Abstract
In the present paper we propose a novel analytical technique to obtain the solution of nonlinear partial differential equations. Additionally, we also implement the proposed method to find out the solution of fractional Kudryashov–Sinelshchikov equation. Since the Kudryashov–Sinelshchikov equation exhorts the pressure waves in mixture of liquid gas bubble while considering the viscosity and heat transport. Here regularized version of Hilfer–Prabhakar derivative of fractional order is utilized to model the problem. The obtained results are presented graphically for some specific values of constants and for distinct values of fractional order at different stages of time. The graphical behaviour of results show that the proposed method is very efficient to solve the nonlinear partial differential equations and reliable.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abdel Kader, A.H., AbdelLatif, M.S., Nour, H.M.: Some exact solutions of the Kudryashov-Sinelshchikov equation using point transformations. Int. J. Appl. Comput. Math 5(27), 1–10 (2019). https://doi.org/10.1007/s40819-019-0612-4
Akram, G., Sadaf, M., Anum, N.: Solutions of time fractional Kudryashov–Sinelshchikov equation arising in the pressure waves in the liquid with gas bubbles. Opt. Quant. Electron. 49(373), 1–16 (2017). https://doi.org/10.1007/s11082-017-1202-5
Ali, K.K., Maneea, M.: New approximation solution for time-fractional Kudryashov-Sinelshchikov equation using novel technique. Alex. Eng. J. 72, 559–572 (2023). https://doi.org/10.1016/j.aej.2023.04.027
Aty, A.H.A., Raza, N., Batool, A., Mahmoud, E.E., Yahia, I.S., Yousef, E.S.: Pattern formation of a bubbly fluid mixture under the effect of thermodynamics via Kudryashov-Sinelshchikov model. J. Math. 9546205 (2022). https://doi.org/10.1155/2022/9546205
Boulaaras, S.M., Urrehman, H., Iqbal, I., Sallah, M., Qayyum, A.: Unveiling optical solitons: solving two forms of nonlinear Schrodinger equations with unified solver method. Optik (stuttg). (2023). https://doi.org/10.1016/j.ijleo.2023.171535
Das G, Ray SS (2023) Numerical simulation of time fractional Kudryashov–Sinelshchikov equation describing the pressure waves in a mixture of liquid and gas bubbles. https://doi.org/10.48550/arXiv2310.11019
Dubey, V.P., Singh, J., Dubey, S., Kumar, D.: Analysis of Cauchy problems and diffusion equations associated with the Hilfer-Prabhakar fractional derivative via Kharrat-Toma transform. Fractal Fract. 7(5), 413 (2023). https://doi.org/10.3390/fractalfract7050413
Fahad, A., Boulaaras, S.M., urRehman, H., Iqbal, I., Saleem, M.S., Chou, D.: Analysing soliton dynamics and a comparative study of fractional derivatives in the nonlinear fractional Kudryashov’s equation. Results Phys. 5 (2023). https://doi.org/10.1016/j.rinp.2023.107001
Garra, R., Gorenflo, R., Polito, F., Tomovski, Z.: Hilfer-Prabhakar derivatives and some applications. Appl. Math. Comput. 242, 576–589 (2014)
Giusti, A., Colombaro, I., Garra, R., Garrappa, R., Polito, F., Popolizio, M., Mainardi, F.: A practical guide to prabhakar fractional calculus. Fract. Calc. Appl. Anal. 23(1), 9–54 (2020). https://doi.org/10.1515/fca-2020-002
Gupta, A.K., Ray, S.S.: On the solitary wave solution of fractional Kudryashov-Sinelshchikov equation describing nonlinear wave processes in a liquid containing gas bubbles. Appl. Math. Comput. 298, 1–12 (2017). https://doi.org/10.1016/j.amc.2016.11.003
