Year 2019, Volume 48 , Issue 6, Pages 1601 - 1619 2019-12-08

An efficient numerical method for solving nonlinear astrophysics equations of arbitrary order

Mehdi DELKHOSH [1] , Kourosh PARAND [2]


The Lane-Emden type equations of arbitrary (fractional and integer) order and the white dwarf equation are employed in the modeling of several phenomena in the areas of mathematical physics and astrophysics. In this paper, an efficient numerical algorithm based on the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) and the collocation method to solve these well-known differential equations is presented. The operational matrices of the fractional derivative and the product of order $\alpha$ in the Caputo's definition for the GFCFs are used. The obtained results are compared with other results to verify the accuracy and efficiency of the presented method. The obtained numerical results are better than other proposed methods.
fractional order of the Chebyshev functions, Lane-Emden type equations, operational matrix, Tau-Collocation method, white dwarf equation
  • [1] A. Akgul, M. Inc, E. Karatas, and D. Baleanu, Numerical solutions of fractional differential equations of Lane-Emden type by an accurate technique, Advan. Difference Equ., 2015, Article No: 220, 12 pages, 2015.
  • [2] A.H. Bhrawy and M.A. Zaky, Shifted fractional-order Jacobi orthogonal functions: Application to a system of fractional differential equations, Appl. Math. Modelling 40, 832-845, 2016.
  • [3] A.H. Bhrawy and A.S. Alofi, The operational matrix of fractional integration for shifted Chebyshev polynomials, Appl. Math. Letters 26, 25-31, 2013.
  • [4] J.P. Boyd, Chebyshev spectral methods and the Lane-Emden problem, Numer. Math. Theor. Method. Appl. 4, 142-157, 2011.
  • [5] J.P. Boyd, Chebyshev and Fourier Spectral Methods, Dover Publications, Mineola, New York, 2000.
  • [6] E.A. Butcher, H. Ma, E. Bueler, V. Averina, and Z. Szabo, Stability of linear time-periodic delay-differential equations via Chebyshev polynomials, Int. J. Numer. Method. Eng. 59, 895-922, 2004.
  • [7] S. Chandrasekhar, Introduction to the Study of Stellar Structure, Dover Publications, New York, 1967.
  • [8] M.A. Darani and M. Nasiri, A fractional type of the Chebyshev polynomials for approximation of solution of linear fractional differential equations, Comp. Meth. Diff. Equ. 1, 96-107, 2013.
  • [9] H.T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover Publications, New York, 1962.
  • [10] J. Davila, L. Dupaigne, and J. Wei, On the fractional Lane-Emden equation, Trans. Amer. Math. Soc. 369, 6087-6104, 2017.
  • [11] M. Dehghan, E. Hamedi, and H. Khosravian-Arab, A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials, J. Vib. Control 22 (6), 1547-1559, 2016.
  • [12] M. Delkhosh, Introduction of Derivatives and Integrals of Fractional order and Its Applications, Appl. Math. and Phys. 1 (4), 103-119, 2013.
  • [13] K. Diethelm, The analysis of fractional differential equations, Springer-Verlag, Berlin, 2010.
  • [14] S. Esmaeili, M. Shamsi, and Y. Luchko, Numerical solution of fractional differential equations with a collocation method based on Muntz polynomials, Comput. Math. Appl. 62, 918-929, 2011.
  • [15] M. Fazly and J. Wei, On Finite Morse Index Solutions of Higher Order Fractional Lane-Emden Equations, American J. Math. 139 (2), 433-460, 2017.
  • [16] B. Fischer and F. Peherstorfer Chebyshev approximation via polynomial mappings and the convergence behavior of Krylov subspace methods, Electron. Trans. Numer. Anal. 12, 205-215, 2001.
  • [17] I. Hashim, O. Abdulaziz, and S. Momani, Homotopy analysis method for fractional IVPs, Commun. Nonlinear Sci. Numer. Simul. 14, 674-684, 2009.
  • [18] J. He, Nonlinear oscillation with fractional derivative and its applications, in: International Conference on Vibrating Engineering 98, 288-291, Dalian, China, 1998.
  • [19] R.W. Ibrahim, Existence of nonlinear Lane-Emden Equation of Fractional Order, Misk. Math. Notes 13 (1), 39-52, 2012.
  • [20] P. Junghanns and A. Rathsfeld, A polynomial collocation method for Cauchy singular integral equations over the interval, Electron. Trans. Numer. Anal. 14, 79-126, 2002.
  • [21] S. Kazem, S. Abbasbandy, and S. Kumar, Fractional-order Legendre functions for solving fractional-order differential equations, Appl. Math. Model. 37, 5498-5510, 2013.
  • [22] E. Keshavarz, Y. Ordokhani, and M. Razzaghi, A numerical solution for fractional optimal control problems via Bernoulli polynomials, J. Vib. Control 22 (18), 3889- 3903, 2016.
  • [23] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, San Diego, 2006.
  • [24] K.M. Kolwankar, Studies of fractal structures and processes using methods of the fractional calculus, Ph.D. thesis, University of Pune, Pune, India, 1998.
  • [25] G.W. Leibniz, Letter from Hanover, Germany, to G.F.A. L’Hopital, September 30; 1695, in Mathematische Schriften, 1849; reprinted 1962, Olms verlag; Hidesheim, Germany, 2, 301-302, 1965.
  • [26] H.R. Marasi, N. Sharifi, and H. Piri, Modified Differential Transform Method For Singular Lane-Emden Equations in Integer and Fractional Order, TWMS J. App. Eng. Math. 5 (1), 124-131, 2015.
