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Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method. (English) Zbl 1178.65085
Summary: This paper investigates the numerical solutions of singular second order three-point boundary value problems using reproducing kernel Hilbert space method. It is a relatively new analytical technique. The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel Hilbert space method cannot be used directly to solve a singular second order three-point boundary value problem, so we convert it into an equivalent integro-differential equation, which can be solved using reproducing kernel Hilbert space method. Four numerical examples are given to demonstrate the efficiency of the present method. The numerical results demonstrate that the method is quite accurate and efficient for singular second order three-point boundary value problems.

##### MSC:
 65L10 Boundary value problems for ODE (numerical methods) 34B16 Singular nonlinear boundary value problems for ODE 34B10 Nonlocal and multipoint boundary value problems for ODE 46E22 Hilbert spaces with reproducing kernels 45J05 Integro-ordinary differential equations 65R20 Integral equations (numerical methods)
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##### References:
 [1] Agarwal, R. P.; O’regan, D.; Rachunkova, I.; Stanek, S.: Two-point higher-order BVPs with singularities in phase variables, Computer and mathematics with applications 46, 1799-1826 (2003) · Zbl 1057.34005 · doi:10.1016/S0898-1221(03)90238-0 [2] Agarwal, R. P.; Stanek, S.: Nonnegative solutions of singular boundary value problems with sigh changing nonlinearities, Computer and mathematics with applications 46, 1827-1837 (2003) · Zbl 1156.34310 · doi:10.1016/S0898-1221(03)90239-2 [3] Kadalbajoo, M. K.; Aggarwal, V. K.: Numerical solution of singular boundary value problems via Chebyshev polynomial and B-spline, Applied mathematics and computation 160, 851-863 (2005) · Zbl 1062.65077 · doi:10.1016/j.amc.2003.12.004 [4] Kanth, A. S. V. Ravi; Aruna, K.: Solution of singular two-point boundary value problems using differential transformation method, Physics letters A 372, 4671-4673 (2008) · Zbl 1221.34060 · doi:10.1016/j.physleta.2008.05.019 [5] Kanth, A. S. V. Ravi; Reddy, Y. N.: Higher order finite difference method for a class of singular boundary value problems, Applied mathematics and computation 155, 249-258 (2004) · Zbl 1058.65078 · doi:10.1016/S0096-3003(03)00774-4 [6] Kanth, A. S. V. Ravi; Reddy, Y. N.: Cubic spline for a class of singular boundary value problems, Applied mathematics and computation 170, 733-740 (2005) · Zbl 1103.65086 · doi:10.1016/j.amc.2004.12.049 [7] Kanth, A. S. V. Ravi; Bhattacharya, Vishnu: Cubic spline for a class of non-linear singular boundary value problems arising in physiology, Applied mathematics and computation 189, No. 2, 2017-2022 (2007) · Zbl 1122.65376 [8] Kanth, A. S. V. Ravi: Cubic spline polynomial for non-linear singular two-point boundary value problems, Applied mathematics and computation 174, No. 1, 768-774 (2006) · Zbl 1089.65075 [9] Mohanty, R. K.; Sachder, P. L.; Jha, N.: An $O(h4)$ accurate cubic spline TAGE method for nonlinear singular two point boundary value problem, Applied mathematics and computation 158, 853-868 (2004) · Zbl 1060.65080 · doi:10.1016/j.amc.2003.08.145 [10] Cui, M. G.; Geng, F. Z.: Solving singular two-point boundary value problem in reproducing kernel space, Journal of computational and applied mathematics 205, 6-15 (2007) · Zbl 1149.65057 · doi:10.1016/j.cam.2006.04.037 [11] Geng, F. Z.; Cui, M. G.: Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space, Applied mathematics and computation 192, 389-398 (2007) · Zbl 1193.34017 · doi:10.1016/j.amc.2007.03.016 [12] Geng, F. Z.; Cui, M. G.: Solving singular nonlinear two-point boundary value problems in the reproducing kernel space, Journal of the korean mathematical society 45, No. 3, 77-87 (2008) · Zbl 1154.34012 · doi:10.4134/JKMS.2008.45.3.631 [13] Agarwal, R. P.; Thompson, H. B.; Tisdell, C. C.: