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Folded novel accurate analytical and semi-analytical solutions of a generalized Calogero【-逻*辑*与-】nda

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Mostafa M A Khater,1,2,, S K Elagan3, M A El-Shorbagy4,5, S H Alfalqi6, J F Alzaidi6, Nawal A Alshehri31Department of Mathematics, Faculty of Science, Jiangsu University, 212013, Zhenjiang, China
2Department of Mathematics, Obour High Institute For Engineering and Technology, 11828, Cairo, Egypt
3Department of Mathematics, Faculty of Science, Taif University P.O.Box 11099, Taif 21944, Saudi Arabia
4Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
5Department of Basic Engineering Science, Faculty of Engineering, Menoufia University, Shebin El-Kom 32511, Egypt
6Department of Mathematics, Faculty of Science and Arts, Mahayil Asir King Khalid University, Abha, Saudi Arabia

First author contact: Author to whom any correspondence should be addressed.
Received:2021-02-20Revised:2021-04-2Accepted:2021-05-25Online:2021-07-16


Abstract
This paper studies the analytical and semi-analytic solutions of the generalized Calogero–Bogoyavlenskii–Schiff (CBS) equation. This model describes the (2 + 1)–dimensional interaction between Riemann-wave propagation along the y-axis and the x-axis wave. The extended simplest equation (ESE) method is applied to the model, and a variety of novel solitary-wave solutions is given. These solitary-wave solutions prove the dynamic behavior of soliton waves in plasma. The accuracy of the obtained solution is verified using a variational iteration (VI) semi-analytical scheme. The analysis and the match between the constructed analytical solution and the semi-analytical solution are sketched using various diagrams to show the accuracy of the solution we obtained. The adopted scheme’s performance shows the effectiveness of the method and its ability to be applied to various nonlinear evolution equations.
Keywords: generalized Calogero–Bogoyavlenskii–Schiff equation;accurate analytical;semi-analytical solutions


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Cite this article
Mostafa M A Khater, S K Elagan, M A El-Shorbagy, S H Alfalqi, J F Alzaidi, Nawal A Alshehri. Folded novel accurate analytical and semi-analytical solutions of a generalized Calogero–Bogoyavlenskii–Schiff equation. Communications in Theoretical Physics, 2021, 73(9): 095003- doi:10.1088/1572-9494/ac049f

1. Introduction

The soliton is one of the most notable characteristics that characterize the integrability of the nonlinear evolution equation [1, 2]. Recently, stray or monster waves, weird waves, and abnormal waves have been regarded as solitary waves. Solitary waves are an uncommon type of nonlinear wave that is restricted in only one direction and has important significance in various branches of physics [3, 4]. These waves were first found in the deep sea [5]. The applications of this wave appear in oceanography and optical fibers [6, 7]. Nowadays, a new type of wandering wave is known as the lumped wave, which is defined as a wave whose wandering is restricted in each direction in space [8].

Nowadays, based on the computer revolution, which has a great effect on the derivation of computational, semi-analytical, and numerical schemes, many schemes have derived, such as the sech-tanh expansion method, the auxiliary equation method, the direct algebraic equation method, the iteration method, the exponential expansion method, B-spline schemes, Kudryashov methods, the Adomian decomposition method, Khater methods, $\left(\tfrac{\varphi ^{\prime} (\zeta )}{\varphi (\zeta )}\right)$- expansion methods, and so on [915]; in addition, the lump solutions of many nonlinear phenomena have been investigated [1620].