  • [27] A. Martinez-Finkelshtein, P. Martinez-Gonzalez, and R. Orive, Asymptotic of polynomial solutions of a class of generalized Lame differential equations, Electron. Trans. Numer. Anal. 19, 18-28, 2005.
  • [28] M.S. Mechee and N. Senu Numerical Study of Fractional Differential Equations of Lane-Emden Type by Method of Collocation, Appl. Math. 3, 851-856, 2012.
  • [29] M. Mohseni Moghadam and H. Saeedi Application of differential transforms for solving the Volterra integro-partial differential equations, Iranian J. Sci. Tech. Tran. A 34, 59-70, 2010.
  • [30] S. Momani and N.T. Shawagfeh, Decomposition method for solving fractional Riccati differential equations, Appl. Math. Comput. 182, 1083-1092, 2006.
  • [31] K. Parand and M. Nikarya, Application of Bessel functions for solving differential and integro-differential equations of the fractional order, Appl. Math. Model. 38, 4137-4147, 2014.
  • [32] K. Parand and M. Hemami, Numerical Study of Astrophysics Equations by Meshless Collocation Method Based on Compactly Supported Radial Basis Function, Int. J. Appl. Comput. Math. 3 (2), 1053-1075, 2017.
  • [33] K. Parand, A. Taghavi, and M. Shahini, Comparison between rational Chebyshev and modified generalized Laguerre functions pseudospectral methods for solving Lane- Emden and unsteady gas equations, Acta Phys. Polo. B 40 (12), 1749-1763, 2009.
  • [34] K. Parand, M.M. Moayeri, S. Latifi, and M. Delkhosh, A numerical investigation of the boundary layer flow of an Eyring-Powell fluid over a stretching sheet via rational Chebyshev functions, Euro. Phys. J. Plus 132, Article No: 325, 11 pages, 2017.
  • [35] K. Parand and M. Shahini, Rational Chebyshev collocation method for solving nonlinear ordinary differential equations of Lane-Emden type, Int. J. Info. Sys. Sci. 6, 72-83, 2010.
  • [36] K. Parand, H. Yousefi, and M. Delkhosh M. A Numerical Approach to Solve Lane- Emden Type Equations by the Fractional Order of Rational Bernoulli Functions, Romanian J. Phys. 62 (104), 1-24, 2017.
  • [37] K. Parand and M. Delkhosh, Operational Matrices to Solve Nonlinear Volterra- Fredholm Integro-Differential Equations of Multi-Arbitrary Order, Gazi Uni. J. Sci. 29 (4), 895-907, 2016.
  • [38] K. Parand and M. Delkhosh, Solving the nonlinear Schlomilch’s integral equation arising in ionospheric problems, Afr. Mat. 28 (3), 459-480, 2017.
  • [39] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • [40] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal. 5, 367-386, 2002.
  • [41] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • [42] J.A. Rad, S. Kazem, M. Shaban, K. Parand, and A. Yildirim, Numerical solution of fractional differential equations with a Tau method based on Legendre and Bernstein polynomials, Math. Meth. in the Appl. Sci. 37, 329-342, 2014.
  • [43] A. Saadatmandi, M. Dehghan, and M.R. Azizi, The Sinc-Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients, Commun. Nonlinear. Sci. Numer. Simulat. 17, 4125-4136, 2012.
  • [44] A. Saadatmandi and M. Dehghan, Numerical solution of hyperbolic telegraph equation using the Chebyshev Tau method, Num. Meth. Partial Diff. Equ. 26 (1), 239-252, 2010.
  • [45] U. Saeed, Haar Adomian method for the solution of fractional nonlinear Lane-Emden type equations arising in astrophysics, Taiwanese J. Math. 21 (5), 1175-1192, 2017.
  • [46] H. Saeedi and G.N. Chuev, Triangular functions for operational matrix of nonlinear fractional Volterra integral equations,J. Appl. Math. Comput. 49 (1), 213-232, 2015.
  • [47] H. Saeedi, Application of Haar wavelets in solving nonlinear fractional Fredholm integro-differential equations, J. Mahani Math. Res. Center 2 (1), 15-28, 2013.
  • [48] J. Shen, T. Tang, and L.L. Wang Spectral Methods Algorithms, Analytics and Applications, Springer, New York, 2001.
  • [49] O.P. Singh, R.K. Pandey, and V.K. Singh, An analytic algorithm of Lane-Emden type equations arising in astrophysics using modified homotopy analysis method, Comput. Phys. Commun. 180, 1116-1124, 2009.
  • [50] G. Szego, orthogonal polynomials, American Mathematical Society Providence, Rhode Island, 1975.
  • [51] R.A. Van Gorder, Relation between Lane-Emden solutions and radial solutions to the elliptic Heavenly equation on a disk, New Astro. 37, 42-47, 2015.
  • [52] A.M.Wazwaz, A new algorithm for solving differential equations of Lane-Emden type, Appl. Math. Compu. 118, 287-310, 2001.
  • [53] S.A. Yousefi, A. Lotfi, and M. Dehghan, The use of a Legendre multiwavelet collocation method for solving the fractional optimal control problems, J. Vib. Control 17 (13), 2059-2065, 2011.
  • [54] S. Yuzbasi, A numerical approximation based on the Bessel functions of first kind for solutions of Riccati type differential-difference equations, Comput. Math. Appl. 64 (6), 1691-1705, 2012.
  • [55] S. Yuzbasi, A numerical approach for solving a class of the nonlinear LaneEmden type equations arising in astrophysics, Math. Method. Appl. Sci. 34 (18), 2218-2230, 2011.
Primary Language en
Subjects Mathematics
Journal Section Mathematics
Authors

