE Congress on Evolutionary Computation (CEC), (2017), 145–152. https://doi.org/10.1109/CEC.2017.7969307
[73]	S. Zhao, Y. Wu, S. Tan, J. Wu, Z. Cui, Y. G. Wang, QQLMPA: A quasi-opposition learning and Q-learning based marine predators algorithm, Exp. Syst. Appl., 213 (2023), 119246. https://doi.org/10.1016/j.eswa.2022.119246 doi: 10.1016/j.eswa.2022.119246
[74]	Y. Xu, Z. Yang, X. Li, H. Kang, X. Yang, Dynamic opposite learning enhanced teaching-learning-based optimization, Knowl. Based Syst., 188 (2020), 104966. https://doi.org/10.1016/j.knosys.2019.104966 doi: 10.1016/j.knosys.2019.104966
[75]	R. W. Morrison, Designing Evolutionary Algorithms for Dynamic Environments, Springer, 2004. https://doi.org/10.1007/978-3-662-06560-0
[76]	K. Hussain, M. N. M. Salleh, S. Cheng, Y. Shi, On the exploration and exploitation in popular swarm-based metaheuristic algorithms, Neural Comput. Appl., 31 (2019), 7665–7683. https://doi.org/10.1007/s00521-018-3592-0 doi: 10.1007/s00521-018-3592-0
[77]	S. Cheng, Y. Shi, Q. Qin, Q. Zhang, R. Bai, Population diversity maintenance in brain storm optimization algorithm, J. Artif. Intell. Soft Comput. Res., 4 (2014), 83–97. https://doi.org/10.1515/jaiscr-2015-0001 doi: 10.1515/jaiscr-2015-0001
[78]	S. Mirjalili, A. Lewis, S-shaped versus V-shaped transfer functions for binary particle swarm optimization, Swarm Evol. Comput., 9 (2013), 1–14. https://doi.org/10.1016/j.swevo.2012.09.002 doi: 10.1016/j.swevo.2012.09.002
[79]	J. Too, A. R. Abdullah, N. Mohd-Saad, Hybrid binary particle swarm optimization differential evolution-based feature selection for emg signals classification, Axioms, 8 (2019), 79. https://doi.org/10.3390/axioms8030079 doi: 10.3390/axioms8030079
[80]	J. Too, A. R. Abdullah, N. Mohd-Saad, N. Mohd-Ali, W. Tee, A new competitive binary grey wolf optimizer to solve the feature selection problem in emg signals classification, Computers, 7 (2018), 58. https://doi.org/10.3390/computers7040058 doi: 10.3390/computers7040058
[81]	M. H. Aghdam, N. Ghasem-Aghaee, M. E. Basiri, Text feature selection using ant colony optimization, Exp. Syst. Appl., 36 (2009), 6843–6853. https://doi.org/10.1016/j.eswa.2008.08.022 doi: 10.1016/j.eswa.2008.08.022
[82]	J. Too, A. R. Abdullah, N. Mohd-Saad, A new quadratic binary harris hawk optimization for feature selection, Electronics, 8 (2019), 1130. https://doi.org/10.3390/electronics8101130 doi: 10.3390/electronics8101130

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