ArticleNo Access

NEW PROPERTIES OF THE FRACTAL BOUSSINESQ–KADOMTSEV–PETVIASHVILI-LIKE EQUATION WITH UNSMOOTH BOUNDARIES

KANGLE WANG

https://orcid.org/0000-0002-6637-5687

School of Mathematics and Information Science, Henan Polytechnic University, 454000 JiaoZuo, P. R. China

Search for more papers by this author

CHUNFU WEI

https://orcid.org/0000-0002-3303-8043

School of Mathematics and Information Science, Henan Polytechnic University, 454000 JiaoZuo, P. R. China

E-mail Address: chunfuwei123@163.com

Corresponding author.

Search for more papers by this author

, and

FENG REN

Henan College of Industry and Information Technology, 454000 JiaoZuo, P. R. China

Search for more papers by this author

https://doi.org/10.1142/S0218348X22501754Cited by:2 (Source: Crossref)

Abstract

The Boussinesq–Kadomtsev–Petviashvili-like model is a famous wave equation which is used to describe the shallow water waves in ocean beaches and lakes. When shallow water waves propagate in microgravity or with unsmooth boundaries, the Boussinesq–Kadomtsev–Petviashvili-like model is modified into its fractal model by the local fractional derivative (LFD). In this paper, we mainly study the fractal Boussinesq–Kadomtsev–Petviashvili-like model (FBKPLM) based on the LFD on Cantor sets. Two efficient and reliable mathematical approaches are successfully implemented to obtain the different types of fractal traveling wave solutions of the FBKPLM, which are fractal variational method (FVM) and fractal Yang wave method (FYWM). Finally, some three-dimensional (3D) simulation graphs are employed to elaborate the properties of the fractal traveling wave solutions.

Keywords:

