Abstract
A one-step hybrid block method which is applied for solving Lane-Emden second-order singular differential equations arising in astrophysics is developed. The present technique is obtained by considering three intra-step nodal points, which are chosen appropriately in order to get optimized errors of the main formulas approximating the solution and the first derivative at the final position of the block, and another approximation of the solution at an intermediate point. The stability properties and characteristics of the proposed strategy are analyzed. The new approach is combined with an appropriate algorithm which is applied to the first subinterval to circumvent the singular behaviour at the left endpoint of the integration interval. Some second-order singular differential problems are solved numerically to show the effectiveness and validity of the proposed technique, which has been compared with other strategies available in the recent literature. The presented results support the better performance of the developed scheme.
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References
Agarwal, R.P., O’Regan, D.: Second order initial value problems of Lane-Emden type. Appl. Math. Lett. 20(12), 1198–1205 (2007)
Bender, C.M., Milton, K.A., Pinsky, S.S., Simmons, L.M.: A new perturbative approach to nonlinear problems. J. Math. Phys. 30, 1447–1455 (1989)
Chandrasekhar, S.: Introduction to Study of Stellar Structure. Dover, New York (1967)
Chowdhury, M.S.H., Hashim, I.: Solution of a class of singular second-order IVPs by homotopy-perturbation method. Phys. Lett. A 365, 439–447 (2007)
Davis, H.T.: Introduction to Nonlinear Differential and Integral Equations. Dover, New York (1962)
Horedt, G.P.: Seven-digit tables of Lane-Emden functions. Astrophys. Space Sci. 126, 357–408 (1986)
Jator, S.N., Oladejo, H.B.: Block Nyström method for singular differential equations of the Lane-Emden type and problems with highly oscillatory solutions. Int. J. Appl. Comput. Math. 3, 1385–1402 (2017)
Koch, O., Kofler, P., Weinmüller, E.B.: The implicit Euler method for the numerical solution of singular initial value problems. Appl. Numer. Math. 34, 231–252 (2000)
Lambert, J.D., Watson, I.A.: Symmetric multistep methods for periodic initial value problems. IMA J. Appl. Math. 18, 189–202 (1976)
Mandelzweig, V.B., Tabakin, F.: Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs. Comput. Phys. Commun. 141(2), 268–281 (2001)
Momoniat, E., Harley, C.: Approximate implicit solution of a Lane-Emden equation. New Astron. 11, 520–526 (2006)
Öztürk, Y.: An efficient numerical algorithm for solving system of Lane-Emden type equations arising in engineering. Nonlinear Eng. 8(1), 429–437 (2019)
Pandey, R.K., Kumar, N., Bhardwaj, A., Dutta, G.: Solution of Lane-Emden type equations using Legendre operational matrix of differentiation. Appl. Math. Comput. 218(14), 7629–7637 (2012)
Parand, K., Pirkhedri, A.: Sinc-collocation method for solving astrophysics equations. New Astron. 15, 533–537 (2010)
Parand, K., Ghaderi-Kangavari, A., Delkosh, M.: Two efficient computational algorithms to solve the nonlinear singular Lane-Emden equations. Astrophysics 63, 133–150 (2020). https://doi.org/10.1007/s10511-020-09621-8
Ramos, H., Rufai, M.A.: Third derivative modification of \(k\)-step block Falkner methods for the numerical solution of second order initial-value problems. Appl. Math. Comput. 333, 231–245 (2018)
Ramos, H., Rufai, M.A.: Numerical solution of boundary value problems by using an optimized two-step block method. Numer. Algorithms (2019). https://doi.org/10.1007/s11075-019-00753-3
Ramos, H., Singh, G.: An optimized two-step hybrid block method formulated in variable step-size mode for integrating \(y''= f (x, y, y')\) numerically. Numer. Math., Theory Methods Appl. 12, 640–660 (2019)
Ramos, H., Kalogiratou, Z., Monovasilis, T., Simos, T.E.: An optimized two-step hybrid block method for solving general second order initial-value problems. Numer. Algorithms 72, 1089–1102 (2016)
Ramos, H., Mehta, S., Vigo-Aguiar, J.: A unified approach for the development of \(k\)-step block Falkner-type methods for solving general second-order initial-value problems in ODEs. J. Comput. Appl. Math. 318, 550–564 (2017)
Rismani, A.M., Monfared, H.: Numerical solution of singular IVPs of Lane-Emden type using a modified Legendre-spectral method. Appl. Math. Model. 36, 4830–4836 (2012)
Rosenau, P.: A note on integration of the Emden-Fowler equation. Int. J. Non-Linear Mech. 19, 303–308 (1984)
Rufai, M.A., Areo, E.A.: An efficient one-eight step hybrid block method for solving second order initial value problems of ODEs. Int. J. Differ. Equ. Appl. 15(2), 117–139 (2016)
Shawagfeh, N.T.: Nonperturbative approximate solution for Lane-Emden equation. J. Math. Phys. 34, 4364–4369 (1993)
Shiralashetti, S.C., Deshi, A.B., Desai, P.B.: Haar wavelet collocation method for the numerical solution of singular initial value problems. Ain Shams Eng. J. 7(2), 663–670 (2016)
Singh, H.: An efficient computational method for the approximate solution of nonlinear Lane-Emden type equations arising in astrophysics. Astrophys. Space Sci. 363, 71 (2018). https://doi.org/10.1007/s10509-018-3286-1
Singh, H., Srivastava, H.M., Kumar, D.: A reliable algorithm for the approximate solution of the nonlinear Lane-Emden type equations arising in astrophysics. Numer. Methods Partial Differ. Equ. 33, 1524–1555 (2017)
Wazwaz, A.: A new algorithm for solving differential equations of Lane-Emden type. Appl. Math. Comput. 118, 287–310 (2001)
Wazwaz, A.M., Rach, R., Duan, J.S.: A study on the systems of the Volterra integral forms of the Lane-Emden equations by the Adomian decomposition method. Math. Methods Appl. Sci. 37(1), 10–19 (2014)
Yildirim, A., Özis, T.: Solutions of singular IVPs of Lane-Emden type by the variational iteration method. Nonlinear Anal., Ser. A Theory Methods Appl. 70, 2480–2484 (2009)
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Rufai, M.A., Ramos, H. Numerical solution of second-order singular problems arising in astrophysics by combining a pair of one-step hybrid block Nyström methods. Astrophys Space Sci 365, 96 (2020). https://doi.org/10.1007/s10509-020-03811-8
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DOI: https://doi.org/10.1007/s10509-020-03811-8