Orcid: 0000-0001-6632-4743
Author: Mehdi DELKHOSH
Institution: Shahid Beheshti University
Country: Iran


Orcid: 0000-0001-5946-0771
Author: Kourosh PARAND (Primary Author)
Institution: Shahid Beheshti University
Country: Iran


Dates

Publication Date : December 8, 2019

Bibtex @research article { hujms656646, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2019}, volume = {48}, pages = {1601 - 1619}, doi = {}, title = {An efficient numerical method for solving nonlinear astrophysics equations of arbitrary order}, key = {cite}, author = {DELKHOSH, Mehdi and PARAND, Kourosh} }
APA DELKHOSH, M , PARAND, K . (2019). An efficient numerical method for solving nonlinear astrophysics equations of arbitrary order. Hacettepe Journal of Mathematics and Statistics , 48 (6) , 1601-1619 . Retrieved from https://dergipark.org.tr/en/pub/hujms/issue/50516/656646
MLA DELKHOSH, M , PARAND, K . "An efficient numerical method for solving nonlinear astrophysics equations of arbitrary order". Hacettepe Journal of Mathematics and Statistics 48 (2019 ): 1601-1619 <https://dergipark.org.tr/en/pub/hujms/issue/50516/656646>
Chicago DELKHOSH, M , PARAND, K . "An efficient numerical method for solving nonlinear astrophysics equations of arbitrary order". Hacettepe Journal of Mathematics and Statistics 48 (2019 ): 1601-1619
RIS TY - JOUR T1 - An efficient numerical method for solving nonlinear astrophysics equations of arbitrary order AU - Mehdi DELKHOSH , Kourosh PARAND Y1 - 2019 PY - 2019 N1 - DO - T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1601 EP - 1619 VL - 48 IS - 6 SN - 2651-477X-2651-477X M3 - UR - Y2 - 2018 ER -
EndNote %0 Hacettepe Journal of Mathematics and Statistics An efficient numerical method for solving nonlinear astrophysics equations of arbitrary order %A Mehdi DELKHOSH , Kourosh PARAND %T An efficient numerical method for solving nonlinear astrophysics equations of arbitrary order %D 2019 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 48 %N 6 %R %U
ISNAD DELKHOSH, Mehdi , PARAND, Kourosh . "An efficient numerical method for solving nonlinear astrophysics equations of arbitrary order". Hacettepe Journal of Mathematics and Statistics 48 / 6 (December 2019): 1601-1619 .
AMA DELKHOSH M , PARAND K . An efficient numerical method for solving nonlinear astrophysics equations of arbitrary order. Hacettepe Journal of Mathematics and Statistics. 2019; 48(6): 1601-1619.
Vancouver DELKHOSH M , PARAND K . An efficient numerical method for solving nonlinear astrophysics equations of arbitrary order. Hacettepe Journal of Mathematics and Statistics. 2019; 48(6): 1619-1601.