In this context, our paper aims to study the new soliton-wave solutions of the generalized CBS equation through the perspective of ESE, and MKud analysis technology [21, 22]. Another goal of the manuscript is to limit the examination of the semi-analytical and numerical solutions of the considered model to an explanation of the accuracy of the analytical solutions obtained and the analytical schemes used [23, 24]. The generalized CBS equation is given by [2530]:$\begin{eqnarray}{{ \mathcal B }}_{t}+{{ \mathcal B }}_{{xxy}}+3{ \mathcal B }{{ \mathcal B }}_{y}+3{{ \mathcal B }}_{x}{{ \mathcal C }}_{y}+{r}_{1}{{ \mathcal B }}_{y}+{r}_{2}{{ \mathcal C }}_{{yy}}=0,\end{eqnarray}$where ${ \mathcal B }={ \mathcal B }(x,t,t),{ \mathcal C }={ \mathcal C }(x,y,t)$ describe the dynamics of solitons and nonlinear waves, while ri, i = 1, 2 are arbitrary constants to be determined later. Using the next relation between ${ \mathcal C }\ \& \ { \mathcal B }$ in the following formula, ${ \mathcal B }={{ \mathcal C }}_{x}$, yields$\begin{eqnarray}{{ \mathcal C }}_{{tx}}+{{ \mathcal C }}_{{xxxy}}+3{{ \mathcal C }}_{x}{{ \mathcal C }}_{{xy}}+3{{ \mathcal C }}_{{xx}}{{ \mathcal C }}_{y}=0.\end{eqnarray}$Eq. (2) is given in (2+1) dimensions, and has the following formula$\begin{eqnarray}{{ \mathcal B }}_{t}+{ \mathcal B }{{ \mathcal B }}_{z}+\displaystyle \frac{1}{2}{{ \mathcal B }}_{x}{\partial }_{x}^{-1}{{ \mathcal B }}_{z}+\displaystyle \frac{1}{4}{{ \mathcal B }}_{{xxz}}=0,\end{eqnarray}$where ${\partial }_{x}^{-1}{{ \mathcal B }}_{z}=\in {{ \mathcal B }}_{z}{dx}$. Equation (2) was derived by Bogoyavlenskii and Schiff in different ways, such as the modified Lax formalism and a reduction of the self-dual Yang–Mills equation. Additionally, the nonlocal form of the (2+1)-dimensional model is given by$\begin{eqnarray}\left\{\begin{array}{c}{{ \mathcal C }}_{x}-{{ \mathcal B }}_{z}=0,\\ 2{{ \mathcal B }}_{x}{ \mathcal C }+{{ \mathcal B }}_{{xxz}}+4{{ \mathcal B }}_{t}+4{ \mathcal B }{{ \mathcal B }}_{z}=0,\end{array}\right.\end{eqnarray}$while the potential form is given by$\begin{eqnarray}4{{ \mathcal E }}_{{tx}}+4{{ \mathcal E }}_{x}{{ \mathcal E }}_{{xz}}+2{{ \mathcal E }}_{{xx}}{{ \mathcal E }}_{z}+{{ \mathcal E }}_{{xxxz}}=0.\end{eqnarray}$Equations (2), (4), and (5) describe many nonlinear phenomena in plasma physics. Employing the next wave transformation ${ \mathcal C }={ \mathcal C }(x,y,t)={ \mathcal S }({\mathfrak{P}})$, where ${\mathfrak{P}}={r}_{1}x+{r}_{2}y\,+{r}_{3}t$ and ri, i = 1, 2, 3 are arbitrary constants, converts equation (2) into the following ordinary differential equation. Integrating the obtained equation once with the zero constant of integration, we get:$\begin{eqnarray}{r}_{3}{{ \mathcal S }}^{{\prime} }+{r}_{2}{r}_{1}^{2}{{ \mathcal S }}^{(3)}+3{r}_{2}{r}_{1}{\left({{ \mathcal S }}^{{\prime} }\right)}^{2}=0.\end{eqnarray}$Handling equation (6) through the perspective of the abovementioned analytical schemes and the homogeneous balance principle gives n = 1. Thus, the general solutions of equation (6) are evaluated by$\begin{eqnarray}{ \mathcal S }({\mathfrak{P}})=\left\{\begin{array}{c}{\sum }_{i=-n}^{n}{a}_{i}{\mathfrak{C}}{\left({\mathfrak{P}}\right)}^{i}={a}_{1}{\mathfrak{C}}({\mathfrak{P}})+\displaystyle \frac{{a}_{-1}}{{\mathfrak{C}}({\mathfrak{P}})}+{a}_{0},\\ {\sum }_{i=0}^{n}{a}_{i}{\mathfrak{L}}{\left({\mathfrak{P}}\right)}^{i}={a}_{1}{\mathfrak{L}}({\mathfrak{P}})+{a}_{0},\end{array}\right.\end{eqnarray}$where a0, a1, and a−1 are arbitrary constants.

The other sections of this manuscript are as follows: section 2 studies the general solution of the generalized CBS model. In addition, the accuracy of the obtained solution is checked using the abovementioned semi-analytical scheme. Section 3 introduces the solutions and achieves the goals of our research paper. Section 4 provides a summary of the manuscript.

