Research article

Development of analytical solution for a generalized Ambartsumian equation

  • Received: 11 May 2019 Accepted: 25 September 2019 Published: 06 November 2019
  • Based on the conformable derivative, a generalized model of the Ambartsumian equation is analyzed in this paper. The solution is expressed as a power series of arbitrary powers. In addition, the convergence of the obtained series solution is theoretically proven. Furthermore, it is shown that the current series reduces to the corresponding one in the literature as the conformable derivative tends to one. It is also revealed that the obtained results are of acceptable accuracy. It is found that the residuals tend to zero in specific sub-domains. The diagonal Pade approximants are implemented to extend the domain of converge to include the whole domain.

    Citation: Ebrahem A. Algehyne, Essam R. El-Zahar, Fahad M. Alharbi, Abdelhalim Ebaid. Development of analytical solution for a generalized Ambartsumian equation[J]. AIMS Mathematics, 2020, 5(1): 249-258. doi: 10.3934/math.2020016

    Related Papers:

  • Based on the conformable derivative, a generalized model of the Ambartsumian equation is analyzed in this paper. The solution is expressed as a power series of arbitrary powers. In addition, the convergence of the obtained series solution is theoretically proven. Furthermore, it is shown that the current series reduces to the corresponding one in the literature as the conformable derivative tends to one. It is also revealed that the obtained results are of acceptable accuracy. It is found that the residuals tend to zero in specific sub-domains. The diagonal Pade approximants are implemented to extend the domain of converge to include the whole domain.


