An efficient numerical method for solving nonlinear astrophysics equations of arbitrary order
Yıl 2019,
Cilt: 48 Sayı: 6, 1601 - 1619, 08.12.2019
Mehdi Delkhosh
Kourosh Parand
Öz
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.
Kaynakça
- [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.
Yıl 2019,
Cilt: 48 Sayı: 6, 1601 - 1619, 08.12.2019
Mehdi Delkhosh
Kourosh Parand
Kaynakça
- [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.