Hilfer, R.: Fractional calculus and regular variation in thermodynamics. In: Hilfer, R. (ed.) applications of fractional calculus in physics, p. 429. World Scientific, Singapore (2000). https://doi.org/10.1142/9789812817747-0009
Hosseini, K., Bekir, A., Kaplan, M., Güner, O.: On a new technique for solving the nonlinear conformable time fractional differential equations. Opt. Quant. Electron. 49(343), 1–12 (2017). https://doi.org/10.1007/s11082-017-1178-1
Jaradat, M.M.M., Batool, A., Butt, A.R., Raza, N.: New solitary wave and computational solitons for Kundu-Eckhaus equation. Results Phys. 43, 106084 (2022). https://doi.org/10.1016/j.rinp.2022106084
Kaplan, M., Alqahtani, R.T.: Exploration of new solitons for the fractional perturbated Radhakrishnan-Kundu-Lakshmana model. Mathematics 11, 2562 (2023). https://doi.org/10.3390/math11112562
Kaplan, M., Akbulut, A., Bekir, A.: Solving space-time fractional differential equations by using modified simple equation method. Commu. Theor. Phys. 65(5), 563–568 (2016). https://doi.org/10.1088/0253-6102/65/5/563
Kharrat, B.N., Toma, G.A.: A new integral transform: Kharrat-Toma transform and its properties. World Appl. Sci. J. 38(5), 436–443 (2020). https://doi.org/10.5829/idosi.wasj.2020.436.443
Kilbas, A.A., Saigo, M., Saxena, R.K.: Generalized Mittag-Leffler function and generalized fractional calculus operators. Integral Transform. Spec. Funct. 15(1), 31–49 (2004). https://doi.org/10.1080//0652460310001600717
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. Elsevier, Amsterdam, The Netherlands (2006)
Kudryashov, N.A., Sinelshchikov, D.I.: Nonlinear waves in bubbly liquids with consideration for viscosity and heat transfer. Phys. Lett. A 374(19–20), 2011–2016 (2010). https://doi.org/10.1016/j.physleta.2010.02.067
Liao, S.J.: An approximate solution technique not depending on small parameters: a special example. Int. J. Nonlinear Mech. 30(3), 371–380 (1995)
Liao, S.J.: Beyond perturbation: introduction to homotopy analysis method. Chapman and Hall/CRC Press, Boca Raton (2003)
Liao, S. (ed.): Homotopy analysis method in nonlinear differential equations, pp. 153–165. Springer, Berlin (2012)
Lu, J.: New exact solutions for Kudryashov-Sinelshchikov equation. Adv. Differequ. 2018(374), 1–17 (2018). https://doi.org/10.1186/s13662-018-1769-6
Miller, K.S., Ross, B.: An introduction to the fractional calculus and fractional differential equations. Wiley, New York (1993)
Odibat, Z., Bataineh, A.S.: An adaptation of homotopy analysis method for reliable treatment of strongly nonlinear problems: construction of homotopy polynomials. Math. Methods Appl. Sci. 38(5), 991–1000 (2014). https://doi.org/10.1002/mma.3136
Oldham, K.B., Spanier, J.: The fractional calculus: theory and applications of differentiation and integration to arbitrary order. Elsevier, Amsterdam (1974)
Podlubny, I.: Fractional differential equations. Academic Press, New York (1999)
Prabhakar, T.R.: A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yolohama Math. J. 19, 7–15 (1971)
Prakash, P.: On group analysis, conservation laws and exact solutions of time-fractional Kudryashov-Sinelshchikov equation. Comp. Appl. Math. 40, 162 (2021). https://doi.org/10.1007/s40314-021-01550-2
Raza, N., Rani, B., Chahlaoui, Y., Shah, N.A.: A variety of new rogue wave patterns for three coupled nonlinear Maccari’s models in complex form. Nonlinear Dyn. 111, 18419–18437 (2023a). https://doi.org/10.1007/s11071-023-08839-3
