Solving singular two-point boundary value problem in reproducing kernel space. (English) Zbl 1149.65057

Authors’ summary: We present a new method for solving singular two-point boundary value problem for certain ordinary differential equation having singular coefficients. Its exact solution is represented in the form of series in reproducing kernel space. In the mean time, the \(n\)-term approximation \(u_{n}(x)\) to the exact solution \(u(x)\) is obtained and is proved to converge to the exact solution. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other.


65L10 Numerical solution of boundary value problems involving ordinary differential equations
34K10 Boundary value problems for functional-differential equations
34B05 Linear boundary value problems for ordinary differential equations
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
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[1] Kadalbajoo, M. K.; Aggarwal, V. K., Numerical solution of singular boundary value problems via Chebyshev polynomial and B-spline, Appl. Math. Comput., 160, 851-863 (2005) · Zbl 1062.65077
[2] Kanth, A. S.V. R.; Reddy, Y. N., Higher order finite difference method for a class of singular boundary value problems, Appl. Math. Comput., 155, 249-258 (2004) · Zbl 1058.65078
[3] Kanth, A. S.V. R.; Reddy, Y. N., Cubic spline for a class of singular boundary value problems, Appl. Math. Comput., 170, 733-740 (2005) · Zbl 1103.65086
[4] Kelevedjiev, P., Existence of positive solutions to a singular second order boundary value problem, Nonlinear Anal., 50, 1107-1118 (2002) · Zbl 1014.34013
[5] Li, C.; Cui, M., The exact solution for solving a class nonlinear operator equations in the reproducing kernel space, Appl. Math. Comput., 143, 2-3, 393-399 (2003) · Zbl 1034.47030
[6] Liu, Y.; Yu, H., Existence and uniqueness of positive solution for singular boundary value problem, Comput. Math. Appl., 50, 133-143 (2005) · Zbl 1094.34015
[7] Mohanty, R. K.; Sachder, P. L.; Jha, N., An \(O(h^4)\) accurate cubic spline TAGE method for nonlinear singular two point boundary value problem, Appl. Math. Comput., 158, 853-868 (2004) · Zbl 1060.65080
[8] Wong, F.; Lian, W., Positive solution for singular boundary value problems, Comput. Math. Appl., 32, 9, 41-49 (1996) · Zbl 0868.34019
[9] Xu, X.; Ma, J., A note on singular nonlinear boundary value problems, J. Math. Anal. Appl., 293, 108-124 (2004) · Zbl 1057.34007
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