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Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space. (English) Zbl 1193.34017
Summary: We present a new method for solving a singular nonlinear second-order periodic boundary value problem. Its analytical solution is represented in the form of series in the reproducing kernel space. In the mean time, the $$n$$-term approximation $$u_n(x)$$ to the analytical solution $$u(x)$$ is obtained and is proved to converge to the analytical solution. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective.

##### MSC:
 34A45 Theoretical approximation of solutions to ordinary differential equations 34B16 Singular nonlinear boundary value problems for ordinary differential equations
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##### References:
 [1] Li, Fuyi; Liang, Zhanping, Existence of positive periodic solutions to nonlinear second-order differential equations, Applied mathematics letters, 18, 1256-1264, (2005) · Zbl 1088.34038 [2] Merdivenci Atici, F.; Guseinov, G.sh., On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions, Journal of computational and applied mathematics, 132, 341-356, (2001) · Zbl 0993.34022 [3] Baslandze, S.R.; Kiguradze, I.T., On the unique solvability of a periodic boundary value problems for third-order linear differential equations, Differential equations, 42, 2, 165-171, (2006) · Zbl 1132.34015 [4] Nieto, J.J., Nonlinear second-order periodic boundary value problems, Journal of mathematical analysis and applications, 130, 22-29, (1988) · Zbl 0678.34022 [5] Seda, V.; Nieto, J.J.; Gera, M., Periodic boundary value problem for nonlinear higher ordinary differential equations, Applied mathematics and computation, 48, 71-82, (1992) · Zbl 0748.34014 [6] Agarwal, R.P., On the periodic solutions of nonlinear second-order differential systems, Journal of computational and applied mathematics, 5, 117-123, (1979) · Zbl 0407.34021 [7] Agarwal, R.P.; Filippakis, M.E.; O’Regan, D.; Papageorgiou, N.S., Degree theoretic methods in the study of nonlinear periodic problems with nonsmooth potentials, Differential and integral equations, 19, 279-296, (2006) · Zbl 1212.34036 [8] Wang, J.Y.; Jiang, D.Q., A singular nonlinear second-order periodic boundary value problems, Tohohu mathematics journal, 50, 203-210, (1998) · Zbl 0916.34027 [9] Rachunková, Irena, Existence of two positive solutions of a singular nonlinear periodic boundary value problem, Journal of computational and applied mathematics, 113, 27-34, (2000) · Zbl 0944.34014 [10] Jiang, Daqing; Chu, Jifeng; o’Regan, Donal; Agarwal, Ravi P., Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces, Journal of mathematical analysis and applications, 286, 563-576, (2003) · Zbl 1042.34047 [11] Zhang, Zhongxin; Wang, Junyu, On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second-order differential equations, Journal of mathematical analysis and applications, 281, 99-107, (2003) · Zbl 1030.34024 [12] Li, Chunli; Cui, Minggen, The exact solution for solving a class nonlinear operator equations in the reproducing kernel space, Applied mathematics and computation, 143, 2-3, 393-399, (2003) · Zbl 1034.47030
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