2. Comparison of analytical and semi-analytical approaches to the generalized CBS equation

This section studies the analytical, semi-analytical, and numerical simulation of the generalized CBS equation. The headings of this section can be summarized in the following order:We apply the ESE and MKud schemes to equation (6) to get solitary-wave solutions, then obtain the initial and boundary conditions to investigate semi-analytical solutions through the VI technique.
We check the accuracy of the obtained solution and calculate the absolute error value between the exact and semi-analytical solutions.


2.1. ESE analytical vs. VI numerical techniques, along with the generalized CBS equation

Applying the ESE scheme’s framework and its auxiliary, $\left({{\mathfrak{C}}}^{{\prime} }({\mathfrak{P}})={h}_{3}{\mathfrak{C}}{\left({\mathfrak{P}}\right)}^{2}+{h}_{2}{\mathfrak{C}}({\mathfrak{P}})+{h}_{1}\right)$ (where hi, i = 1, 2, 3 are arbitrary constants to be determined later), to equation (6) obtains the following sets of abovementioned parameters:

Set I$\begin{eqnarray*}{a}_{-1}\to \displaystyle \frac{2{h}_{1}}{{r}_{1}},{a}_{1}\to 0,{r}_{3}\to -\left({h}_{2}^{2}-4{h}_{1}{h}_{3}\right){r}_{2}.\end{eqnarray*}$

Set II$\begin{eqnarray*}{a}_{-1}\to 0,{a}_{1}\to -\displaystyle \frac{2{h}_{3}}{{r}_{1}},{r}_{3}\to -\left({h}_{2}^{2}-4{h}_{1}{h}_{3}\right){r}_{2}.\end{eqnarray*}$Therefore, the solitary solutions of the considered model are constructed in the following formulas:

For h2 = 0, h1h3 > 0, we get$\begin{eqnarray}{{ \mathcal C }}_{{\rm{I}},1}={a}_{0}+\displaystyle \frac{2\sqrt{{h}_{1}{h}_{3}}\cot \left(\sqrt{{h}_{1}{h}_{3}}\left(\eta +{r}_{2}\left(4{h}_{1}{h}_{3}t+y\right)+{r}_{1}x\right)\right)}{{r}_{1}},\end{eqnarray}$$\begin{eqnarray}{{ \mathcal C }}_{{\rm{I}},2}={a}_{0}+\displaystyle \frac{2\sqrt{{h}_{1}{h}_{3}}\tan \left(\sqrt{{h}_{1}{h}_{3}}\left(\eta +{r}_{2}\left(4{h}_{1}{h}_{3}t+y\right)+{r}_{1}x\right)\right)}{{r}_{1}},\end{eqnarray}$$\begin{eqnarray}{{ \mathcal C }}_{\mathrm{II},1}={a}_{0}-\displaystyle \frac{2\sqrt{{h}_{1}{h}_{3}}\tan \left(\sqrt{{h}_{1}{h}_{3}}\left(\eta +{r}_{2}\left(4{h}_{1}{h}_{3}t+y\right)+{r}_{1}x\right)\right)}{{r}_{1}},\end{eqnarray}$$\begin{eqnarray}{{ \mathcal C }}_{\mathrm{II},2}={a}_{0}-\displaystyle \frac{2\sqrt{{h}_{1}{h}_{3}}\cot \left(\sqrt{{h}_{1}{h}_{3}}\left(\eta +{r}_{2}\left(4{h}_{1}{h}_{3}t+y\right)+{r}_{1}x\right)\right)}{{r}_{1}}.\end{eqnarray}$

