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Application of reproducing kernel method to third order three-point boundary value problems. (English) Zbl 1204.65090
Summary: We investigate the analytical approximate solutions of third order three-point boundary value problems using the reproducing kernel method. The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel method can not be used directly to solve third order three-point boundary value problems, since there is no method of obtaining reproducing kernel satisfying three-point boundary conditions.
This paper presents a method for solving reproducing kernel satisfying three-point boundary conditions so that reproducing kernel method can be used to solve third order three-point boundary value problems. Results of numerical examples 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
34B05 Linear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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[1] 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
[2] 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
[3] 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
[4] 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
[5] Sun, Y.P., Positive solutions for third-order three-point nonhomogeneous boundary value problems, Applied mathematics letters, 22, 1, 45-51, (2009) · Zbl 1163.34313
[6] Sun, Y.P., Existence of triple positive solutions for a third-order three-point boundary value problem, Journal of computational and applied mathematics, 221, 1, 194-201, (2008) · Zbl 1157.34311
[7] Moorti, V.R.G.; Garner, J.B., Existence and uniqueness theorems for three-point boundary value problems for third order differential equations, Journal of mathematical analysis and applications, 70, 2, 370-385, (1979) · Zbl 0418.34025
[8] Anderson, D.R.; Smyrlis, G., Solvability for a third-order three-point BVP on time scales, Mathematical and computer modelling, (2009) · Zbl 1171.34306
[9] Cui, Minggen; Geng, Fazhan, Solving singular two-point boundary value problem in reproducing kernel space, Journal of computational and applied mathematics, 205, 6-15, (2007) · Zbl 1149.65057
[10] Geng, Fazhan; Cui, Minggen, 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
[11] Geng, Fazhan; Cui, Minggen, Solving singular nonlinear two-point boundary value problems in the reproducing kernel space, Journal of the Korean mathematical society, 45, 3, 77-87, (2008) · Zbl 1154.34012
[12] Geng, Fazhan; Cui, Minggen, Solving a nonlinear system of second order boundary value problems, Journal of mathematical analysis and applications, 327, 1167-1181, (2007) · Zbl 1113.34009
[13] Cui, Minggen; Geng, Fazhan, A computational method for solving one-dimensional variable-coefficient Burgers equation, Applied mathematics and computation, 188, 1389-1401, (2007) · Zbl 1118.35348
[14] Cui, Minggen; Lin, Yingzhen, 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
[15] Cui, Minggen; Chen, Zhong, The exact solution of nonlinear age-structured population model, Nonlinear analysis: real world applications, 8, 1096-1112, (2007) · Zbl 1124.35030
[16] 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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