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A reliable hybrid numerical method for a time dependent vibration model of arbitrary order

  • Received: 15 October 2019 Accepted: 19 December 2019 Published: 08 January 2020
  • MSC : 35Q99, 44A99

  • In this article, the solution of vibration equation of fractional order is found numerically for the large membranes using a powerful technique namely q-homotopy analysis Sumudu transform technique. The parameter ħ suggests a convenient way to control convergence region. The given numerical examples depict competency and accuracy of this scheme. The results are discussed using figures taking diverse wave velocities and initial conditions. Results are also compared with other methods. The outcome divulges that q-HASTM is highly reliable, more efficient, attractive, easier to use as well as highly effective.

    Citation: Amit Prakash, Manish Goyal, Haci Mehmet Baskonus, Shivangi Gupta. A reliable hybrid numerical method for a time dependent vibration model of arbitrary order[J]. AIMS Mathematics, 2020, 5(2): 979-1000. doi: 10.3934/math.2020068

    Related Papers:

  • In this article, the solution of vibration equation of fractional order is found numerically for the large membranes using a powerful technique namely q-homotopy analysis Sumudu transform technique. The parameter ħ suggests a convenient way to control convergence region. The given numerical examples depict competency and accuracy of this scheme. The results are discussed using figures taking diverse wave velocities and initial conditions. Results are also compared with other methods. The outcome divulges that q-HASTM is highly reliable, more efficient, attractive, easier to use as well as highly effective.