For h2 = 0, h1h3 < 0, we get$\begin{eqnarray}{{ \mathcal C }}_{{\rm{I}},3}={a}_{0}-\displaystyle \frac{2\sqrt{-{h}_{1}{h}_{3}}\coth \left(\sqrt{-{h}_{1}{h}_{3}}\left({r}_{2}\left(4{h}_{1}{h}_{3}t+y\right)+{r}_{1}x\right)\mp \tfrac{\mathrm{log}(\eta )}{2}\right)}{{r}_{1}},\end{eqnarray}$$\begin{eqnarray}{{ \mathcal C }}_{{\rm{I}},4}={a}_{0}-\displaystyle \frac{2\sqrt{-{h}_{1}{h}_{3}}\tanh \left(\sqrt{-{h}_{1}{h}_{3}}\left({r}_{2}\left(4{h}_{1}{h}_{3}t+y\right)+{r}_{1}x\right)\mp \tfrac{\mathrm{log}(\eta )}{2}\right)}{{r}_{1}},\end{eqnarray}$$\begin{eqnarray}{{ \mathcal C }}_{\mathrm{II},3}={a}_{0}-\displaystyle \frac{2\sqrt{-{h}_{1}{h}_{3}}\tanh \left(\sqrt{-{h}_{1}{h}_{3}}\left({r}_{2}\left(4{h}_{1}{h}_{3}t+y\right)+{r}_{1}x\right)\mp \tfrac{\mathrm{log}(\eta )}{2}\right)}{{r}_{1}},\end{eqnarray}$$\begin{eqnarray}{{ \mathcal C }}_{\mathrm{II},4}={a}_{0}-\displaystyle \frac{2\sqrt{-{h}_{1}{h}_{3}}\coth \left(\sqrt{-{h}_{1}{h}_{3}}\left({r}_{2}\left(4{h}_{1}{h}_{3}t+y\right)+{r}_{1}x\right)\mp \tfrac{\mathrm{log}(\eta )}{2}\right)}{{r}_{1}}.\end{eqnarray}$

For h1 = 0, h2 > 0, we get$\begin{eqnarray}{{ \mathcal C }}_{\mathrm{II},5}={a}_{0}+\displaystyle \frac{2{h}_{2}}{{r}_{1}\left({h}_{3}{e}^{{h}_{2}\left(\eta +{r}_{2}\left(y-{h}_{2}^{2}t\right)+{r}_{1}x\right)}-1\right)}+\displaystyle \frac{2{h}_{2}}{{r}_{1}}.\end{eqnarray}$

For h1 = 0, h2 < 0, we get$\begin{eqnarray}{{ \mathcal C }}_{\mathrm{II},6}={a}_{0}-\displaystyle \frac{2{h}_{3}}{{r}_{1}\left({h}_{3}{e}^{{h}_{2}\left(\eta +{r}_{2}\left(y-{h}_{2}^{2}t\right)+{r}_{1}x\right)}+1\right)}+\displaystyle \frac{2{h}_{3}}{{r}_{1}}.\end{eqnarray}$

For $4{h}_{1}{h}_{3}\gt {h}_{2}^{2}$, we get$\begin{eqnarray}{{ \mathcal C }}_{{\rm{I}},5}={a}_{0}-\displaystyle \frac{4{h}_{1}{h}_{3}}{{r}_{1}\left({h}_{2}-\sqrt{4{h}_{1}{h}_{3}-{h}_{2}^{2}}\tan \left(\tfrac{1}{2}\sqrt{4{h}_{1}{h}_{3}-{h}_{2}^{2}}\left(\eta +{r}_{2}\left({h}_{2}^{2}(-t)+4{h}_{1}{h}_{3}t+y\right)+{r}_{1}x\right)\right)\right)},\end{eqnarray}$$\begin{eqnarray}{{ \mathcal C }}_{{\rm{I}},6}={a}_{0}-\displaystyle \frac{4{h}_{1}{h}_{3}}{{r}_{1}\left({h}_{2}-\sqrt{4{h}_{1}{h}_{3}-{h}_{2}^{2}}\cot \left(\tfrac{1}{2}\sqrt{4{h}_{1}{h}_{3}-{h}_{2}^{2}}\left(\eta +{r}_{2}\left({h}_{2}^{2}(-t)+4{h}_{1}{h}_{3}t+y\right)+{r}_{1}x\right)\right)\right)},\end{eqnarray}$$\begin{eqnarray}{{ \mathcal C }}_{\mathrm{II},7}={a}_{0}-\displaystyle \frac{\sqrt{4{h}_{1}{h}_{3}-{h}_{2}^{2}}\tan \left(\tfrac{1}{2}\sqrt{4{h}_{1}{h}_{3}-{h}_{2}^{2}}\left(\eta +{r}_{2}\left({h}_{2}^{2}(-t)+4{h}_{1}{h}_{3}t+y\right)+{r}_{1}x\right)\right)}{{r}_{1}}+\displaystyle \frac{{h}_{2}}{{r}_{1}},\end{eqnarray}$$\begin{eqnarray}{{ \mathcal C }}_{\mathrm{II},8}={a}_{0}-\displaystyle \frac{\sqrt{4{h}_{1}{h}_{3}-{h}_{2}^{2}}\cot \left(\tfrac{1}{2}\sqrt{4{h}_{1}{h}_{3}-{h}_{2}^{2}}\left(\eta +{r}_{2}\left({h}_{2}^{2}(-t)+4{h}_{1}{h}_{3}t+y\right)+{r}_{1}x\right)\right)}{{r}_{1}}+\displaystyle \frac{{h}_{2}}{{r}_{1}}.\end{eqnarray}$

