Applying He's variational iteration method to FRW cosmology
Keywords:Approximate Solution, FRW Cosmological Models, Dynamical System Analysis, Variational Iteration Method.
This work is devoted to the investigation of Friedmann-Robertson-Walker (FRW) cosmological models with the help of the so-called Variational Iteration Method (VIM). For this end, we briefly recall the main equations of the cosmological models and the basic idea of VIM. In order to approbate the VIM in FRW cosmology and demonstrate the main steps in solving by this method, we consider the test example of the universe with dust for which the exact solution of the model is known. Then, a solution for the spatially flat FRW model of the universe filled with the dust and quintessence is obtained when the exact analytic solution cannot be found. A comparison of our solution with the corresponding numerical solution shows that it is of a high degree of accuracy. Moreover, the Dynamical System Analysis to the dynamics of the homogeneous and isotropic FRW universes is used as a special case of generalized Lotkaâ€“Volterra system where the competitive species are the barotropic fluids filling the Universe. With the help of VIM, we have found the iterative formulae for the density parameters of the cosmological analog of the generalized Lotkaâ€“Volterra set of equations. All solutions illustrated graphically by means of Maple software.
 A. G. Riess, et al., "Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant", Astronomical Journal, Vol. 116 (1998), 1009. https://doi.org/10.1086/300499.
 S. Perlmutter, et al., "Measurements of Omega and Lambda from 42 High-Redshift Supernovae", Astrophysical Journal, Vol. 517 (1999), 565. https://doi.org/10.1086/307221.
 N. W. Halverson, et al., "Degree Angular Scale Interferometer First Results: A Measurement of the Cosmic Microwave Background Angular Power Spectrum", Astrophys. J., 568 (2002) 38-45. https://doi.org/10.1086/338879.
 D. N. Spergel, et al., "First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters", Astrophys. J. Suppl. Ser., 148 (2003) 175-194. https://doi.org/10.1086/377226.
 M. Tegmark, M. A. Strauss, et al., "Cosmological parameters from SDSS and WMAP", Phys. Rev. D, 69 (2004) 103501. https://doi.org/10.1103/PhysRevD.69.103501.
 S. W. Allen, R. W. Schmidt, H. Ebeling, et al., "Constraints on dark energy from Chandra observations of the largest relaxed galaxy clusters", Mon. Not. Roy. Astron. Soc., 353 (2004) 457-467. https://doi.org/10.1111/j.1365-2966.2004.08080.x.
 L.D. Landau, E.M. Lifshitz. The Classical Theory of Fields, Vol. 2 (3rd ed.), Pergamon Press, 1971.
 A. Guth, The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems, Phys. Rev. D, Vol. 23, 347, 1981. https://doi.org/10.1103/PhysRevD.23.347.
 A. D. Linde, A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems, Phys. Lett. B, Vol. 108, 389, 1982. https://doi.org/10.1016/0370-2693(82)91219-9.
 E. J. Copeland, A. R. Liddle, D. Wands, "Exponential potentials and cosmological scaling solutions", Physical Review D, 57(8), (1998) 4686â€“4690. https://doi.org/10.1103/PhysRevD.57.4686.
 L. Amendola, "Scaling solutions in general nonminimal coupling theorie", Physical Review D, 60(4) (1999) 043501, https://doi.org/10.1103/PhysRevD.60.043501.
 J.-H. He, "Homotopy perturbation technique", Computer Methods in Applied Mechanics and Engineering, 178 (1999), 257-262. https://doi.org/10.1016/S0045-7825(99)00018-3.
 J.-H. He, "A coupling method of homotopy technique and perturbation technique for nonlinear problems", International Journal of Non-Linear Mechanics, 35 (1) (2000), 37-43. https://doi.org/10.1016/S0020-7462(98)00085-7.
 V. Shchigolev, "Homotopy Perturbation Method for Solving a Spatially Flat FRW Cosmological Model", Universal Journal of Applied Mathematics, 2(2) (2014), 99-103.
 V. Shchigolev, "Analytical Computation of the Perihelion Precession in General Relativity via the Homotopy Perturbation Method", Universal Journal of Computational Mathematics, 3(4) (2015), 45-49.
 V. K. Shchigolev, "Calculating Luminosity Distance versus Redshift in FLRW Cosmology via Homotopy Perturbation Method", Gravitation and Cosmology, 23 (2017) 142. https://doi.org/10.1134/S0202289317020098.
