A Petrov–Galerkin approach for the numerical analysis of soliton and multi-soliton solutions of the Kudryashov–Sinelshchikov equation

Document Type : Research Article

Authors

• H. Samy ¹
• W. Adel ^{2, 3}
• I. Hanafy ¹
• M. Ramadan ¹

¹ Department of Mathematics and Computer Sciences, Faculty of Science, Port-Said University, Egypt.

² Laboratoire Interdisciplinaire de l’Universite Francaise d’Egypte (UFEID Lab), Universite Francaise d’Egypte, Cairo 11837, Egypt.

³ . Department of Mathematics and Engineering Physics, Faculty of Engineering, Mansoura University, Mansoura, 35516, Egypt.

10.22067/ijnao.2024.87548.1422

Abstract

This study delves into the potential polynomial and rational wave solutions of the Kudryashov–Sinelshchikov equation. This equation has multiple applications including the modeling of propagation for nonlinear waves in various physical systems. Through detailed numerical simulations using the finite element approach, we present a set of accurate solitary and soliton solutions for this equation. To validate the effectiveness of our proposed method, we utilize a collocation finite element approach based on quintic B-spline functions. Error norms, including L2 and L∞, are employed to assess the precision of our numerical solutions, ensuring their reliability and accuracy. Visual representations, such as graphs derived from tabulated data, offer valuable insights into the dynamic changes of the equation over time or in response to varying parameters. Furthermore, we compute conservation quantities of motion and investigate the stability of our numerical scheme using Von Neumann theory, providing a comprehensive analysis of the Kudryashov–Sinelshchikov equation and the robustness of our computational approach. The strong alignment between our analytical and numerical results underscores the efficacy of our methodology, which can be extended to tackle more complex nonlinear models with direct relevance to various fields of science and engineering.

Keywords

• Quintic B-spline
• Finite element method
• Error analysis

Main Subjects

• Numerical methods for differential equations and its applications

---

References

[1] Ak, T., Osman, M.S., and Kara, A.H. Polynomial and rational wave solutions of Kudryashov–Sinelshchikov equation and numerical simulations for its dynamic motions, J. Appl. Anal. Comput. 10(5), (2020), 2145–2162.
[2] Ali, L., Liu, X., Ali, B., Mujeed, S., Abdal, S., and Mutahir, A. The impact of nanoparticles due to applied magnetic dipole in micropolar fluid flow using the finite element method, Symmetry, 12(4), (2020), 520.
[3] Belhocine, A., and Abdullah, O.I. Thermomechanical model for the analysis of disc brake using the finite element method in frictional contact, Multiscale Science and Engineering, 2(1), (2020), 27–41.
[4] Bildik, N., and Deniz, S. New approximate solutions to the nonlinear Klein–Gordon equations using perturbation iteration techniques, Discrete Continuous Dyn. Syst. Ser S, 13(3), (2020), 503–518.
[5] Chettri, K., Tamang, J., Chatterjee, P., and Saha, A. Dynamics of nonlinear ion-acoustic waves in Venus’ lower ionosphere, Astrophys. Space Sci. 369(5), (2024).
[6] El-shamy, O., El-barkoki, R., Ahmed, H.M., Abbas, W., and Samir, I. Exploration of new solitons in optical medium with higher-order dispersive and nonlinear effects via improved modified extended tanh function method, Alex. Eng. J. 68, (2023), 611–618.
[7] Feng, Y.-Y., and Wang, C.-H. Discontinuous finite element method applied to transient pure and coupled radiative heat transfer, Int. Commun. Heat Mass Transf. 122, (2021), 105156.
[8] Gepreel, K. A. Analytical methods for nonlinear evolution equations in mathematical physics, Mathematics, 8(12), (2020), 2211.
[9] Ghanbari, B., Kumar, S., Niwas, M., and Baleanu, D. The Lie symmetry analysis and exact Jacobi elliptic solutions for the Kawahara–KdV type equations, Results Phys. 23, (2021), 104006. 

[10] Gohar, M., Li, C., and Li, Z. Finite difference methods for Caputo–Hadamard fractional differential equations, Mediterr. J. Math. 17(6), (2020), 194.
[11] He, J.-H., Qie, N., and He, C.-H. Solitary waves travelling along an unsmooth boundary, Results Phys. 24, (2021), 104104.
[12] Hyder, A.-A., and Barakat, M.A. General improved Kudryashov method for exact solutions of nonlinear evolution equations in mathematical physics, Phys. Scr. 95(4), (2020), 045212.
[13] Jamshed, W. Finite element method in thermal characterization and streamline flow analysis of electromagnetic silver-magnesium oxide nanofluid inside grooved enclosure, Int. Commun. Heat Mass Transf. 130, (2022), 105795.
[14] Jin, Y.-T., and Chen, A.-H. Resonant solitary wave and resonant periodic wave solutions of the Kudryashov–Sinelshchikov equation, Phys. Scr. 95(8), (2020), 085208.
[15] Jisha, C.R., Dubey, R.K., Benton, D., and Rashid, A. The exact solutions for Kudryashov and Sinelshchikov equation with variable coefficients, Phys. Scr. 97(9), (2022), 095212.
[16] Johnson, R.S. Water waves and Korteweg–de Vries equations, J. Fluid Mech. 97(04), (1980), 701.
[17] Karakoc, S.B.G., Saha, A., Bhowmik, S.K., and Sucu, D.Y. Numerical and dynamical behaviors of nonlinear traveling wave solutions of the Kudryashov–Sinelshchikov equation, Wave Motion, 118, (2023), 103121.
[18] Karakoc, S.B.G., Saha, A., and Sucu, D. A novel implementation of Petrov–Galerkin method to shallow water solitary wave pattern and superperiodic traveling wave and its multistability: Generalized Kortewegde Vries equation, Chin. J. Phys. 68, (2020), 605–617.
[19] Kruglov, V.I., and Triki, H. Propagation of coupled quartic and dipole multi-solitons in optical fibers medium with higher-order dispersions, Chaos Soliton. Fract. 172, (2023), 113526.