2.1.1. Comparison of the analytical and numerical solutions

Applying the VI method to equation (2), we get the following semi-analytical solutions:$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal C }}_{n+1} & = & {{ \mathcal C }}_{n}-{\displaystyle \int }_{0}^{t}\left(-{\left({{ \mathcal C }}_{n}\right)}_{{xs}}+{\left({{ \mathcal C }}_{n}\right)}_{{xxxz}}\right.\\ & & \left.+4{\left({{ \mathcal C }}_{n}\right)}_{x}{\left({{ \mathcal C }}_{n}\right)}_{{xz}}+2{\left({{ \mathcal C }}_{n}\right)}_{{xx}}{\left({{ \mathcal C }}_{n}\right)}_{z}+3{\left({{ \mathcal C }}_{n}\right)}_{{yy}}\right){ds}.\end{array}\end{eqnarray}$Using equation (22),$\begin{eqnarray}{{ \mathcal C }}_{0}=1-2\tanh (x+2y),\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{{ \mathcal C }}_{1}(\zeta ,t)=1-2\tanh (x+2y)\\ \quad \times \left(8t\left(\cosh (2(x+2y))-11\right){{\rm{sech}} }^{4}(x+2y)+1\right),\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{{ \mathcal C }}_{2}(\zeta ,t)=2{\rm{sech}} (x)\sinh (2y){\rm{sech}} (x+2y)\\ \quad \times \left(32t{{\rm{sech}} }^{2}(x+2y)\left(2{{\rm{sech}} }^{2}(x+2y)\left(12t{{\rm{sech}} }^{2}(x+2y)\right.\right.\right.\\ \quad \times \left(9056t-270){{\rm{sech}} }^{2}(x+2y)-3600t\left(3\cosh (2(x+2y))-1\right)\right.\\ \quad \left.\left.\left.\left.\times {{\rm{sech}} }^{6}(x+2y)-864t+243\right)+4t(64t-117)+3\right)+8t-1\right)-1\right)\\ \quad +2\tanh (x)\left(32t{{\rm{sech}} }^{2}(x+2y)\left(2{{\rm{sech}} }^{2}(x+2y)\right.\right.\\ \quad \times \left(12t{{\rm{sech}} }^{2}(x+2y)\left(9056t-270){{\rm{sech}} }^{2}(x+2y)\right.\right.\\ \quad \left.-3600t\left(3\cosh (2(x+2y))-1\right){{\rm{sech}} }^{6}(x+2y)-864t+243\right)\\ \quad \left.\left.\left.+4t(64t-117)+3\right)+8t-1\right)-1\right)+1.\end{array}\end{eqnarray}$Using same steps, we can get ${{ \mathcal C }}_{i}$, i = 3, 4, 5, ⋯.

2.2. MKud analytical vs. VI numerical techniques, along with the generalized CBS equation
Applying the MKud scheme’s framework and its auxiliary, $\left({{\mathfrak{L}}}^{{\prime} }({\mathfrak{P}})=\mathrm{ln}(k)\left({\mathfrak{L}}{\left({\mathfrak{P}}\right)}^{2}-{\mathfrak{L}}({\mathfrak{P}})\right)\right)$ (where k is an arbitrary constant to be evaluated later), to equation (6) obtains the following sets of abovementioned parameters:

Set I$\begin{eqnarray*}{a}_{1}\to -\displaystyle \frac{2\mathrm{log}(k)}{{r}_{1}},{r}_{3}\to {r}_{2}\left(-{\mathrm{log}}^{2}(k)\right).\end{eqnarray*}$Therefore, the solitary solutions of the considered model are constructed in the following formula:$\begin{eqnarray}{ \mathcal C }={a}_{0}-\displaystyle \frac{2\mathrm{log}(k)}{{r}_{1}\left(1\pm {k}^{{r}_{2}\left(y-t{\mathrm{log}}^{2}(k)\right)+{r}_{1}x}\right)}.\end{eqnarray}$