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    [1] H. M. Srivastava, D. Kumar, J. Singh, An efficient analytical technique for fractional model of vibration equation, Appl. Math. Model., 45 (2017), 192-204. doi: 10.1016/j.apm.2016.12.008
    [2] H. Singh, Approximate solution of fractional vibration equation using Jacobi polynomials, Appl. Math. Comput., 317 (2018), 85-100.
    [3] H. Singh, H. M. Srivastava, D. Kumar, A reliable numerical algorithm for the fractional vibration equation, Chaos Soliton. Fract., 103 (2017), 131-138. doi: 10.1016/j.chaos.2017.05.042
    [4] S. Das, A numerical solution of the vibration equation using modified decomposition method, J. Sound Vib., 320 (2009), 576-583. doi: 10.1016/j.jsv.2008.08.029
    [5] S. Das, P. K. Gupta, Application of homotopy perturbation method and homotopy analysis method for fractional vibration equation, Int. J. Comput. Math., 88 (2011), 430-441. doi: 10.1080/00207160903474214
    [6] S. T. Mohyud-Din, A. Yildirim, An algorithm for solving the fractional vibration equation, Comput. Math. Model., 23 (2012), 228-237.
    [7] H. M. Baskonus, T. Mekkaoui, Z. Hammouch, et al. Active control of a chaotic fractional order economic system, Entropy, 17 (2015), 5771-5783. doi: 10.3390/e17085771
    [8] H. M. Baskonus, H. Bulut, On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method, Open Math., 13 (2015), 547-556.
    [9] H. M. Baskonus, G. Yel, H. Bulut, Novel wave surfaces to the fractional Zakharov- Kuznetsov Benjamin-Bona-Mahony equation, AIP Conference Proceedings, 1863 (2017), 560084.
    [10] C. Ravichandran, K. Jothimani, H. M. Baskonus, et al. New results on nondensely characterized integrodifferential equations with fractional order, Eur. Phys. J. Plus, 133 (2018), 109.
    [11] R. Panda, M. Dash, Fractional generalized splines and signal processing, Signal Process, 86 (2006), 2340-2350. doi: 10.1016/j.sigpro.2005.10.017
    [12] R. L. Bagley, P. J. Torvik, A theoretical basis for the application of fractional calculus to Viscoelasticity, J. Rheol., 27(1983), 201-210. doi: 10.1122/1.549724
    [13] A. Prakash, M. Goyal, S. Gupta, q-homotopy analysis method for fractional Bloch model arising in nuclear magnetic resonance via the Laplace transform, Ind. J. Phys., (2019), http://doi.org/10.1007/s12648-019-01487-7.
    [14] A. Prakash, M. Goyal, S. Gupta, Numerical simulation of space-fractional Helmholtz equation arising in Seismic wave propagation, imaging and inversion, Pramana, 93 (2019), 28.
    [15] M. Goyal, H. M. Baskonus, A. Prakash, An efficient technique for a time fractional model of lassa hemorrhagic fever spreading in pregnant women, Eur. Phys. J. Plus, 134 (2019), 482.
    [16] S. Das, Solution of fractional vibration equation by the variational iteration method and modified decomposition method, Int. J. Nonlin. Sci. Num., 9 (2008), 361-366.
    [17] S. J. Liao, On the homotopy analysis method for non-linear problems, Appl. Math. Comput., 147 (2004), 499-513.
    [18] M. A. El-Tawil, S. N. Huseen, The q-homotopy analysis method (q-HAM), Int. J. Appl. Math. Mech., 8 (2012), 51-75.
    [19] M. A. El-Tawil, S. N. Huseen, On convergence of the q-homotopy analysis method, Int. J. Contemp. Math. Sci., 8 (2013), 481-497. doi: 10.12988/ijcms.2013.13048
    [20] A. Prakash, P. Veeresha, D. G. Prakasha, et al. A homotopy technique for a fractional order multi-dimensional telegraph equation via the Laplace Transform, Eur. Phys. J. Plus, 134 (2019), 19.
    [21] A. Prakash, P. Veeresha, D. G. Prakasha, et al. A new efficient technique for solving fractional coupled Navier-Stokes equations using q-homotopy analysis transform method, Pramana, 93 (2019), 6.
    [22] A. Prakash, M. Goyal, S. Gupta, Fractional variational iteration method for solving time-fractional Newell-Whitehead-Segel equation, Nonlinear Eng., 8 (2019), 164-171. doi: 10.1515/nleng-2018-0001
    [23] M. Goyal, A. Prakash, S. Gupta, Numerical simulation for time-fractional nonlinear coupled dynamical model of romantic and interpersonal relationships, Pramana, 92 (2019), 82.
    [24] A. Prakash, M. Goyal, S. Gupta, A reliable algorithm for fractional Bloch model arising in magnetic resonance imaging, Pramana, 92 (2019), 18.
    [25] A. M. S. Mahdy, A. S. Mohamed, A. A. H. Mtawa, Implementation of the Homotopy perturbation Sumudu transform method for solving Klein-Gordon equation, Appl. Math., 6 (2015), 617-628. doi: 10.4236/am.2015.63056
    [26] A. A. Elbeleze, A. Kilicman, B. M. Taib, Homotopy perturbation method for fractional Black-Scholes European option pricing equations using Sumudu transform, Math. Probl. Eng., 2013 (2013), 524852.
    [27] G. K. Watugala, Sumudu transform: A new integral transform to solve differential equations and control engineering problems, Integr. Educ., 24 (1993), 35-43.
    [28] G. K. Watugala, The Sumudu transform for functions of two variables, Math. Eng. Ind., 8 (2002), 293-302.
    [29] M. A. Asiru, Further properties of the Sumudu transform and its applications, Int. J. Math. Educ. Sci. Tech., 33 (2002), 441-449. doi: 10.1080/002073902760047940
    [30] S. Weerakoon, Applications of Sumudu transform to partial differential equations, Int. J. Math. Educ. Sci. Tech., 25 (1994), 277-283. doi: 10.1080/0020739940250214
    [31] S. Weerakoon, Complex inversion formula for Sumudu transforms, Int. J. Math. Educ. Sci. Tech., 29 (1998), 618-621.
    [32] J. Singh, D. Kumar, D. Baleanu, et al. An efficient numerical algorithm for the fractional Drinfeld-Sokolov-Wilson equation, Appl. Math. Comput., 335 (2018), 12-24.
    [33] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
    [34] M. Caputo, Elasticita e Dissipazione, Zani-Chelli, 1969.
    [35] K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag, 2004.
    [36] A. Prakash, M. Kumar, D. Baleanu, A new iterative technique for a fractional model of nonlinear Zakharov-Kuznetsov equations via Sumudu transform, Appl. Math. Comput., 334 (2004), 30-40.
    [37] J. Choi, D. Kumar, J. Singh, et al. Analytical techniques for system of time fractional nonlinear differential equations, J. Korean Math. Soc., 54 (2017), 1209-1229.
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