Three-point boundary value problems for second-order discrete equations, Computer and mathematics with applications 45, 1429-1435 (2003) · Zbl 1055.39024 · doi:10.1016/S0898-1221(03)00098-1 [14] Thompson, H. B.; Tisdell, C.: Three-point boundary value problems for second-order, ordinary, differential equation, Mathematical and computer modelling 34, 311-318 (2001) · Zbl 0998.34011 · doi:10.1016/S0895-7177(01)00063-2 [15] Lepin, A. Ya.; Ponomarev, V. D.: On a positive solution of a three-point boundary value problem, Differential equations 42, No. 2, 291-293 (2006) · Zbl 1294.34021 [16] Zhang, Z. G.; Liu, L. S.; Wu, C. X.: Nontrival solution of third-order nonlinear eigenvalue problems, Applied mathematics and computation 176, 714-721 (2006) · Zbl 1107.34015 · doi:10.1016/j.amc.2005.10.017 [17] Pei, M. H.; Chang, S. K.: A quasilinearization method for second-order four-point boundary value problems, Applied mathematics and computation 202, 54-66 (2008) · Zbl 1161.65059 · doi:10.1016/j.amc.2008.01.026 [18] Li, X. F.: Multiple positive solutions for some four-point boundary value problems with p-Laplacian, Applied mathematics and computation 202, 413-426 (2008) · Zbl 1157.34015 · doi:10.1016/j.amc.2008.02.006 [19] Agarwal, R. P.; Kiguradze, I.: On multi-point boundary value problems for linear ordinary differential equations with singularities, Journal of mathematical analysis and applications 297, 131-151 (2004) · Zbl 1058.34012 · doi:10.1016/j.jmaa.2004.05.002 [20] Zhang, Z. X.; Wang, J. Y.: The upper and lower solution method for a class of singular nonlinear second order three-point boundary value problems, Journal of computational and applied mathematics 147, 41-52 (2002) · Zbl 1019.34021 · doi:10.1016/S0377-0427(02)00390-4 [21] Ma, R. Y.; O’regan, D.: Solvability of singular second order m-point boundary value problems, Journal of mathematical analysis and applications 301, 124-134 (2005) · Zbl 1062.34018 · doi:10.1016/j.jmaa.2004.07.009 [22] Zhang, Q. M.; Jiang, D. Q.: Upper and lower solutions method and a second order three-point singular boundary value problems, Computer and mathematics with applications 56, 1059-1070 (2008) · Zbl 1155.34305 · doi:10.1016/j.camwa.2008.01.033 [23] Du, X. S.; Zhao, Z. Q.: Existence and uniqueness of positive solutions to a class of singular m-point boundary value problems, Applied mathematics and computation 198, 487-493 (2008) · Zbl 1158.34315 · doi:10.1016/j.amc.2007.08.080 [24] Liu, B. M.; Liu, L. S.; Wu, Y. H.: Positive solutions for a singular second-order three-point boundary value problem, Applied mathematics and computation 196, 532-541 (2008) · Zbl 1138.34015 · doi:10.1016/j.amc.2007.06.013 [25] Daniel, A.: Reproducing kernel spaces and applications, (2003) · Zbl 1021.00005 [26] Berlinet, A.; Thomas-Agnan, Christine: Reproducing kernel Hilbert space in probability and statistics, (2004) · Zbl 1145.62002 [27] Geng, F. Z.; Cui, M. G.: Solving a nonlinear system of second order boundary value problems, Journal of mathematical analysis and applications 327, 1167-1181 (2007) · Zbl 1113.34009 · doi:10.1016/j.jmaa.2006.05.011 [28] Cui, M. G.; Geng, F. Z.: A computational method for solving one-dimensional variable-coefficient Burgers equation, Applied mathematics and computation 188, 1389-1401 (2007) · Zbl 1118.35348 · doi:10.1016/j.amc.2006.11.005 [29] Cui, M. G.; Lin, Y. Z.: A new method of solving the coefficient inverse problem of differential equation, Science in China series A 50, No. 4, 561-572 (2007) · Zbl 1125.35418 · doi:10.1007/s11425-007-0013-8 [30] Cui, M. G.; Chen, Z.: The exact solution of nonlinear age-structured population model, Nonlinear analysis: real world applications 8, 1096-1112 (2007) · Zbl 1124.35030 · doi:10.1016/j.nonrwa.2006.06.004 [31] Li, C. L.; Cui, M. G.: The exact solution for solving a class nonlinear operator equations in the reproducing kernel space, Applied mathematics and computation 143, No. 2-3, 393-399 (2003) · Zbl 1034.47030 · doi:10.1016/S0096-3003(02)00370-3