2.2.1. Comparison of the analytical and semi-analytical solutions
Applying the VI method along with equation (22) to equation (2), one obtains$\begin{eqnarray}{{ \mathcal C }}_{0}=3-\displaystyle \frac{2}{{e}^{x+2y}+1},\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal C }}_{1} & = & \displaystyle \frac{1}{{\left({{\rm{e}}}^{x+2y}+1\right)}^{3}}\\ & & \times \left({{\rm{e}}}^{x+2y}\left({{\rm{e}}}^{x+2y}\left(4t+3{{\rm{e}}}^{x+2y}+7\right)-4t+5\right)+1\right),\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal C }}_{2} & = & \frac{1}{{\left({{\rm{e}}}^{x+2y}+1\right)}^{9}}\left({{\rm{e}}}^{x+2y}\left(\left(4t\left(896{t}^{2}-77t+10\right)+266\right)\right.\right.\\ & & \times {{\rm{e}}}^{4x+8y}+\left(238-4t\left(896{t}^{2}-77t+10\right)\right)\\ & & \times {{\rm{e}}}^{3x+6y}+(25-4(t-2)t){{\rm{e}}}^{7(x+2y)}\\ & & -4\left(3t(5t(16t+3)-6)-49\right){{\rm{e}}}^{5(x+2y)}\\ & & +4\left(3t(5t(16t+3)-6)+35\right){{\rm{e}}}^{2x+4y}\\ & & -4\left(t(t(16t+31)+10)-13\right){{\rm{e}}}^{x+2y}\\ & & +4\left(t(t(16t+31)+10)+23\right){{\rm{e}}}^{6(x+2y)}\\ & & \left.\left.+4(t-2)t+3{{\rm{e}}}^{8(x+2y)}+11\right)+1\right).\end{array}\end{eqnarray}$Using same steps, we can get ${{ \mathcal C }}_{i}$, i = 3, 4, 5, ⋯.

3. Interpretation of the results

This section highlights the novelty of our research papers. It also shows the accuracy of the analytical solution obtained. The ESE and MKud calculation schemes have been applied to the generalized CBS equation for the constructed solitary-wave solution. Many different wave solutions have been obtained, some of which are demonstrated in 2D, 3D, and some sketches in profile drawings to explain solitary waves’ dynamic behavior in nonlinear phenomena in plasma physics. Figures 1 and 3 show the periodic-kink solitary-wave solution. While figure 2 show the semi-analytical solution in three different forms (2D, 3D, and contour plots). Comparing our solution with the solutions obtained in previously published papers, it can be seen that our solution is completely different from the solution evaluated in [2529].

Figure 1.

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Figure 1.Solitary-wave-solution equation (13) in 3D, 2D, and contour plots.


Figure 2.

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Figure 2.Semi-analytical solutions of equation (6) in 3D and contour plots.


The VI scheme was applied to the considered model based on the computational solution obtained. The absolute error between the analytical and semi-analytic solutions was calculated, to show the accuracy of the solution and the method used (tables 1 and 2 and figures 3 and 4). This calculation shows that the MKud method is superior to the ESE method, and its absolute error value is much smaller than the absolute value obtained by the ESE method (figure 5).

Figure 3.

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Figure 3.Solitary-wave solutions of equation (26) in 3D, 2D, and contour plots.


Figure 4.

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Figure 4.Semi-analytical solutions of equation (2) in 3D and contour plots.


Figure 5.

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Figure 5.Absolute errors between the ESE and MKud analytical schemes and the TQBS numerical scheme.