[20] Kumar, S., Niwas, M., and Dhiman, S.K. Abundant analytical soliton solutions and different wave profiles to the Kudryashov–Sinelshchikov equation in mathematical physics, J. Ocean Eng. Sci. 7(6), (2022), 565–577.
[21] Leissa, A. The historical bases of the Rayleigh and Ritz methods, J. Sound Vib. 287(4-5), (2005), 961–978.
[22] Liu, H., Zheng, X., Wang, H., and Fu, H. Error estimate of finite element approximation for two-sided space-fractional evolution equation with variable coefficient, J. Sci. Comput. 90(1), (2022), 15.
[23] Lu, J. (2018). New exact solutions for Kudryashov–Sinelshchikov equation, Adv. Diff. Equ. 2018(1), 374. [24] Ma, Y.-L., Wazwaz, A.-M., and Li, B.-Q. Soliton resonances, soliton molecules, soliton oscillations and heterotypic solitons for the nonlinear Maccari system, Nonlinear Dyn. 111(19), (2023), 18331–18344.
[25] Mohebbi, A., and Dehghan, M. High-order compact solution of the onedimensional heat and advection–diffusion equations, Appl. Math. Model. 34(10), (2010), 3071–3084.
[26] Nakazawa, S. Computational Galerkin methods, Comput. Methods Appl. Mech. Eng. 50(2), (1985), 199–200.
[27] Ozisik, M., Secer, A., Bayram, M., Cinar, M., Ozdemir, N., Esen, H., and Onder, I. Investigation of optical soliton solutions of higher-order nonlinear Schrödinger equation having Kudryashov nonlinear refractive index, Optik, 274, (2023), 170548.
[28] Pal, N.K., Chatterjee, P., and Saha, A. Solitons, multi-solitons and multi-periodic solutions of the generalized Lax equation by Darboux transformation and its quasiperiodic motions, Int. J. Mod. Phys. B, (2023).
[29] Rabinowitz, P.H. On a class of nonlinear Schrodinger equations, Z. fur Angew. Math. Phys., 43(2), (1992), 270–291.
[30] Ramadan, M., and Aly, H. New approach for solving of extended KdV equation, Alfarama Journal of Basic & Applied Sciences, (2022). 

[31] Rezaei, S., Harandi, A., Moeineddin, A., Xu, B.-X., and Reese, S. A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: Comparison with finite element method, Comput. Methods Appl. Mech. Eng. 401, (2022), 115616.
[32] Rubinstein, J. Sine-Gordon equation, J. Math. Phys. 11(1), (1970), 258–266.
[33] Ryabov, P.N. Exact solutions of the Kudryashov–Sinelshchikov equation, Appl. Math. Comput. 217(7), (2010), 3585–3590.
[34] Saeed, T., and Abbas, I. Finite element analyses of nonlinear DPL bioheat model in spherical tissues using experimental data, Mechanics Based Design of Structures and Machines, 50(4), (2022), 1287–1297.
[35] Shaikhova, G., Kutum, B., Altaybaeva, A., and Rakhimzhanov, B. Exact solutions for the (3+1)-dimensional Kudryashov–Sinelshchikov equation, J. Phys. Conf. Ser. 1416(1), (2019), 012030.
[36] Sultana, M., Arshad, U., Abdel-Aty, A.-H., Akgül, A., Mahmoud, M., and Eleuch, H. New numerical approach of solving highly nonlinear fractional partial differential equations via fractional novel analytical method, Fract. Fract. 6(9), (2022), 512.
[37] Weickert, J., Romeny, B., and Viergever, M. Efficient and reliable schemes for nonlinear diffusion filtering, IEEE Trans. Image Process. 7(3), (1998), 398–410.
[38] Yang, X., and He, X. A fully-discrete decoupled finite element method for the conserved Allen–Cahn type phase-field model of three-phase fluid flow system, Comput. Methods Appl. Mech. Eng. 389, (2022), 114376.
[39] Yin, X., Xu, L., and Yang, L. Evolution and interaction of soliton solutions of Rossby waves in geophysical fluid mechanics, Nonlinear Dyn. 111(13), (2023), 12433–12445.
[40] Yong, W., Zhang, W., Nguyen, H., Bui, X.-N., Choi, Y., Nguyen-Thoi, T., Zhou, J., and Tran, T.T. Analysis and prediction of diaphragm wall  deflection induced by deep braced excavations using finite element method and artificial neural network optimized by metaheuristic algorithms, Reliab. Eng. Syst. Saf., 221, (2022), 108335.
[41] Zahra, W.K., Ouf, W.A., and El-Azab, M.S. An effective scheme based on quartic B-spline for the solution of Gardner equation and Harry Dym equation, Communications on Advanced Computational Science with Applications, 2016(2), (2016), 82–94.