Mostf M.A.Khter ,,Smir A.Slm

a Department of Mathematics,Faculty of Science,Jiangsu University,Zhenjiang,212013,China

b Department of Mathematics,Obour High Institute For Engineering and Technology,Cairo,11828,Egypt

c Division of Biochemistry,Department of Pharmacology,College of Pharmacy,Taif University,P.O.Box 11099,Taif 21944,Saudi Arabia

Keywords:Modified Benjamin–Bona–Mahony equation Approximate solutions

ABSTRACT In this article,semi-analytical and numerical simulations of the well-known modified Benjamin–Bona–Mahony (mBBM) equation are processed.This study targets to check the accuracy of the obtained analytical solutions of the mBBM model that have been obtained in [1]through three recent analytical schemes (extended simplest equation (ESE) method,modified kudryashov (mKud) method,and Sech-Tanh (ST) expansion method).The considered model describes the propagation of long waves in the nonlinear dispersive media in a visual illusion.The Homotopy iteration (HI) method,exponential cubic-B-spline (ECBS) method,and trigonometric-quantic-B-spline (TQBS) method are employed to construct novel semi-analytical and accurate numerical solutions.The obtained solutions’ accuracy has been checked through some different types of two-dimensional graphs

1.Introduction

Recently,many analytical,semi-analytical,numerical schemes have been formulated,such as sech-tanh expansion method,modified Riccati-expansion method,modified Khater method,generalized Khater method,extended simplest equation method,exponential method,ansatz method,unified method,direct algebraic method,modified Kudryashov method,Adomian decomposition method,iteration method,B-spline schemes,Homotopy iteration method,and so on [2–11].These methods are employed to handle formulated complex phenomena in nonlinear evolution equations[12–16].Many complex phenomena have been analytical,semianalytical,and numerical researched,and many novel properties have been discovered and have been used in the models’ applications [17–21].

In this context,we study the approximate solutions of the mBBM model which is given by [1,9,22,23]

whereris arbitrary constant.Eq.(1) has been analytically studied by three recent analytical schemes (ESE,mKud,and ST expansion methods) [1].That research paper has used the next wave transformation [V(x,t)=U(ζ),ζ=kx+λt,wherek,λare arbitrary constants.],to convert Eq.(1) to the following ODE

Integrating Eq.(2) once with zero constant of integration obtain

Many novel solutions have been constructed and demonstrated in some distinct figures.We select three different solutions for our research for investigating their methods’ accuracy.The three selected analytical solutions have been constructed by

The paper’s rest sections are given as follows; Section 2 investigates semi-analytical and approximate solutions of the mBBM model through AD,ECBS,TQBS schemes [18,24–27].Section 3 shows the accuracy of the obtained solutions.Section 4 gives the conclusion of the whole paper.

2.Numerical solutions

This section investigates the obtained analytical solutions of the researched model to check their accuracy through applying the HI,ECBS,TQBS schemes.The requested conditions of these schemes are constructed through the solutions as mentioned earlier by using the following values of the shown-arbitrary constantsThus,the initial conditions of this model are given by 1.

Fig.1.Breath wave graphs of Eq.(9) when ζ=k x+λt in three (a),two (b),contour (c) representations.

2.1.Semi–analytical solutions

Employing the HI method through the above-conditions,obtains

Fig.2.Kink wave graphs of Eq.(10) when ζ=k x+λt in three (a),two (b),contour (c) representations.

Fig.3.Solitary wave graphs of Eq.(11) when ζ=k x+λt in three (a),two (b),contour (c) representations.

Fig.4.Matching between analytical,semi-analytical,and approximate solutions based on ESE method and HI (a),ECBS (b),TQBS (c) schemes.

Consequently,we construct the semi-analytical solutions of the mBBM model in the following formula

Fig.5.Matching between analytical,semi-analytical,and approximate solutions based on m Kud method and HI (a),ECBS (b),TQBS (c) schemes.

The following graphs 1,2,3 show the dynamical behavior of long waves in the nonlinear dispersive media in a visual illusion through two,three,and contour graphs

While the following Tables 2,3,4 show the semi-analytical,relative error and absolute error based on the obtained semianalytical solutions and above-mentioned analytical solutions through the ESE,mKud,and ST expansion methods,respectively.