Table 1.
Table 1.Absolute errors between the analysis obtained using the ESE method and the constructed semi-analytical solution using the VI scheme for different values of x and the following values of the abovementioned parameters: a0 = 1, h2 = 0, h1 = −1, h3 = 1, r1 = −1, and r2 = −2.
Value of xt = 1t = 2t = 3t = 4t = 5t = 6t = 7t = 8t = 9t = 10
00.071 941 143.999 959 593.999 963 583.999 934 563.999 897 13.999 851 193.999 796 843.999 734 053.999 662 8193.999 583 142
10.009 889 993.999 816 273.999 995 073.999 991 143.999 986 073.999 979 863.999 972 513.999 964 013.999 954 3683.999 943 585
20.001 341 333.998 658 313.999 999 333.999 998 83.999 998 123.999 997 273.999 996 283.999 995 133.999 993 8243.999 992 365
30.000 181 583.990 109 473.999 999 913.999 999 843.999 999 743.999 999 633.999 999 53.999 999 343.999 999 1643.999 998 967
42.4575E-053.928 055 153.999 999 983.999 999 983.999 999 973.999 999 953.999 999 933.999 999 913.999 999 8873.999 999 86
53.3259E-063.523 188 313.999 999 94443.999 999 993.999 999 993.999 999 993.999 999 9853.999 999 981
64.5012E-0723.999 999 55444443.999 999 9983.999 999 997
76.0917E-080.476 811 693.999 996 674444444
88.2442E-090.071 944 843.999 975 424444444
91.1157E-090.009 890 493.999 818 414444444
101.51E-100.001 341 43.998 658 64444444
112.0435E-110.000 181 593.990 109 514444444
122.7656E-122.4577E-053.928 055 163.999 999 99444444
133.7437E-133.3261E-063.523 188 313.999 999 94444444
145.0626E-144.5014E-0723.999 999 55444444
156.8834E-156.092E-080.476 811 693.999 996 67444444
168.8818E-168.2446E-090.071 944 843.999 975 42444444
172.2204E-161.1158E-090.009 890 493.999 818 41444444
1801.5101E-100.001 341 43.998 658 6444444
1902.0436E-110.000 181 593.990 109 51444444
2002.7658E-122.4577E-053.928 055 163.999 999 9944444
2103.7437E-133.3261E-063.523 188 313.999 999 9444444
2205.0626E-144.5014E-0723.999 999 5544444
2306.8834E-156.092E-080.476 811 693.999 996 6744444
2408.8818E-168.2446E-090.071 944 843.999 975 4244444
2502.2204E-161.1158E-090.009 890 493.999 818 4144444
26001.5101E-100.001 341 43.998 658 644444
27002.0436E-110.000 181 593.990 109 5144444
28002.7658E-122.4577E-053.928 055 163.999 999 994444
29003.7437E-133.3261E-063.523 188 313.999 999 944444
30005.0626E-144.5014E-0723.999 999 554444