Raza, N., Butt, A.R., Arshed, S., Kaplan, M.: A new exploration of some explicit soliton solutions of q-deformed Sinh-Gordon equation utilizing two novel techniques. Opt. Quant. Electron. 55, 200 (2023b). https://doi.org/10.1007/s11082-022-04461-6
Rezazadeh, H., Korkmaz, A., Raza, N., Ali, K.K., Eslami, M.: Soliton solution of generalized Zakharov-Kuznetsov and Zakharov-Kuznetsov-Benjamin-Bona-Mahony equations with conformable temporal evolution. Gravit. Math. Phys. Field Theory (2021). https://doi.org/10.31349/RevMexFis.67.050701
Ryabov, P.N.: Exact solutions of the Kudryashov-Sinelshchikov equation. Appl. Maths. Comput. 217, 3585–3590 (2010). https://doi.org/10.48550/arxiv.1011.5069
Singh, J., Gupta, A.: Computational analysis of fractional modified Degasperis-Procesi equation with Caputo-Katugampola derivative. AIMS Math. 8(1), 194–212 (2022). https://doi.org/10.3934/math.2023009
Singh, J., Gupta, A., Baleanu, D.: Fractional dynamics and analysis of coupled Schrödinger-KdV equation with Caputo-Katugampola type memory, ASME. J. Comput. Nonlinear Dynam. 18(9), 091001 (2023a). https://doi.org/10.1115/1.4062391
Singh, J., Gupta, A., Kumar, D.: Computational analysis of the fractional Riccati differential equation with Prabhakar type memory. Mathematics 11(3), 644 (2023b). https://doi.org/10.3390/math11030644
Singh, J., Alshehri, A.M., Sushila, Kumar, D.: Computational analysis of fractional Lineard’s equation with exponential memory. ASME J. Comput. Nonlinear Dyn. 18(4), 041004 (2023c). https://doi.org/10.1115/1.4056858
Tu, J.M., Tian, S.F., Xu, M.J., Zhang, T.T.: On Lie symmetries, optimal systems and explicit solutions to the Kudryashov-Sinelshchikov equation. Appl. Math. Comput. 275, 345–352 (2016). https://doi.org/10.1016/j.amc.2015.11.072
Ullah, N., Asjad, M.I., Ur Rehman, H., Akgül, A.: Construction of optical solitons of Radhakrishnan–Kundu–Lakshmanan equation in birefringentfibers. Nonlinear Eng. Model 11, 80–91 (2022). https://doi.org/10.1515/nleng-2022-0010
H. Ur Rehman, M. Inc, M.I. Asjad, A. Habib, Q. Munir, New soliton solutions for the space-time fractional modified third order Korteweg-de Vries equation, JOES, (2022).https://doi.org/10.1016/j.joes.2022.05.032
urRehman, H., Awan, A.U., Habib, A., Gamaoun, F., ElSayaed Din, D., Galal, A.M.: Solitary wave solutions for a strain wave equation in a microstructured solid. Results Phys. 39, 105755 (2022a). https://doi.org/10.1016/j.rinp.2022.105755
Yue, C., Khater, M.M.A., Attia, R.A.M., Lu, D.: The plethora of explicit solutions of the fractional KS equation through liquid-gas bubbles mix under the thermodynamics conditions via Atangana-Baleanu derivative operator. Adv. Differequ. 2020(62), 1–12 (2020). https://doi.org/10.1186/s13662-020-2540-3
Funding
The authors have not disclosed any funding.
Author information
Authors and Affiliations
Contributions
JS and AG wrote the main manuscript text and prepared figures. All authors reviewed the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Singh, J., Gupta, A. Novel computational technique for fractional Kudryashov–Sinelshchikov equation associated with regularized Hilfer–Prabhakar derivative. Opt Quant Electron 56, 749 (2024). https://doi.org/10.1007/s11082-024-06594-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11082-024-06594-2