Solitary wave solutions of the time fractional Benjamin Bona Mahony Burger equation.
Atangana Baleanu Caputo derivative
Benjamin Bona Mahony Burger equation
Caputo Fabrizio derivative
Caputo derivative
Journal
Scientific reports
ISSN: 2045-2322
Titre abrégé: Sci Rep
Pays: England
ID NLM: 101563288
Informations de publication
Date de publication:
25 Jun 2024
25 Jun 2024
Historique:
received:
13
02
2024
accepted:
20
06
2024
medline:
26
6
2024
pubmed:
26
6
2024
entrez:
25
6
2024
Statut:
epublish
Résumé
The present study examines the approximate solutions of the time fractional Benjamin Bona Mahony Burger equation. This equation is critical for characterizing the dynamics of water waves and fluid acoustic gravity waves, as well as explaining the unidirectional propagation of long waves in nonlinear dispersive systems. This equation also describes cold plasma for hydromagnetic and audio waves in harmonic crystals. The natural transform decomposition method is used to obtain the analytical solution to the time fractional Benjamin Bona Mahony Burger equation. The proposed method uses the Caputo, Caputo Fabrizio, and Atangana Baleanu Caputo derivatives to describe the fractional derivative. We utilize a numerical example with appropriate initial conditions to assess the correctness of our findings. The results of the proposed method are compared to those of the exact solution and various existing techniques, such as the fractional homotopy analysis transform method and the homotopy perturbation transform technique. As a result, bell shaped solitons are discovered under the influence of hyperbolic functions. By comparing the outcomes with tables and graphs, the findings demonstrate the efficacy and effectiveness of the suggested approach.
Identifiants
pubmed: 38918464
doi: 10.1038/s41598-024-65471-w
pii: 10.1038/s41598-024-65471-w
doi:
Types de publication
Journal Article
Langues
eng
Sous-ensembles de citation
IM
Pagination
14596Informations de copyright
© 2024. The Author(s).
Références
Oldham, K. & Spanier, J. The fractional calculus theory and applications of differentiation and integration to arbitrary order (Elsevier, 1974).
Arikoglu, A. & Ozkol, I. Solution of fractional differential equations by using differential transform method. Chaos Solitons Fract. 34(5), 1473–1481 (2007).
doi: 10.1016/j.chaos.2006.09.004
Meena, M., Purohit, M., Purohit, S. D. & Nisar, K. S. A novel investigation of the hepatitis B virus using a fractional operator with a non-local kernel. Part. Differ. Equ. Appl. Math. 8, 100577 (2023).
Gour, M. M., Yadav, L. K., Purohit, S. D. & Suthar, D. L. Homotopy decomposition method to analysis fractional hepatitis B virus infection model. Appl. Math. Sci. Eng. 31(1), 2260075 (2023).
doi: 10.1080/27690911.2023.2260075
Pavani, K. & Raghavendar, K. A novel technique to study the solutions of time fractional nonlinear smoking epidemic model. Sci. Rep. 14(1), 4159 (2024).
doi: 10.1038/s41598-024-54492-0
pubmed: 38378902
Pavani, K., Raghavendar, K. & Aruna, K. Soliton solutions of the time-fractional Sharma-Tasso-Olver equations arise in nonlinear optics. Opt. Quant. Electron. 56(5), 748 (2024).
doi: 10.1007/s11082-024-06384-w
Podlubny, I. Fractional Differential Equations (Academic Press, San Diego, 1999).
Ray, S. S. New exact solutions of nonlinear fractional acoustic wave equations in ultrasound. Comput. Math. Appl. 71(3), 859–868 (2016).
doi: 10.1016/j.camwa.2016.01.001
Atangana, A. & Baleanu, D. New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. arXiv preprint arXiv:1602.03408 (2016).
Prakasha, D. G., Saadeh, R., Kachhia, K., Qazza, A. & Malagi, N. S. A new computational technique for analytic treatment of time-fractional nonlinear equations arising in magneto-acoustic waves. Math. Probl. Eng. 2023 (2023).
