Abstract
The aim of this paper is to present a twofold approach for the numerical solution of a class of singular second-order nonlinear differential equations. The first is based on a modified version of an adaptive spline collocation method (ASCM). The second is a patching approach (PASCM) that splits the problem domain into two subintervals: Chebyshev economization procedure is implemented in the vicinity of the singular point and outside this domain the resulting initial or boundary value problem is handled by the (ASCM). The second strategy is based on the linearization of the nonlinear term about the given initial condition at the singular point. The choice of either technique relies on the specified boundary or initial conditions. Performance of the approach is investigated numerically through a number of application examples that demonstrate the efficiency of the approach and that it has O(h 4) rate of convergence. Results confirm that the scheme yields highly accurate results when compared with the exact and/or numerical solutions that exist in the literature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Benko D., Biles D.C., Robinson M.P., Spraker J.S.: Numerical approximation for singular second order differential equations. Math. Comput. Model. 49(5-6), 1109–1114 (2009)
Benko D., Bykes D.C., Robinson M.P., Spraker J.S.: Nyström methods and singular second-order differential equations. Comput. Math. Appl. 56, 1975–1980 (2008)
Caglar H.N., Caglar S.H., Twizell E.H.: The numerical solution of third-order boundary value problems with fourth-degree B-spline functions. Int. J. Comput. Math. 71, 373–381 (1999)
Caglar H.N., Caglar S.H., Twizell E.H.: The numerical solution of fifth-order boundary value problems with sixth-degree B-spline functions. Appl. Math. Lett. 12, 25–30 (1999)
Çaǧlar H., Çaǧlar N., Özer M.: B-spline solution of non-linear singular boundary value problems arising in physiology. Chaos Solitons Fractals 39(3), 1232–1237 (2009)
Chawla M.M., Jain M.K., Subramanian R.: The application of explicit Nyström methods to singular second order differential equations. Comput. Math. Appl. 19, 47–51 (1990)
Christara C.C., Ng Kit Sun: Adaptive techniques for spline collocation. Computing 76, 259–277 (2006)
Chou Jung-Hua, Wu Raylin: A generalization of the Frobenius method for ordinary differential equations with regular singular points. J. Math. Stat. 1(1), 3–7 (2005)
Deeba E., Khuri S.A.: Nonlinear equations, Wiley Encyclopedia of Electrical and Electronics Engineering, vol. 14, pp. 562–570. John Wiley and Sons, NewYork (1999)
Kadalbajoo, Mohan K., Aggarwal, Vivek K.: Cubic spline for solving singular two-point boundary value problems. Appl. Math. Comput. 156, 249–259 (2004)
Kadalbajoo, Mohan K., Aggarwal, Vivek K.: Numerical solution of singular boundary value problems via Chebyshev polynomial and B-spline. Appl. Math. Comput. 160, 851–863 (2005)
Kadalbajoo, Mohan K., Aggarwal, Vivek K.: Fitted mesh B-spline collocation method for solving self-adjoint singularly perturbed boundary value problems. Appl. Math. Comput. 161, 973–987 (2005)
Ravi Kanth A.S.V., Aruna K.: He’s variational iteration method for treating nonlinear singular boundary value problems. Comput. Math. Appl. 60, 821–829 (2010)
Liao Shijun: A new analytic algorithm of Lane-Emden type equations. Appl. Math. Comput. 142, 1–16 (2003)
Pandey R.K., Singh Arvind K.: On the convergence of a finite difference method for a class of singular boundary value problems arising in physiology. J. Comput. Appl. Math. 166, 553–564 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Khuri, S.A., Sayfy, A. A Twofold Spline Chebyshev Linearization Approach for a Class of Singular Second-Order Nonlinear Differential Equations. Results. Math. 63, 817–835 (2013). https://doi.org/10.1007/s00025-012-0235-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-012-0235-0