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    [1] V. A. Ambartsumian, On the fluctuation of the brightness of the milky way, Doklady Akad. Nauk USSR, 44 (1944), 223-226.
    [2] J. Patade, S. Bhalekar, On analytical solution of Ambartsumian equation, Natl. Acad. Sci. Lett., 40 (2017), 291-293. doi: 10.1007/s40009-017-0565-2
    [3] T. Kato, J. B. McLeod, The functional-differential equation y'(x) = ayx)+by(x), Bull. Am. Math. Soc., 77 (1971), 891-935.
    [4] H. O. Bakodah, A. Ebaid, Exact solution of Ambartsumian delay differential equation and comparison with Daftardar-Gejji and Jafari approximate method, Mathematics, 6 (2018), 331.
    [5] D. Kumar, J. Singh, D. Baleanu, et al. Analysis of a fractional model of the Ambartsumian equation, Eur. Phys. J. Plus, 133 (2018), 133-259. doi: 10.1140/epjp/i2018-11954-7
    [6] Q. Feng, A new approach for seeking coefficient function solutions of conformable fractional partial differential equations based on the Jacobi elliptic equation, Chin. J. Phys., 56 (2018), 2817-2828. doi: 10.1016/j.cjph.2018.08.006
    [7] A. Ebaid, B. Masaedeh, E. El-Zahar, A new fractional model for the falling body problem, Chin. Phys. Lett., 34 (2017), 020201.
    [8] H. C. Yaslan, Numerical solution of the conformable space-time fractional wave equation, Chin. J. Phys., 56 (2018), 2916-2925. doi: 10.1016/j.cjph.2018.09.026
    [9] H. Rezazadeh, H. Tariq, M. Eslami, et al. New exact solutions of nonlinear conformable timefractional Phi-4 equation, Chin. J. Phys., 56 (2018), 2805-2816. doi: 10.1016/j.cjph.2018.08.001
    [10] G. Adomian, R. Rach, Algebraic equations with exponential terms, J. Math. Anal. Appl., 112 (1985), 136-140. doi: 10.1016/0022-247X(85)90280-X
    [11] G. Adomian, R. Rach, Algebraic computation and the decomposition method, Kybernetes, 15 (1986), 33-37. doi: 10.1108/eb005727
    [12] A. M. Wazwaz, Adomian decomposition method for a reliable treatment of the Bratu-type equations, Appl. Math. Comput., 166 (2005), 652-663.
    [13] A. M. Wazwaz, The combined Laplace transform-Adomian decomposition method for handling nonlinear Volterra integro-differential equations, Appl. Math. Comput., 216 (2010), 1304-1309.
    [14] A. Ebaid, Approximate analytical solution of a nonlinear boundary value problem and its application in fluid mechanics, Z. Naturforschung A., 66 (2011), 423-426.
    [15] A. Ebaid, A new analytical and numerical treatment for singular two-point boundary value problems via the Adomian decomposition method, J. Comput. Appl. Math., 235 (2011), 1914-1924. doi: 10.1016/j.cam.2010.09.007
    [16] E. H. Ali, A. Ebaid, R. Rach, Advances in the Adomian decomposition method for solving two-point nonlinear boundary value problems with Neumann boundary conditions, Comput. Math. Appl., 63 (2012), 1056-1065. doi: 10.1016/j.camwa.2011.12.010
    [17] C. Chun, A. Ebaid, M. Lee, et al. An approach for solving singular two point boundary value problems: Analytical and numerical treatment, ANZIAM J., 53 (2012), 21-43.
    [18] A. M. Wazwaz, R. Rach, J. S. Duan, Adomian decomposition method for solving the Volterra integral form of the Lane-Emden equations with initial values and boundary conditions, Appl. Math. Comput., 219 (2013), 5004-5019.
    [19] H. Triki, Solitons and periodic solutions to the dissipation-modified KdV equation with timedependent coefficients, Rom. J. Phys., 59 (2014), 421-432.
    [20] A. Ebaid, M. D. Aljoufi, A. M. Wazwaz, An advanced study on the solution of nanofluid flow problems via Adomian's method, Appl. Math. Lett., 46 (2015), 117-122. doi: 10.1016/j.aml.2015.02.017
    [21] A. Alshaery, A. Ebaid, Accurate analytical periodic solution of the elliptical Kepler equation using the Adomian decomposition method, Acta Astronaut., 140 (2017), 27-33. doi: 10.1016/j.actaastro.2017.07.034
    [22] A. M. Wazwaz, M. A. Z. Raja, M. I. Syam, Reliable treatment for solving boundary value problems of pantograph delay differential equation, Rom. Rep. Phys., 69 (2017), 69-102.
    [23] A. A. Gaber, A. Ebaid, Analytical study on the slip flow and heat transfer of nanofluids over a stretching sheet using Adomian's method, Rom. Rep. Phys., 70 (2018), 1-15.
    [24] A. K. Golmankhaneh, A. K. Golmankhaneh, D. Baleanu, Homotopy perturbation method for solving a system of Schrodinger-Korteweg-De Vries equations, Rom. Rep. Phys., 63 (2011), 609-623.
    [25] A. Patra, S. S. Ray, Homotopy perturbation sumudu transform method for solving convective radial fins with temperature-dependent thermal conductivity of fractional order energy balance equation, Int. J. Heat Mass Tran., 76 (2014), 162-170. doi: 10.1016/j.ijheatmasstransfer.2014.04.020
    [26] Z. Ayati, J. Biazar, On the convergence of Homotopy perturbation method, J. Egypt. Math. Soc., 23 (2015), 424-428. doi: 10.1016/j.joems.2014.06.015
    [27] S. M. Khaled, E. R. El-Zahar, A. Ebaid, Solution of Ambartsumian delay differential equation with conformable eerivative, Mathematics, 7 (2019), 425.
    [28] M. Turkyilmazoglu, Accelerating the convergence of decomposition method of Adomian, J. Comput. Sci., 31 (2019), 54-59. doi: 10.1016/j.jocs.2018.12.014
    [29] M. Turkyilmazoglu, Convergence accelerating in the homotopy analysis method: A new approach, Adv. Appl. Math. Mech., 10 (2018), 925-947. doi: 10.4208/aamm.OA-2017-0196
    [30] M. Turkyilmazoglu, Is homotopy perturbation method the traditional Taylor series expansion, Hacet. J. Math. Stat., 44 (2015), 651-657.
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