Malagi, N. S., Veeresha, P., Prasannakumara, B. C., Prasanna, G. D. & Prakasha, D. G. A new computational technique for the analytic treatment of time-fractional Emden-Fowler equations. Math. Comput. Simul. 190, 362–376 (2021).
doi: 10.1016/j.matcom.2021.05.030
Veeresha, P., Malagi, N. S., Prakasha, D. G. & Baskonus, H. M. An efficient technique to analyze the fractional model of vector-borne diseases. Phys. Scr. 97(5), 054004 (2022).
doi: 10.1088/1402-4896/ac607b
Sunitha, M. et al. An efficient analytical approach with novel integral transform to study the two-dimensional solute transport problem. Ain Shams Eng. J. 14(3), 101878 (2023).
doi: 10.1016/j.asej.2022.101878
Nadeem, M., Li, F. & Ahmad, H. Modified Laplace variational iteration method for solving fourth-order parabolic partial differential equation with variable coefficients. Comput. Math. Appl. 78(6), 2052–2062 (2019).
doi: 10.1016/j.camwa.2019.03.053
Singh, J., Kumar, D. & Sushila, D. Homotopy perturbation Sumudu transform method for nonlinear equations. Adv. Theor. Appl. Mech 4(4), 165–175 (2011).
Kanth, A. R. & Aruna, K. Solution of fractional third-order dispersive partial differential equations. Egypt. J. Basic Appl. Sci. 2(3), 190–199 (2015).
Alquran, M., Al-Khaled, K. & Chattopadhyay, J. Analytical solutions of fractional population diffusion model: Residual power series. Nonlinear Stud. 22(1), 31–39 (2015).
Yadav, L. K., Agarwal, G., Gour, M. M., Akgül, A., Misro, M. Y. & Purohit, S. D. A hybrid approach for non-linear fractional Newell-Whitehead-Segel model. Ain Shams Eng. J. 102645 (2024).
Yadav, L. K., Agarwal, G., Gour, M. M. & Kumari, M. Analytical approach to study weakly nonlocal fractional Schrödinger equation via novel transform. Int. J. Dyn. Control 12(1), 271–282 (2024).
doi: 10.1007/s40435-023-01246-x
Yi, M. & Huang, J. Wavelet operational matrix method for solving fractional differential equations with variable coefficients. Appl. Math. Comput. 230, 383–394 (2014).
Mohan, L. & Prakash, A. Stability and numerical analysis of fractional BBM-Burger equation and fractional diffusion-wave equation with Caputo derivative. Opt. Quant. Electron. 56(1), 26 (2024).
doi: 10.1007/s11082-023-05608-9
Benjamin, T. B., Bona, J. L. & Mahony, J. J. Model Equations for Long Waves in Nonlinear Dispersive Systems. Philosophical Transactions of the Royal Society of London. Ser. A Math. Phys. Sci. 272(1220), 47–78 (1972).
Abbasbandy, S. & Shirzadi, A. The first integral method for modified Benjamin-Bona-Mahony equation. Commun. Nonlinear Sci. Numer. Simul. 15(7), 1759–1764 (2010).
doi: 10.1016/j.cnsns.2009.08.003
Zhang, J., Wei, Z., Yong, L. & Xiao, Y. Analytical solution for the time fractional BBM-Burger equation by using modified residual power series method. Complexity, 2018 (2018).
Shen, X. & Zhu, A. A Crank-Nicolson linear difference scheme for a BBM equation with a time fractional nonlocal viscous term. Adv. Diff. Equ. 2018(1), 1–12 (2018).
doi: 10.1186/s13662-018-1815-4
Li, C. Linearized difference schemes for a BBM equation with a fractional nonlocal viscous term. Appl. Math. Comput. 311, 240–250 (2017).
