Research article

New computations for the two-mode version of the fractional Zakharov-Kuznetsov model in plasma fluid by means of the Shehu decomposition method

  • Received: 06 August 2021 Accepted: 24 October 2021 Published: 05 November 2021
  • MSC : 26A33, 26A51, 26D07, 26D10, 26D15

  • In this research, the Shehu transform is coupled with the Adomian decomposition method for obtaining the exact-approximate solution of the plasma fluid physical model, known as the Zakharov-Kuznetsov equation (briefly, ZKE) having a fractional order in the Caputo sense. The Laplace and Sumudu transforms have been refined into the Shehu transform. The action of weakly nonlinear ion acoustic waves in a plasma carrying cold ions and hot isothermal electrons is investigated in this study. Important fractional derivative notions are discussed in the context of Caputo. The Shehu decomposition method (SDM), a robust research methodology, is effectively implemented to generate the solution for the ZKEs. A series of Adomian components converge to the exact solution of the assigned task, demonstrating the solution of the suggested technique. Furthermore, the outcomes of this technique have generated important associations with the precise solutions to the problems being researched. Illustrative examples highlight the validity of the current process. The usefulness of the technique is reinforced via graphical and tabular illustrations as well as statistics theory.

    Citation: Maysaa Al-Qurashi, Saima Rashid, Fahd Jarad, Madeeha Tahir, Abdullah M. Alsharif. New computations for the two-mode version of the fractional Zakharov-Kuznetsov model in plasma fluid by means of the Shehu decomposition method[J]. AIMS Mathematics, 2022, 7(2): 2044-2060. doi: 10.3934/math.2022117

    Related Papers:

  • In this research, the Shehu transform is coupled with the Adomian decomposition method for obtaining the exact-approximate solution of the plasma fluid physical model, known as the Zakharov-Kuznetsov equation (briefly, ZKE) having a fractional order in the Caputo sense. The Laplace and Sumudu transforms have been refined into the Shehu transform. The action of weakly nonlinear ion acoustic waves in a plasma carrying cold ions and hot isothermal electrons is investigated in this study. Important fractional derivative notions are discussed in the context of Caputo. The Shehu decomposition method (SDM), a robust research methodology, is effectively implemented to generate the solution for the ZKEs. A series of Adomian components converge to the exact solution of the assigned task, demonstrating the solution of the suggested technique. Furthermore, the outcomes of this technique have generated important associations with the precise solutions to the problems being researched. Illustrative examples highlight the validity of the current process. The usefulness of the technique is reinforced via graphical and tabular illustrations as well as statistics theory.