 V. K. Shchigolev, D. N. Bezbatko, "Studying gravitational deflection of light by Kiselev black hole via homotopy perturbation method", General Relativity and Gravitation, 51 (2019) 34. https://doi.org/10.1007/s10714-019-2521-6.
 J.-H. He, "Variational iteration method - a kind of non-linear analytical technique: some examples", International Journal of Non-Linear Mechanics, 34(4) (1999), 699-708. https://doi.org/10.1016/S0020-7462(98)00048-1.
 J.-H. He, "Variational iteration method for autonomous ordinary differential systems", Applied Mathematics and Computation, 114(2-3) (2000), 115-123. https://doi.org/10.1016/S0096-3003(99)00104-6.
 J.-H. He, "Variational iteration method-Some recent results and new interpretations", Journal of Computational and Applied Mathematics, 207(1) (2007), 3-17. https://doi.org/10.1016/j.cam.2006.07.009.
 M. Tatari and M. Dehghan, "On the Convergence of Heâ€™s Variational Iteration Method", Journal of Computational and Applied Mathematics, Vol. 207, No. 1, 121-128, 2007. https://doi.org/10.1016/j.cam.2006.07.017.
 J. I. Ramos, "On the Variational Iteration Method and Other Iterative Techniques for Nonlinear Differential Equations", Applied Mathematics and Computation, Vol. 199, No. 1, 39-69, 2008. https://doi.org/10.1016/j.amc.2007.09.024.
 Ernest Scheiber, "On the Convergence of the Variational Iteration Method", arxiv: 1509.01779.
 V. K. Shchigolev, "Variational iteration method for studying perihelion precession and deflection of light in General Relativity", International Journal of Physical Research, 4 (2) (2016) 52-57. https://doi.org/10.14419/ijpr.v4i2.6530.
 V. K. Shchigolev, "Analytic Approximation of Luminosity Distance in Cosmology via Variational Iteration Method", Universal Journal of Computational Mathematics, 5(3) (2017) 68-74.
 N. Roy, Dynamical Systems Analysis of Various Dark Energy Models. PhD Thesis (2015). arXiv: 1511.07978 [gr-qc].
 P. Shah, G.C. Samanta, S. Capozziello, "Qualitative behavior of cosmological models combining various matter fields", Int. J. Mod. Phys. A , 33 (2018) 1850116. https://doi.org/10.1142/S0217751X18501166.
 P. Shah, G. C. Samanta, "Stability analysis for cosmological models in f(R) gravity using dynamical system analysis", The European Physical Journal C, 79(5) (2019), 414. https://doi.org/10.1140/epjc/s10052-019-6934-x.
 J. Wainright and G. F. R. Ellis, Dynamical Systems in Cosmology (Cambridge University Press, Cambridge, 1997). https://doi.org/10.1017/CBO9780511524660.
 A. A. Coley, Dynamical Systems and Cosmology (Springer, 2003). https://doi.org/10.1007/978-94-017-0327-7.
 J. Perez, A. Fuzfa. T. Carletti, L. Melot, L. Guedezounme, "The Jungle Universe: coupled cosmological models in a Lotkaâ€“Volterra framework", General Relativity & Gravitation, 46 (2014) 1753. https://doi.org/10.1007/s10714-014-1753-8.
 A. Simon-Petit, H.-H. Yap, J. Perez, "Refinements in the Jungle Universes", http://arxiv.org/abs/1603.02267
 B. D. Yuliyanto and S. Mungkasi, "Variational iteration method for solving the population dynamics model of two species", Journal of Physics: Conf. Series, 795 (2017) 012044. https://doi.org/10.1088/1742-6596/795/1/012044.
 S. M. Goh, M. S. M. Noorani, I. Hashim, "Introducing variational iteration method to a biochemical reaction model", Nonlinear Analysis: Real World Applications, 11 (2010) 2264-2272. https://doi.org/10.1016/j.nonrwa.2009.06.015.
 B. Batiha, M. S. M. Noorani and I. Hashim, "Variational iteration method for solving multispecies Lotkaâ€“Volterra equations", Computers and Mathematics with Applications, 54 (2007) 903. https://doi.org/10.1016/j.camwa.2006.12.058.
 M. Rafei, H. Daniali and D. D. Ganji, "Variational iteration method for solving the epidemic model and the prey and predator problem", Applied Mathematics and Computation, 186 (2007) 1701. https://doi.org/10.1016/j.amc.2006.08.077.
 F. Shakeri and M. Dehghan, "Numerical solution of a biological population model using He's variational iteration method", Computers and Mathematics with Applications, 54 (2007) 1197. https://doi.org/10.1016/j.camwa.2006.12.076.