A meshless discrete Galerkin method for solving the universe evolution differential equations based on the moving least squares approximation

Authors

1 Department of Mathematics, Faculty of Sciences, Bu-Ali Sina University, Hamedan 65178, Iran

2 Department of Physic, Faculty of Sciences, Bu-Ali Sina University, Hamedan 65178, Iran

Abstract

In terms of observational data, there are some problems in the standard Big Bang cosmological model. Inflation era, early accelerated phase of the evolution of the universe, can successfully solve these problems. The inflation epoch can be explained by scalar inflaton field. The evolution of this field is presented by a non-linear differential equation. This equation is considered in FLRW model. In FLRW model, we consider the universe as the warped product of real line with a three dimensional homogeneous and isotropic manifold _ which could have positive, negative or zero curvature. The main aim of this paper is the numerical solution of the inflation evolution differential equations using of a meshless discrete Galerkin method. The method reduces the solution of these types of differential equations to the solution of Volterra integral equations of the second kind. Therefore, we solve these integral equations using moving least squares method. Finally, a numerical example is included to show the validity and efficiency of the new technique.

Keywords


 

[1] P. Ade et al.: Planck 2015 results. XX. Constraints on inflation, arXiv:1502.02114 (2015).

[2] P. Ade et al.: A Joint Analysis of BICEP2/Keck Array and Planck Data, Phys. Rev. Lett. 114, 101301 (2015).

[3] P. Ade et al.: Planck 2013 results. XXII. Constraints on inflation, Astron. Astrophys. 571, A22 (2014).

[4] A. Albrecht and P. J. Steinhardt, Cosmology for grand unified theories with radiatively induced symmetry breaking, Phys. Rev. Lett. 48, 1220 (1982).

[5] L. Godinho and J. Natario: An introduction to riemannian geometry: With applications to Mechanics and Relativity, Springer, 2014.

[6] A. H. Guth, Inflationary universe: A possible solution to the horizon and flatness problems, Phys. Rev. D23, 347 (1981).

[7] S. Hawking: The occurrence of singularities in cosmology. III. Causality and singularities. Proc. Roy. Soc. Lon. 300, 187–201 (1967)

[8] S. Hawking, R. Penrose, : The singularities of gravitational collapse and cosmology. Proc. Roy. Soc. Lon. A. 314, 529–548 (1970)

[9] H. Wendland. Scattered Data Approximation. Cambridge University Press, New York, 2005.

[10] C. Zuppa. Good quality point sets and error estimates for moving least square approximations. Appl. Numer. Math., 47(3-4):575–585, 2003.

[11] K.E. Atkinson. The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge, 1997.

[12] W. Fang,Y. Wang andY. Xu. An implementation of fast wavelet Galerkin methods for integral equations of the second kind. J. Sci. Comput., 20(2):277–302, 2004.

[13]P. Assari, H. Adibi and M. Dehghan. A meshless method for solving nonlinear two-dimensional integral equations of the second kind on non-rectangular domains using radial basis functions with error analysis. J. Comput. Appl. Math., 239(1):72–92, 2013.

[14]P. Assari, H. Adibi and M. Dehghan. A numerical method for solving linear integral equations of the second kind on the non-rectangular domains based on the meshless method. Appl. Math. Model., 37(22):9269–9294, 2013.

[15]P. Assari, H. Adibi and M. Dehghan. A meshless discrete Galerkin (MDG) method for the numerical solution of integral equations with logarithmic kernels. J. Comput. Appl. Math., 267:160–181, 2014.

[16]P. Assari, H. Adibi and M. Dehghan. A meshless method based on the moving least squares (MLS) approximation for the numerical solution of two-dimensional nonlinear integral equations of the second kind on non-rectangular domains. Numer. Algor., 67(2):423–455, 2014.

[17]P. Assari, H. Adibi and M. Dehghan. The numerical solution of weakly singular integral equations based on the meshless product integration (MPI) method with error analysis. Appl. Numer. Math., 81:76–93, 2014.

[18]P. Assari and M. Dehghan. A meshless method for the numerical solution of nonlinear weakly singular integral equations using radial basis functions. Eur. Phys. J. Plus., 132:1–23, 2017.

[19]P. Assari and M. Dehghan. The numerical solution of two-dimensional logarithmic integral equations on normal domains using radial basis functions with polynomial precision. Eng. Comput., 2017.

[20] A. Bejancu Jr. Local accuracy for radial basis function interpolation on finite uniform grids. J. Approx. Theory, 99(2):242–257, 1999.

[21] M. Dehghan and R. Salehi. The numerical solution of the non-linear integro-differential equations based on the meshless method. J. Comput. Appl. Math., 236(9):2367–2377, 2012.

[22] J. Duchon. Splines minimizing rotation-invariant semi-norms in Sobolev spaces, pp. 85–100. Springer Berlin Heidelberg, Berlin, Heidelberg, 1977.

[23] G. E. Fasshauer. Meshfree methods. In Handbook of Theoretical and Computational Nanotechnology. American Scientific Publishers, 2005.

[24] R. Franke. Scattered data interpolation: Tests of some methods. Math. Comput,

38(157):181–200, 1982.

[25] A. Golbabai and S. Seifollahi. Numerical solution of the second kind integral equations using radial basis function networks. Appl. Math. Comput., 174(2):877–883, 2006.

[26] H. Kaneko and Y. Xu. Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind. Math. Comp., 62(206):739–753, 1994.

[27] E.J. Kansa. Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics-i surface approximations and partial derivative estimates. Comput. Math. Appl., 19(8-9):127–145, 1990.

[28] X. Li and J. Zhu. A Galerkin boundary node method and its convergence analysis. J. Comput. Appl. Math., 230(1):314–328, 2009.

[29] X. Li and J. Zhu. A Galerkin boundary node method for biharmonic problems. Eng. Anal. Bound. Elem., 33(6):858–865, 2009.

[30] X. Li and J. Zhu. A meshless Galerkin method for Stokes problems using boundary integral equations. Comput. Methods Appl. Mech. Engrg., 198:2874–2885, 2009.

[31] D. Mirzaei and M. Dehghan. A meshless based method for solution of integral equations. Appl. Numer. Math., 60(3):245–262, 2010.

[32] K. Parand and J. A. Rad. Numerical solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via collocation method based on radial basis functions. Appl. Math. Comput., 218(9):5292–5309, 2012.

[33] G. Wahba. Convergence rate of ”thin plate” smoothing splines when the data are noisy (preliminary report). Springer Lecture Notes in Math., 757, 1979.

[34] R.L. Hardy. Hardy, multiquadric equations of topography and other irregular surfaces. J. Geophys. Res., 176(8):1905–1915, 2006.

[35] D. Shepard, A two-dimensional interpolation function for irregularly spaced points, in: Proc. 23rd Nat. Conf. ACM, ACM Press, New York, 1968, pp. 517-524.

[36] P. Lancaster, K. Salkauskas, Surfaces generated by moving least squares methods, Math. Comput. 37 (1981) 141-158.