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    [1] S. Kumar, A. Atangana, A numerical study of the nonlinear fractional mathematical model of tumor cells in presence of chemotherapeutic treatment, Int. J. Biomath., 13 (2020), 2050021. doi: 10.1142/S1793524520500217. doi: 10.1142/S1793524520500217
    [2] B. Ghanbari, A. Atangana, A new application of fractional Atangana-Baleanu derivatives: designing ABC-fractional masks in image processing, Physica A, 542 (2020), 123516. doi: 10.1016/j.physa.2019.123516. doi: 10.1016/j.physa.2019.123516
    [3] I. Podlubny, Fractional differential equations, Academic Press, 1999.
    [4] R. Hilfer, Applications of fractional calculus in physics, Word Scientific, 2000.
    [5] A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and application of fractional differential equations, Elsevier, 2006.
    [6] R. L. Magin, Fractional calculus in bioengineering, Begell House, 2006.
    [7] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives: Theory and applications, Gordon and Breach, 1993.
    [8] S. Maitama, W. Zhao, New integral transform: Shehu transform a generalization of Sumudu and Laplace transform for solving differential equations, Int. J. Anal. Appl., 17 (2019), 167–190.
    [9] S. Rashid, K. T. Kubra, S. Ullah, Fractional spatial diffusion of a biological population model via a new integral transform in the settings ofpower and Mittag–Leffler nonsingular kernel, Phys. Scr., 96 (2021), 114003. doi: 10.1088/1402-4896/ac12e5. doi: 10.1088/1402-4896/ac12e5
    [10] S. Rashid, R. Ashraf, A. O. Akdemir, M. A. Alqudah, T. Abdeljawad, M. S. Mohamed, Analytic fuzzy formulation of a time-fractional Fornberg-Whitham model with power and Mittag–Leffler kernels, Fractal Fract., 5 (2021), 113. doi: 10.3390/fractalfract5030113. doi: 10.3390/fractalfract5030113
    [11] S. Rashid, Z. Hammouch, H. Aydi, A. G. Ahmad, A. M. Alsharif, Novel computations of the time-fractional Fisher's model viageneralized fractional integral operators by means of the Elzaki transform, Fractal Fract., 5 (2021), 94. doi: 10.3390/fractalfract5030094. doi: 10.3390/fractalfract5030094
    [12] S. Rashid, K. T. Kubra, J. L. G. Guirao, Construction of an approximate analytical solution for multi-dimensional fractionalZakharov-Kuznetsov equation via Aboodh Adomian decomposition method, Symmetry, 13 (2021), 1542. doi: 10.3390/sym13081542. doi: 10.3390/sym13081542
    [13] S. S. Zhou, S. Rashid, A. Rauf, K. T. Kubra, A. M. Alsharif, Initial boundary value problems for a multi-term time fractional diffusion equation with generalized fractional derivatives in time, AIMS Mathematics, 6 (2021), 12114–12132. doi: 10.3934/math.2021703. doi: 10.3934/math.2021703
    [14] S. Rashid, F. Jarad, K. M. Abualnaja, On fuzzy Volterra-Fredholm integrodifferential equation associated with Hilfer-generalizedproportional fractional derivative, AIMS Mathematics, 6 (2021), 10920–10946. doi: 10.3934/math.2021635. doi: 10.3934/math.2021635
    [15] S. Rashid, K. T. Kubra, A. Rauf, Y. M. Chu, Y. S. Hamed, New numerical approach for time-fractional partial differential equations arising in physical system involving natural decomposition method, Phys. Scr., 96 (2021), 105204. doi: 10.1088/1402-4896/ac0bce. doi: 10.1088/1402-4896/ac0bce
    [16] M. A. Alqudah, R. Ashraf, S. Rashid, J. Singh, Z. Hammouch, T. Abdeljawad, Novel numerical investigations of fuzzy Cauchy reaction–diffusion models via generalized fuzzy fractional derivative operators, Fractal Fract., 5 (2021), 151. doi: 10.3390/fractalfract5040151. doi: 10.3390/fractalfract5040151
    [17] S. El-Sayed, D. Kaya. An application of the ADM to seven-order Sawada-Kotara equations, Appl. Math. Comput., 157 (2004), 93–101. doi: 10.1016/j.amc.2003.08.104. doi: 10.1016/j.amc.2003.08.104
    [18] M. T. Darvishia, S. Kheybaria, F. Khanib, A numerical solution of the Lax's 7th-order KdV equation by Pseudo spectral method and Darvishi's Preconditioning, Int. J. Contemp. Math. Sciences, 2 (2007), 1097–1106. doi: 10.12988/ijcms.2007.07111