Fakhari, A. & Domairry, G. Approximate explicit solutions of nonlinear BBMB equations by homotopy analysis method and comparison with the exact solution. Phys. Lett. A 368(1–2), 64–68 (2007).
doi: 10.1016/j.physleta.2007.03.062
Kumar, S. & Kumar, D. Fractional modelling for BBM-Burger equation by using new homotopy analysis transform method. J. Assoc. Arab Univ. Basic Appl. Sci. 16, 16–20 (2014).
Shakeel, M., Ul-Hassan, Q. M., Ahmad, J. & Naqvi, T. Exact solutions of the time fractional BBM-Burger equation by novel-expansion method. Adv. Math. Phys. (2014).
Wu, W. et al. Numerical and analytical results of the 1D BBM equation and 2D coupled BBM-system by finite element method. Int. J. Mod. Phys. B 36(28), 2250201 (2022).
doi: 10.1142/S0217979222502010
Rawashdeh, M. & Maitama, S. Finding exact solutions of nonlinear PDEs using the natural decomposition method. Math. Methods Appl. Sci. 40(1), 223–236 (2017).
doi: 10.1002/mma.3984
Shah, K., Junaid, M. & Ali, N. Extraction of Laplace, Sumudu, Fourier and Mellin transform from the natural transform. J. Appl. Environ. Biol. Sci 5(9), 108–115 (2015).
Koppala, P. & Kondooru, R. An Efficient Technique to Solve Time-Fractional Kawahara and Modified Kawahara Equations. Symmetry 14(9), 1777 (2022).
doi: 10.3390/sym14091777
Zhou, M. X., Kanth, A. S. V., Aruna, K., Raghavendar, K., Rezazadeh, H., Inc, M. & Aly, A. A. Numerical solutions of time fractional Zakharov-Kuznetsov equation via natural transform decomposition method with nonsingular kernel derivatives. J. Funct. Spaces 9884027 (2021).
Pavani, K. & Raghavendar, K. Approximate solutions of time-fractional swift-hohenberg equation via natural transform decomposition method. Int. J. Appl. Comput. Math. 9(3), 29 (2023).
doi: 10.1007/s40819-023-01493-8
Kanth, A. R., Aruna, K., Raghavendar, K., Rezazadeh, H. & Inc, M. Numerical solutions of nonlinear time fractional Klein-Gordon equation via natural transform decomposition method and iterative Shehu transform method. J. Ocean Eng. Sci. (2021).
Pavani, K. & Raghavendar, K. A novel method to study time fractional coupled systems of shallow water equations arising in ocean engineering. AIMS Math. 9(1), 542–564 (2024).
doi: 10.3934/math.2024029
Losada, J. & Nieto, J. J. Properties of a new fractional derivative without singular kernel. Progr. Fract. Differ. Appl 1(2), 87–92 (2015).
Atangana, A. & Koca, I. Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order. Chaos, Solitons Fractals 89, 447–454 (2016).
doi: 10.1016/j.chaos.2016.02.012
Prakasha, D. G., Veeresha, P. & Rawashdeh, M. S. Numerical solution for (2+ 1)-dimensional time fractional coupled Burger equations using fractional natural decomposition method. Math. Methods Appl. Sci. 42(10), 3409–3427 (2019).
doi: 10.1002/mma.5533
Jan, H. U., Ullah, H., Fiza, M., Khan, I. & Eldin, S. M. Fractional view analysis of the diffusion equations via a natural Atangana-Baleanu operator. Alex. Eng. J. 83, 19–26 (2023).
doi: 10.1016/j.aej.2023.10.031
Ravi Kanth, A. S. V., Aruna, K. & Raghavendar, K. Natural transform decomposition method for the numerical treatment of the time fractional Burgers-Huxley equation. Num. Methods Part. Differ. Equ. 39(3), 2690–2718 (2023).
doi: 10.1002/num.22983
Adivi Sri Venkata, R. K., Kirubanandam, A. & Kondooru, R. Numerical solutions of time fractional Sawada Kotera Ito equation via natural transform decomposition method with singular and nonsingular kernel derivatives. Math. Methods Appl. Sci. 44(18), 14025–14040 (2021).
doi: 10.1002/mma.7672