## 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 Numerical solution of boundary value problems involving ordinary differential equations 34B16 Singular nonlinear boundary value problems for ordinary differential equations 34B10 Nonlocal and multipoint 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) 45J05 Integro-ordinary differential equations 65R20 Numerical methods for integral equations
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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 [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 [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 [4] Ravi Kanth, A. S.V.; Aruna, K., Solution of singular two-point boundary value problems using differential transformation method, Physics Letters A, 372, 4671-4673 (2008) · Zbl 1221.34060 [5] Ravi Kanth, A. S.V.; 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 [6] Ravi Kanth, A. S.V.; Reddy, Y. N., Cubic spline for a class of singular boundary value problems, Applied Mathematics and Computation, 170, 733-740 (2005) · Zbl 1103.65086 [7] Ravi Kanth, A. S.V.; Bhattacharya, Vishnu, Cubic spline for a class of non-linear singular boundary value problems arising in physiology, Applied Mathematics and Computation, 189, 2, 2017-2022 (2007) · Zbl 1122.65376 [8] Ravi Kanth, A. S.V., Cubic spline polynomial for non-linear singular two-point boundary value problems, Applied Mathematics and Computation, 174, 1, 768-774 (2006) · Zbl 1089.65075 [9] Mohanty, R. K.; Sachder, P. L.; Jha, N., An $$\text{O}(h^4)$$ accurate cubic spline TAGE method for nonlinear singular two point boundary value problem, Applied Mathematics and Computation, 158, 853-868 (2004) · Zbl 1060.65080 [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 [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 [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, 3, 77-87 (2008) [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 [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 [15] Lepin, A. Ya.; Ponomarev, V. D., On a positive solution of a three-point boundary value problem, Differential Equations, 42, 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 [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 [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 [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 [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 [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 [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 [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 [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 [25] Daniel, A., Reproducing Kernel Spaces and Applications (2003), Springer · Zbl 1021.00005 [26] Berlinet, A.; Thomas-Agnan, Christine, Reproducing Kernel Hilbert Space in Probability and Statistics (2004), Kluwer Academic Publishers · 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 [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 [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, 4, 561-572 (2007) · Zbl 1125.35418 [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 [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, 2-3, 393-399 (2003) · Zbl 1034.47030
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