    [19] M. A. El-Tawil, S. Huseen, On convergence of the q-homotopy analysis method, Int. J. Contemp. Math. Scis., 8 (2013), 481–497. doi: 10.12988/ijcms.2013.13048
    [20] M. I. El-Bahi, K. Hilal, Lie symmetry analysis, exact solutions, and conservation laws for the generalized time-fractional KdV-Like equation, J. Funct. Space., 2021 (2021), 6628130. doi: 10.1155/2021/6628130. doi: 10.1155/2021/6628130
    [21] S. C. Shiralashetti, S. Kumbinarasaiah, Laguerre wavelets collocation method for the numerical solution of the Benjamina–Bona–Mohany equations, J. Taibah Univ. Sci., 13 (2019), 9–15. doi: 10.1080/16583655.2018.1515324. doi: 10.1080/16583655.2018.1515324
    [22] N. A. Lahmar, O. Belhamitib, S. M. Bahric, A new Legendre-Wavelets decomposition method for solving PDEs, Malaya. J. Mat, 1 (2014), 72–81.
    [23] G. A. Birajdar, Numerical solution of time fractional Navier-Stokes equation by discrete Adomian decomposition method, Nonlinear Eng., 3 (2014), 21–26. doi: 10.1515/nleng-2012-0004. doi: 10.1515/nleng-2012-0004
    [24] V. E. Zakharov, E. A. Kuznetsov, Three dimensional solutions, Soviet Phys. JETP, 39 (1974), 285–286.
    [25] D. Kumara, J. Singh, S. Kumar, Numerical computation of nonlinear fractional Zakharov-Kuznetsov equation arising in ion-acoustic waves, J. Egypt. Math. Soc., 22 (2014), 373–378. doi: 10.1016/j.joems.2013.11.004. doi: 10.1016/j.joems.2013.11.004
    [26] S. Monro, E. J. Parkes, The derivation of a modified Zakharov-Kuznetsov equation and the stability of its solutions, J. Plasma Phys., 62 (1999), 305–317. doi: 10.1017/S0022377899007874. doi: 10.1017/S0022377899007874
    [27] S. Monro, E. J. Parkes, Stability of solitary-wave solutions to a modified Zakharov-Kuznetsov equation, J. Plasma Phys., 64 (2000), 411–426. doi: 10.1017/S0022377800008771. doi: 10.1017/S0022377800008771
    [28] I. P. Akpan, Adomian decomposition approach to the solution of the Burger's equation, Am. J. Comput. Math., 5 (2015), 329–335. doi: 10.4236/ajcm.2015.53030. doi: 10.4236/ajcm.2015.53030
    [29] W. Li, Y. Pang, Application of Adomian decomposition method to nonlinear systems, Adv. Differ. Equ., 2020 (2020), 67. doi: 10.1186/s13662-020-2529-y. doi: 10.1186/s13662-020-2529-y
    [30] M. Osman, Z. Gong, A. M. Mustafa, Comparison of fuzzy Adomian decomposition method with fuzzy VIM for solving fuzzy heat-like and wave-like equations with variable coefficients, Adv. Differ. Equ., 2020 (2020), 327. doi: 10.1186/s13662-020-02784-w. doi: 10.1186/s13662-020-02784-w
    [31] A. M. Wazwaz, R. Rach, L. Bougoffa, J. S. Duan, Solving the Lane-Emden-Fowler type equations of higher orders by the Adomian decomposition method, Comput. Model. Eng. Sci., 100 (2014), 507–529.
    [32] D. Rani, V. Mishra, Modification of Laplace Adomian decomposition method for solving nonlinear Volterra integral and integro-differential equations based on Newton Raphson formula, Eur. J. Pure. Appl. Math., 11 (2018), 202–214. doi: 10.29020/nybg.ejpam.v11i1.2645. doi: 10.29020/nybg.ejpam.v11i1.2645
    [33] F. Haq, K. Shah, A. Khan, M. Shahzad, Numerical solution of fractional order epidemic model of a vector born disease by Laplace Adomian decomposition method, Punjab Univ. J. Math., 49 (2017), 13–22.
    [34] R. Y. Molliq, M. S. M. Noorani, I. Hashim, R. R. Ahmad, Approximate solutions of fractional Zakharov-Kuznetsov equations by VIM, J. Comput. Appl. Math., 233 (2009), 103–108. doi: 10.1016/j.cam.2009.03.010. doi: 10.1016/j.cam.2009.03.010
    [35] M. Şenol, M. Alquran, H. D. Kasmaei, On the comparison of perturbation-iteration algorithm and residual power series method to solve fractional Zakharov-Kuznetsov equation, Results Phys., 9 (2018), 321–327. doi: 10.1016/j.rinp.2018.02.056. doi: 10.1016/j.rinp.2018.02.056
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