References

1. B. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman, San Francisco, 1982). Google Scholar
2. T. J. Miao, A. M. Chen, L. W. Zhang and B. M. Yu, A novel fractal model for permeability of damaged tree-like branching networks, Int. J. Heat. Mass. Trans. 127 (2018) 278–285. Crossref, Web of Science, Google Scholar
3. B. M. Yu and P. Cheng, A fractal model for permeability of bi-dispersed porous media, Int. J. Heat Mass Trans. 45(14) (2002) 2983–2993. Crossref, Web of Science, Google Scholar
4. Q. T. Ain, N. Anjum, A. Din, A. Zeb and S. Djilali, On the analysis of Caputo fractional order dynamics of Middle East Lungs Coronavirus (MERS-CoV) model, Alex. Eng. J. 61 (2022) 5123–5131. Crossref, Web of Science, Google Scholar
5. C. J. Xu, M. Farman, A. Akgul, K. S. Nisar and A. Ahmad, Modeling and analysis fractal order cancer model with effects of chemotherapy, Chaos Solitons Fractals 161 (2022) 112325. Crossref, Web of Science, Google Scholar
6. G. Rahman, K. S. Nisar and A. K. Golamankaneh, The nonlocal fractal integral reverse Minkowski’s and other related inequalities on fractal sets, Math. Probl. Eng. 2021 (2021) 4764891. Crossref, Web of Science, Google Scholar
7. X. J. Yang and J. A. Tenreiro Machado, A new fractal nonlinear Burger’s equation arising in the acoustic signals propagation, Math. Methods Appl. Sci. 42(18) (2019) 7539–7544. Crossref, Web of Science, Google Scholar
8. J. Singh, D. Kumar and J. J. Nieto, A reliable algorithm for a local fractional Tricomi equation arising in fractal Transonic flow, Entropy 18(6) (2016) 2061–2068. Crossref, Web of Science, Google Scholar
9. G. S. Chen, J. S. Liang, H. M. Srivastava and L. Chao, Local fractional integral Hölder-Type inequalities and some related results, Fractal Fraction. 6 (2022) 195. Crossref, Web of Science, Google Scholar
10. X. J. Yang, H. M. Srivastava, J. H. He and D. Baleanu, Cantor-type cylindrical method for differential equations with local fractional derivative, Phys. Lett. A 377(28) (2013) 1696–1700. Crossref, Web of Science, Google Scholar
11. F. M. Fernández, On the homotopy perturbation method for Boussinesq-like equations, Appl. Math. Comput. 230 (2014) 208–210. Web of Science, Google Scholar
12. A. M. Wazwaz, The extended tanh method for new compact and noncompact solutions for the KP-BBM and the ZK-BBM equations, Chaos Solitons Fractals 38 (2008) 1505–1516. Crossref, Web of Science, Google Scholar
13. M. K. Elboree, New soliton solutions for a Kadomtsev–Petviashvili (KP) like equation coupled to a Schrödinger equation, Appl. Math. Comput. 218 (2012) 5966–5973. Web of Science, Google Scholar
14. S. Kumar, S. K. Dhiman, D. Baleanu and M. S. Osman, Lie symmetries, closed-form solutions, and various dynamical profiles of solitons for the variable coefficient (2+1)-dimensional KP equations, Symmetry 14 (2022) 597. Crossref, Web of Science, Google Scholar
15. L. Akinyemi, M. Senol, E. Az-Zobi and P. Veeresha, Novel soliton solutions of four sets of generalized (2+1)-dimensional Boussinesq–Kadomtsev–Petviashvili-like equations, Mod. Phys. Lett. B 36(1) (2022) 2150530. Link, Web of Science, Google Scholar
16. S. Kumar, A. Ahmadian, R. Kumar, D. Kumar, J. Singh, D. Baleanu and M. Salimi, An efficient numerical method for fractional SIR epidemic model of infectious disease by using Bernstein wavelets, Mathematics 8 (2020) 558. Crossref, Web of Science, Google Scholar
17. K. L. Wang, Novel approach for fractal nonlinear oscillators with discontinuities by Fourier series, Fractals 30(1) (2022) 2250009. Link, Web of Science, Google Scholar
18. M. Nadeem and J. H. He, The homotopy perturbation method for fractional differential equations: Part 2, two-scale transform, Int. J. Numer. Method H. 32(2) (2022) 559–567. Crossref, Web of Science, Google Scholar
19. Q. T. Ain and J. H. He, On two-scale dimension and its applications, Therm. Sci. 23(3B) (2019) 1707–1712. Crossref, Web of Science, Google Scholar
20. Q. T. Ain, N. Anjum and C. H. He, An analysis of time-fractional heat transfer problem using two-scale approach, Gem. Int. J. Geomath. 12(1) (2021) 1–10. Web of Science, Google Scholar
21. Q. T. Ain, T. Sathiyaraj, S. Karim, M. Nadeem and P. Kandege Mwanakatwe, ABC fractional derivative for the alcohol drinking model using two-scale fractal dimension, Complexity 2022 (2022) 6. Web of Science, Google Scholar
22. M. Nadeem and J. H. He, He–Laplace variational iteration method for solving the nonlinear equations arising in chemical kinetics and population dynamics, J. Math. Chem. 59(5) (2021) 1234–1245. Crossref, Web of Science, Google Scholar
23. J. J. Mao, S. F. Tian, L. Zou and T. T. Zhang, Optical solitons, complexitons, Gaussian soliton and power series solutions of a generalized Hirota equation, Mod. Phys. Lett. B 32(14) (2018) 1850143. Link, Web of Science, Google Scholar
24. K. K. Ali, M. A. Abd El Salam, E. M. H. Mohamed, B. Samet, S. Kumar and M. S. Osman, Numerical solution for generalized nonlinear fractional integro-differential equations with linear functional arguments using Chebyshev series, Adv. Differ. Equations 2020(1) (2020) 494. Crossref, Web of Science, Google Scholar
25. S. S. Ray, New analytical exact solutions of time fractional Kdv-KZK equation by Kudryashov methods, Chin. Phys. B 25(4) (2016) 040204. Crossref, Web of Science, Google Scholar
26. S. Kumar, A. Kumar, Z. Odibat, M. Aldhaifallah and K. S. Nisar, A compassion study of two modified analytical approach for the solution of nonlinear fractional shallow water equations in fluid flow, AIMS Math. 5(4) (2020) 3035–3055. Crossref, Web of Science, Google Scholar
27. U. Ali, H. Ahmad, J. Baili, T. Botmart and M. A. Aldahlan, Exact analytical wave solutions for space-time variable-order fractional modified equal width equation, Results Phys. 33 (2022) 105216. Crossref, Web of Science, Google Scholar
28. S. S. Ray, New exact solutions of nonlinear fractional acoustic wave equations in ultrasound, Comput. Math. Appl. 71(3) (2026) 859–868. Google Scholar
29. H. Ahmad, A. R. Seadawy, Khan and A. Tufail, Study on numerical solution of dispersive water wave phenomena by using a reliable modification of variational iteration algorithm, Math. Comput. Simulat. 17(17) (2020) 13–23. Crossref, Web of Science, Google Scholar
30. D. Khan, A. Khan, I. Khan, Y. M. Chu and K. S. Nisar, A new idea of fractal-fractional derivative with power law Kernel for Free Convection Heat Transfer in a Channel Flow between two static upright parallel plates, Comput. Mater. Contin. 65(2) (2020) 1237–1251. Crossref, Web of Science, Google Scholar
31. K. S. Nisar, K. Jothimani, K. Kaliraj and C. Ravichandran, An analysis of controllability results for nonlinear Hilfer neutral fractional derivatives with non-dense domain, Chaos Solitons Fractals 146 (2021) 110915. Crossref, Web of Science, Google Scholar
32. J. G. Liu, X. J. Yang, Y. Y. Feng and P. Cui, On the (N+1)-dimensional local fractional reduced differential transform method and its applications, Math. Methods Appl. Sci. 43(5) (2020) 8856–8866. Crossref, Web of Science, Google Scholar
33. K. L. Wang, Exact travelling wave solutions for the local fractional Kadomtsov–Petviashvili–Benjamin–Bona–Mahony model by variational perspective, Fractals 30(6) (2022) 2250101. Link, Web of Science, Google Scholar
34. K. L. Wang, A novel variational approach to fractal Swift–Hohenberg model arising in fluid dynamics, Fractals 30(7) (2022) 2250156. Link, Web of Science, Google Scholar
35. K. L. Wang, A novel perspective to the local fractional Zakharov–Kuznetsov-modified equal width dynamical model on Cantor sets, Math. Methods Appl. Sci. 2022 (2022), https://doi.org/10.1002/mma.8533. Google Scholar
36. X. J. Yang, J. A. Tenreiro Machado, D. Baleanu and C. Cattani, On exact traveling-wave solutions for local fractional Korteweg–de Vries equation, Chaos 26(8) (2016) 084312. Crossref, Web of Science, Google Scholar