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Table 2.
Table 2.Absolute errors between the analysis obtained using the MKud method and the constructed semi-analytical solution using the VI scheme for different values of x and the following values of the abovementioned parameters a0 = 1, k = e, r1 = −1, and r2 = −2.
Value of xt = 1t = 2t = 3t = 4t = 5t = 6t = 7t = 8t = 9t = 10
00.000 761 90.004 856 6910.035 343 1520.236 875 6940.997 209 5911.757 184 9011.957 641 6821.986 335 2081.987 919 891.985 454 506
10.000 280 2470.001 789 0030.013 152 7350.094 285 7090.536 850 641.460 485 7991.902 784 8581.983 386 091.993 952 211.994 397 482
20.000 103 0910.000 658 4530.004 859 3430.035 763 7630.238 025 350.999 398 5991.760 722 7941.962 837 2171.993 496 3631.997 353 871
33.792 45E-050.000 242 2740.001 790 4730.013 308 8840.094 711 6660.537 661 431.461 796 3421.904 709 9681.986 040 4781.997 450 479
41.395 15E-058.913 34E-050.000 659 0610.004 916 9790.035 920 8730.238 324 3680.999 881 9431.761 432 8691.963 816 4131.994 787 057
55.132 47E-063.279 11E-050.000 242 5070.001 811 7020.013 366 7370.094 821 770.537 839 4071.462 057 8161.905 070 561.986 515 805
61.888 13E-061.206 33E-058.922 03E-050.000 666 8740.004 938 2690.035 961 3920.238 389 8640.999 978 1691.761 565 5731.963 991 345
76.946 04E-074.437 85E-063.282 33E-050.000 245 3820.001 819 5360.013 381 6450.094 845 8680.537 874 8111.462 106 6421.905 134 923
82.5553E-071.6326E-061.207 51E-059.027 79E-050.000 669 7560.004 943 7540.035 970 2570.238 402 8890.999 996 1321.761 589 252
99.400 44E-086.005 99E-074.442 21E-063.321 23E-050.000 246 4420.001 821 5530.013 384 9060.094 850 6590.537 881 421.462 115 353
103.458 23E-082.209 48E-071.6342E-061.221 83E-059.066 79E-050.000 670 4980.004 944 9540.035 972 020.238 405 3210.999 999 336
111.272 21E-088.128 22E-086.011 89E-074.494 87E-063.335 58E-050.000 246 7150.001 821 9950.013 385 5550.094 851 5540.537 882 599
124.6802E-092.990 21E-082.211 65E-071.653 57E-061.227 11E-059.076 84E-050.000 670 6610.004 945 1920.035 972 3490.238 405 754
131.721 75E-091.100 04E-088.136 21E-086.083 15E-074.514 29E-063.339 28E-050.000 246 7750.001 822 0820.013 385 6760.094 851 713
146.333 96E-104.0468E-092.993 14E-082.237 87E-071.660 72E-061.228 46E-059.079 04E-050.000 670 6930.004 945 2370.035 972 408
152.330 11E-101.488 73E-091.101 12E-088.232 66E-086.109 43E-074.519 29E-063.340 09E-050.000 246 7860.001 822 0990.013 385 697
168.572 03E-115.476 75E-104.050 78E-093.028 62E-082.247 54E-071.662 56E-061.228 76E-059.079 47E-050.000 670 6990.004 945 245
173.1537E-112.0148E-101.4902E-091.114 17E-088.268 22E-086.1162E-074.520 38E-063.340 25E-050.000 246 7890.001 822 102
181.160 18E-117.412 03E-115.482 14E-104.0988E-093.041 71E-082.250 03E-071.662 96E-061.228 82E-059.079 56E-050.000 670 7
194.269 03E-122.7268E-112.016 77E-101.507 86E-091.118 98E-088.277 38E-086.117 68E-074.5206E-063.340 28E-050.000 246 789
201.569 41E-121.003 02E-117.419 18E-115.547 12E-104.1165E-093.045 08E-082.250 57E-071.663 04E-061.228 83E-059.079 57E-05
215.768 72E-133.690 38E-122.729 33E-112.040 67E-101.514 38E-091.120 22E-088.279 39E-086.117 98E-074.520 64E-063.340 28E-05
222.144 95E-131.359 36E-121.004 26E-117.507 42E-115.5711E-104.121 07E-093.045 82E-082.250 68E-071.663 05E-061.228 83E-05
237.815 97E-145.004 89E-133.694 82E-122.761 79E-112.049 49E-101.516 06E-091.120 49E-088.279 78E-086.118 03E-074.520 65E-06
242.930 99E-141.838 53E-131.3598E-121.016 03E-117.539 75E-115.577 27E-104.122 06E-093.045 96E-082.2507E-071.663 06E-06
251.065 81E-146.750 16E-144.996E-133.737 45E-122.773 69E-112.051 76E-101.516 42E-091.120 55E-088.279 86E-086.118 04E-07
265.773 16E-152.620 13E-141.856 29E-131.376 23E-121.020 52E-117.548 18E-115.578 63E-104.122 26E-093.045 99E-082.2507E-07
273.552 71E-151.154 63E-146.9722E-145.084 82E-133.756 11E-122.776 93E-112.052 28E-101.5165E-091.120 56E-088.279 87E-08
288.881 78E-163.108 62E-152.4869E-141.860 73E-131.380 67E-121.021 49E-117.549 83E-115.578 88E-104.1223E-093.045 99E-08
291.776 36E-153.108 62E-151.065 81E-147.016 61E-145.0937E-133.759 66E-122.7776E-112.052 37E-101.516 51E-091.120 56E-08
30−1.77636E-15−1.33227E-151.776 36E-152.353 67E-141.851 85E-131.381 12E-121.021 63E-117.550 09E-115.578 91E-104.1223E-09

New window|CSV

4. Conclusions

This research paper has successfully processed the generalized CBS equations generated for plasma physics and weak dispersion media. The ESE, MKud, and VI schemes were used to find accurate and novel solitary-wave solutions for the model under consideration. Three different types of sketch were used to represent the kink solitary-wave solution. The accuracy of the MKud method was verified using the ESE method. This is our fourth paper on the topic of the ESE method, which examined the accuracy of the calculation scheme. This series aims to determine the accuracy of all the calculation schemes.

Acknowledgments

This research was supported by Taif University Researchers Supporting Project Number (TURSP-2020/247), Taif University, Taif, Saudi Arabia. Additionally, the authors extend their appreciation to the Deanship of Scientific Research at the King Khalid University, Abha, KSA, for funding this work through research group under grant number (RGP. 2/ 121/ 42).


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DOI:10.1007/s11071-016-2785-2 [Cited within: 1]

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相关话题/Folded novel accurate

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