Table 1 The mBBM model’s initial conditions.

Table 2 Solutions and error’s value along with different values of ζ for ESE and HI methods’solutions.

Table 3 Solutions and error’s value along with different values of ζ for m Kud and HI methods’solutions.

Table 4 Solutions and error’s value along with different values of ζ for ST expansion and HI methods’ solutions.

Table 5 Numerical Solution and errors’ values along with different values of ζ for ESE and ECBS methods’ solutions.

Table 6 Numerical Solution and errors’ values along with different values of ζ for m Kud and ECBS methods’ solutions.

Table 7 Numerical Solution and errors’ values along with different values of ζ for ST expansion and ECBS methods’ solutions.

Table 8 Numerical Solution and errors’ values along with different values of ζ for ESE and TQBS methods’ solutions.

2.2.Approximate solutions

Applying the ECBS,TQBS schemes to the researched model with the above-initial conditions (Table 1),gets the following values of approximate,relative and absolute error:

2.2.1.ECBS-solutions

Here,we explain the obtained results by applying the ECBS method to the mBBM model (Tables 5,6,7)

Table 9 Numerical Solution and errors’ values along with different values of ζ for m Kud and TQBS methods’solutions.

Table 10 Numerical Solution and errors’ values along with different values of ζ for ST expansion and TQBS methods’ solutions.

Fig.6.Matching between analytical,semi-analytical,and approximate solutions based on ST expansion method and HI (a),ECBS (b),TQBS (c) schemes.

Fig.7.Accuracy of the obtained solutions through the HI method (a),ECBS method (b),TQBS method (c).

2.2.2.TQBS-solutions

Here,we explain the obtained results by applying the TQBS method to the mBBM model (Tables 8,9,10).

3.Results and discussion

This part shows the obtained solutions and discusses their accuracy.We have successfully obtained a novel semi-analytical solution,and we have represented our solutions by some figures to show the dynamical behavior of the long waves in the nonlinear dispersive media [Figs.1,2,3 ].We also have applied two numerical schemes (ECBS and TQBs methods) to the mBBM model and the obtained result have been shown through Tables [2,3,4,5,6,7,8,9,10]and Figures [4,5,6].Comparing the shown data of the above–shown tables show the accuracy of the ST expansion method’s solution over the ESE and m Kud methods’ solutions.This accuracy has been demonstrated in Fig.7.

Investigating our solutions with solutions in [28,29]that have been obtained through employing the modified-expansion method,and an ansatz solution by VikasKumar in 2019 [28],and Ozkan Guner in 2017 [29].They have obtained some good results but their solutions are completely different from that we have constructed.This difference demonstrates our paper’s novelty and contributions.We also plan to use their obtained data for checking the solutions’ accuracy in our future work.

4.Conclusion

This document has applied three semi-analytical and numerical schemes to the mBBM model.The HI method has constructed many accurate semi-analytical solutions and have been represented in 2D,3D,and contour plots.Additionally,the ECBS and TQBS numerical schemes have been successfully employed.The accuracy of the obtained analytical solutions has been investigated by calculating a relative and absolute error between analytical and semi-analytical and numerical solutions.This accuracy has been demonstrated through some two-dimensional graphs.

Authors’ contribution

M.M.A.K.has contributed in the first draft,software,and methodology,M.M.A.K.and S.A.S.have contributed in formal analysis and investigation,while S.A.S.has contributed in writing–review,and editing.All authors have read and agreed to the published version of the manuscript.

Funding

We greatly thank Taif University for providing fund for this work through Taif University Researchers Supporting Project number (TURSP-2020/52),Taif University,Taif,Saudi Arabia.

Availabilityofdataandmaterial

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Codeavailability

The used code of this study is available from the corresponding author upon reasonable request.

Declaration of Competing Interest

There is no conflict of interest.

Acknowledgment

We greatly thank Taif University for providing fund for this work through Taif University Researchers Supporting Project number (TURSP-2020/52),Taif University,Taif,Saudi Arabia.