Abstract
The present article is devoted to scouting invariant analysis and some kind of approximate and explicit solutions of the (3+1)-dimensional Jimbo Miwa system of nonlinear fractional partial differential equations (NLFPDEs). Feasible vector field of the system is obtained by employing the invariance attribute of one-parameter Lie group of transformation. The reduction of the number of independent variables by this method gives the reduction of Jimbo Miwa system of NLFPDES into a system of nonlinear fractional ordinary differential equations (NLFODEs). Explicit solutions in form of power series are scrutinized by using power series method (PSM). In addition, convergence is also examined. The residual power series method (RPSM) is employed for disquisition of solitary pattern (SP) solutions in form of approximate series. A comparative analysis of the obtained results of the considered problem is provided. The conserved vectors are scrutinized in the form of fractional Noether’s operator. Numerical solutions are represented graphically.
Funding source: Council of Scientific and Industrial Research (CSIR)
Award Identifier / Grant number: 09/1051(0007)2017-EMR-1
Award Identifier / Grant number: 25(0257)/16/EMR-II
Acknowledgment
Baljinder Kour, thanks, Council of Scientific and Industrial Research (CSIR) for financial support under JRF grant 09/1051(0007)2017-EMR-1 for carrying out the research work. Dr. Sachin Kumar, thanks, CSIR India for financial support under the grant 25(0257)/16/EMR-II.
Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: This research was supported by the Council of Scientific and Industrial Research (CSIR) under JRF grant 09/1051(0007)2017-EMR-1 and 25(0257)/16/EMR-II.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
[1] V. Kiryakova, Generalized Fractional Calculus and Applications, Longman Scientific & Technical, Harlow; co published in the United States with John Wiley & Sons, New York, 1994.Search in Google Scholar
[2] I. Podlubny, Fractional Differential Equations, CA: Mathematics in Science and Engineering, San Diego, Academic Press, 1999.Search in Google Scholar
[3] V. Daftardar-Gejji and H. Jafari, “Adomian decomposition: a tool for solving a system of fractional differential equations,” J. Math. Anal. Appl., vol. 301, pp. 508–518, 2005, https://doi.org/10.1016/j.jmaa.2004.07.039.Search in Google Scholar
[4] Z. Odibat and S. Momani, “A generalized differential transform method for linear partial differential equations of fractional order,” Appl. Math. Lett., vol. 21, pp. 194–199, 2008, https://doi.org/10.1016/j.aml.2007.02.022.Search in Google Scholar
[5] H. W. Yang, M. Guo, and H. He, “Conservation laws of space-time fractional mZK equation for Rossby solitary waves with complete coriolis force,” Int. J. Nonlinear Sci. Numer. Stimul., vol. 20, pp. 1–16, 2019.10.1515/ijnsns-2018-0026Search in Google Scholar
[6] W. M. Abd-Elhameed and Y. H. Youssri, “Sixth-kind Chebyshev spectral approach for solving fractional differential equations,” Int. J. Nonlinear Sci. Numer. Stimul., vol. 20, pp. 191–203, 2019, https://doi.org/10.1515/ijnsns-2018-0118.Search in Google Scholar
[7] P. K. Pradhan and M. Pandey, “Lie symmetries, one-dimensional optimal system and group invariant solutions for the Ripa system,” Int. J. Nonlinear Sci. Numer. Stimul., vol. 20, pp. 713–723, 2019.10.1515/ijnsns-2018-0311Search in Google Scholar
[8] R. Sahadevan and P. Prakash, “On lie symmetry analysis and invariant subspace methods of coupled time fractional partial differential equations,” Chaos, Solit. Fractals., vol. 104, pp. 107–120, 2017, https://doi.org/10.1016/j.chaos.2017.07.019.Search in Google Scholar
[9] H. Thabet and S. Kendre, “Analytical solutions for conformable space-time fractional partial differential equations via fractional differential transform,” Chaos, Solit. Fractals., vol. 109, pp. 238–245, 2018, https://doi.org/10.1016/j.chaos.2018.03.001.Search in Google Scholar
[10] J. Lu, S. Bilige, and X. Gao, “Abundant Lump solution and interaction phenomenon of (3+1)-dimensional generalized Kadomtsev–Petviashvili equation,” Int. J. Nonlinear Sci. Numer. Stimul., vol. 20, pp. 1–8, 2019, https://doi.org/10.1515/ijnsns-2018-0034.Search in Google Scholar
[11] S. Jamal, A. Kara, A. H. Bokhari, “Symmetries, conservation laws, reductions, and exact solutions for the klein–gordon equation in de sitter space–times,” Can. J. Phys., vol. 90, pp. 667–674, 2012, https://doi.org/10.1139/p2012-065.Search in Google Scholar
