Abstract
Singular and highly oscillatory ordinary differential equations (ODEs) are of great scientific significance, since they frequently arise in practice. Hence, in this paper, we present a Block Nyström method (BNM) that is applied to directly solve the Lane–Emden type equations which belong to a class of nonlinear singular ODEs. The application of the BNM is also extended to solve highly oscillatory ODEs directly without reducing the ODE into an equivalent first order system. The BNM is formulated from its continuous scheme which is constructed from an appropriate power series via collocation and interpolation techniques. The convergence and stability properties of the BNM are discussed. Accuracy and efficiency benefits of the method are demonstrated via several numerical examples.
Similar content being viewed by others
References
Bender, C.M., Milton, K.A., Pinsky, S.S., Simmons, L.M.: A new perturbative approach to nonlinear problems. J. Math. Phys. 30, 144–1455 (1989)
Wazwaz, A.A.: A new algorithm for solving differential equations of Lane–Emden type. Appl. Math. Comput. 118, 287–310 (2001)
Lia, S.: A new analytic algorithm of Lane–Emden type equation. Appl. Math. Comput. 142, 539–541 (2003)
Kaur, H., Mittal, R.C., Mishra, V.: Haar wavelet approximation solutions for the generalized Lane–Emden equations arising from astrophysics. Comput. Phys. Commun. 184, 2169–2177 (2013)
Parand, K., Dehghan, M., Rezaei, A.R., Ghaderi, S.M.: An approximation algorithm for the solution of nonlinear Lane–Emden type equations arising in astrophysics using Hermite function collocation method. Comput. Phys. Commun. 181, 1096–1108 (2010)
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. doi:10.1016/j.asej.2015.06.006
Awoyemi, D.O.: A P-stable linear multistep method for solving general third order ordinary differential equations. Int. J. Comput. Math. 80(8), 987–993 (2003)
Jator, S.N., Lee, L.: Implementing a seventh-order linear multistep method in a predictor–corrector mode or block mode: which is more efficient for the general second order initial value problem. SpringerPlus 3, 447 (2014). http://www.springerplus.com/content/3/1/447
Vigo-Aguiar, J., Ramos, H.: Variable stepsize implementation of multistep methods for \(y^{\prime \prime }=f (x, y, y^{\prime })\). J. Comput. Appl. Math. 192, 114–131 (2006)
Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Springer, New York (1996)
Jator, S.N., Swindle, S., French, R.: Trigonometrically fitted block Numerov type method for \(y^{\prime \prime }=f(x, y, y^{\prime })\). Numer. Algorithms 62, 13–26 (2013)
Jator, S.N.: A sixth order linear multistep method for the direct solution of \(y^{\prime \prime } = f(x, y, y^{\prime })\). Int. J. Pure Appl. Math. 40(4), 457–472 (2007)
Jator, S.N., Li, J.: A self starting linear multistep method for the direct solution of the general second order initial value problems. Int. J. Comput. Math. 86(5), 817–836 (2009)
Jator, S.N.: Implicit third derivative Runge–Kutta–Nyström method with trigonometric coefficients. Numer. Algorithms 70, 133–150 (2015)
Jator, S.N.: A continuous two-step method of order 8 with a block extension for \(y^{\prime \prime }=f(x, y, y^{\prime })\). Appl. Math. Comput. 219, 781–791 (2012)
Wang, B., Iserles, A., Wu, X.: Arbitrary-order trigonometric Fourier collocation methods for multi-frequency oscillatory systems. Found. Comput. Math. 16, 151–181 (2016)
Ding, H., Zhang, Y., Cao, J., Tian, J.: A class of difference scheme for solving telegraph equation by new non-polynomial spline methods. Appl. Math. Comput. 218, 4671–4683 (2012)
Coleman, J.P., Ixaru, G.R.: P-stability and exponential-fitting methods for \(y^{\prime \prime }=f(x, y)\). IMA J. Numer. Anal. 16, 179–199 (1996)
Coleman, J.P., Duxbury, S.C.: Mixed collocation methods for \(y^{\prime \prime }=f(x, y)\). J. Comput. Appl. Math. 126, 47–75 (2000)
Baranwal, V.K., Pandy, R.K., Tripathi, M.P., PSingh, O.: An analytic algorithm of Lane–Emden type equations arising in astrophysics—a hybrid approach. J. Theor. Appl. Phys. 6, 22 (2012)
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)
Brugnano, L., Trigiante, D.: Solving Differential Problems by Multistep Initial and Boundary Value Methods, pp. 280–299. Gordon and Breach Science Publishers, Amsterdam (1998)
Chawla, M.M., Jain, M.K., Subramanian, R.: The application of explicit Nyström methods to singular second order differential equations. Comput. Math. Appl. 19(12), 47–51 (1990)
Mazzia, F., Sestini, A., Trigiante, T.: B-spline linear multistep methods and their continuous extensions. SIAM J. Numer. 44, 1954–1973 (2006)
Sahlan, M.N., Hashemizadeh, E.: Wavelet Galerkin method for solving nonlinear singular boundary value problems in physiology. Appl. Math. Comput. 250, 260–269 (2015)
Fang, Y., Song, Y., Wu, X.: A robust trigonometrically fitted embedded pair for perturbed oscillators. J. Comput. Appl. Math. 225, 347–355 (2009)
Franco, J.M.: Runge–Kutta–Nyström methods adapted to the numerical integration of perturbed oscillators. Comput. Phys. Commun. 147, 770–787 (2002)
Sommeijer, B.P.: Explicit, high-order Runge–Kutta–Nyström methods for parallel computers. Appl. Numer. Math. 13, 221–240 (1993)
Franco, J.M.: New methods for oscillatory systems based on ARKN methods. Appl. Numer. Math. 56, 1040–1053 (2006)
D’Ambrosio, R., Ferro, M., Paternoster, B.: Two-step hybrid collocation methods for \(y^{\prime \prime }=f(x, y)\). Appl. Math. Lett. 22, 1076–1080 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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 (Suppl 1), 1385–1402 (2017). https://doi.org/10.1007/s40819-017-0425-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40819-017-0425-2