[12] B. Kour and S. Kumar, “Space time fractional Drinfel’d-Sokolov-Wilson system with time-dependent variable coefficients: symmetry analysis, power series solutions and conservation laws,” Eur. Phys. J. Plus., vol. 134, p. 467, 2019, https://doi.org/10.1140/epjp/i2019-12986-1.Search in Google Scholar
[13] P. Prakash and R. Sahadevan, “Lie symmetry analysis and exact solution of certain fractional ordinary differential equations,” Nonlinear Dynam., vol. 89, pp. 305–319, 2017, https://doi.org/10.1007/s11071-017-3455-8.Search in Google Scholar
[14] G. Wang, A. H. Kara, K. Fakhar, “Symmetry analysis and conservation laws for the class of time-fractional nonlinear dispersive equation,” Nonlinear Dynam., vol. 82, pp. 281–287, 2015, https://doi.org/10.1007/s11071-015-2156-4.Search in Google Scholar
[15] B. Kour and S. Kumar, “Time fractional Biswas Milovic equation: group analysis, soliton solutions, conservation laws and residual power series solution,” Optik., vol. 183, pp. 1085–1098, 2019 https://doi.org/10.1016/j.ijleo.2019.02.099.Search in Google Scholar
[16] S. Kumar and B. Kour, “Symmetry analysis of some nonlinear generalised systems of space–time fractional partial differential equations with time-dependent variable coefficients,” Pramana – J. Phys. 2019, vol. 93, p. 21, https://doi.org/10.1007/s12043-019-1791-6.Search in Google Scholar
[17] E. Lashkarian, S. R. Hejazi, N. Habibi, and A. Motamednezhad, “Symmetry properties, conservation laws, reduction and numerical approximations of time-fractional cylindrical-Burgers equation,” Commun. Nonlinear Sci. Numer. Simul., vol. 67, pp. 176–191, 2019, https://doi.org/10.1016/j.cnsns.2018.06.025.Search in Google Scholar
[18] M. Hong-Cai, “A simple method to generate Lie point symmetry groups of the (3+ 1)-dimensional Jimbo–Miwa equation,” Chin. Phys. Lett., vol. 22, p. 554, 2005, https://doi.org/10.1088/0256-307x/22/3/010.Search in Google Scholar
[19] M. Usman, A. Nazir, T. Zubair, et al., “Solitary wave solutions of (3+1)-dimensional Jimbo Miwa and Pochhammer-Chree equations by modified Exp-function method,” Int. J. Modern Math. Sci., vol. 5, pp. 27–36, 2013.Search in Google Scholar
[20] T. Ozis and I. Aslan, “Exact and explicit solutions to the (3+1)-dimensional Jimbo Miwa equation via the Exp-function method,” Phys. Lett. A., vol. 372, p. 7011–7015, 2008.10.1016/j.physleta.2008.10.014Search in Google Scholar
[21] T. Su and H. H. Dai, “Theta function solutions of the 3+1-dimensional Jimbo Miwa equation,” Math. Probl Eng., vol. 2017, p. 9, 2017, https://doi.org/10.1155/2017/2924947.Search in Google Scholar
[22] A. M. Wazwaz, “Multiple-soliton solutions for extended (3+1)-dimensional Jimbo-Miwa equations,” Appl. Math. Lett., vol. 64, pp. 21–26, 2017.10.1016/j.aml.2016.08.005Search in Google Scholar
[23] P. J. Olver, Applications of Lie Groups to Differential Equation, New York, Springer Science and Business Media, 2000.Search in Google Scholar
[24] B. Kour and S. Kumar, “Symmetry analysis, explicit power series solutions and conservation laws of the space-time fractional variant Boussinesq system,” Eur. Phys. J. Plus., vol. 133, p. 520, 2018, https://doi.org/10.1140/epjp/i2018-12297-1.Search in Google Scholar
[25] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives, Yverdon, Gordon and Breach Science Publishers, 1993.Search in Google Scholar
[26] C. Y. Qin, S. F. Tian, X. B. Wang, T. T. Zhang, “Lie symmetry analysis, conservation laws and explicit solutions for the time fractional Rosenau-Haynam equation,” Waves Random Complex Media, vol. 27, pp. 308–324, 2017, https://doi.org/10.1080/17455030.2016.1231434.Search in Google Scholar
[27] W. Rudin, Principles of Mathematical Analysis, New York, McGraw-Hill Book Co., 1964.Search in Google Scholar
[28] A. Kumar, S. Kumar, and M. Singh, “Residual power series method for fractional Sharma-Tasso-Olever equation,” Commun. Numer. Anal., vol. 2016, pp. 1–10, 2016, https://doi.org/10.5899/2016/cna-00235.Search in Google Scholar
[29] N. K. Ibragimov and E. D. Avdonina, “Nonlinear selfadjointness, conservation laws, and the construction of solutions to partial differential equations using conservation laws,” Uspekhi Mat. Nauk., vol. 68, pp. 111–146, 2013, https://doi.org/10.1070/rm2013v068n05abeh004860.Search in Google Scholar
[30] N. H. Ibragimov, “A new conservation theorem,” J. Math. Anal. Appl., vol. 333, pp. 311–328, 2007, https://doi.org/10.1016/j.jmaa.2006.10.078.Search in Google Scholar
[31] N. H. Ibragimov, “Nonlinear self-adjointness and conservation laws,” J. Phys. A: Math. Theor., vol. 44, 2011, Art no. 432002, https://doi.org/10.1088/1751-8113/44/43/432002.Search in Google Scholar
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