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The use of sinc-collocation method for solving multi-point boundary value problems. (English) Zbl 1244.65114
Summary: Multi-point boundary value problems have received considerable interest in the mathematical applications in different areas of science and engineering. In this work, our goal is to obtain numerically the approximate solution of these problems by using the sinc-collocation method. Some properties of the sinc-collocation method required for our subsequent development are given and are utilized to reduce the computation of solution of multi-point boundary value problems to some algebraic equations. It is well known that the sinc procedure converges to the solution at an exponential rate. Numerical examples are included to demonstrate the validity and applicability of the new technique.

##### MSC:
 65L10 Numerical solution of boundary value problems involving ordinary differential equations 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations
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##### References:
 [1] Stenger, F., Numerical methods based on sinc and analytic functions, (1993), Springer-Verlag New York · Zbl 0803.65141 [2] Lund, J.; Bowers, K., Sinc methods for quadrature and differential equations, (1992), SIAM PA, Philadelphia. · Zbl 0753.65081 [3] Winter, D.F.; Bowers, K.; Lund, J., Wind-driven currents in a sea with a variable eddy viscosity calculated via a sinc – galerkin technique, Int J numer methods fluids, 33, 1041-1073, (2000) · Zbl 0984.76066 [4] Bialecki, B., Sinc-collocation methods for two-point boundary value problems, IMA J numer anal, 11, 357-375, (1991) · Zbl 0735.65052 [5] Parand, K.; Pirkhedri, A., Sinc-collocation method for solving astrophysics equations, New astron, 15, 533-537, (2010) [6] Parand, K.; Dehghan, M.; Pirkhedri, A., Sinc-collocation method for solving the Blasius equation, Phys lett A, 373, 4060-4065, (2009) · Zbl 1234.76014 [7] Parand, K.; Delafkar, Z.; Pakniat, N.; Pirkhedri, A.; Kazemnasab Haji, M., Collocation method using sinc and rational Legendre functions for solving volterra’s population model, Commun nonlinear sci numer simulat, 16, 1811-1819, (2011) · Zbl 1221.65186 [8] Saadatmandi, A.; Razzaghi, M.; Dehghan, M., Sinc-collocation methods for the solution of hallen’s integral equation, J electromagan waves appl, 19, 2, 245-256, (2005) [9] Saadatmandi, A.; Razzaghi, M., The numerical solution of third-order boundary value problems using sinc-collocation method, Commun numer meth eng, 23, 681-689, (2007) · Zbl 1121.65088 [10] Dehghan, M.; Saadatmandi, A., The numerical solution of a nonlinear system of second-order boundary value problems using the sinc-collocation method, Math comput model, 46, 1434-1441, (2007) · Zbl 1133.65050 [11] El-Gamel, M.; Behiry, S.H.; Hashish, H., Numerical method for the solution of special nonlinear fourth – order boundary value problems, Appl math comput, 145, 717-734, (2003) · Zbl 1033.65065 [12] Dinh Alain, P.N.; Quan, P.H.; Trong, D.D., Sinc approximation of the heat distribution on the boundary of a two-dimensional finite slab, Nonlinear anal: real world appl, 9, 1103-1111, (2008) · Zbl 1146.35376 [13] Abdella, K.; Yu, X.; Kucuk, I., Application of the sinc method to a dynamic elasto-plastic problem, J comput appl math, 223, 626-645, (2009) · Zbl 1153.74051 [14] Lund, J.; Vogel, C.R., A fully-Galerkin method for the numerical solution of an inverse problem in a parabolic partial differential equation, Inverse probl, 6, 205-217, (1990) · Zbl 0709.65104 [15] Shidfar, A.; Zolfaghari, R.; Damirchi, J., Application of sinc-collocation method for solving an inverse problem, J comput appl math, 223, 545-554, (2009) · Zbl 1180.65118 [16] Rashidinia, J.; Zarebnia, M., The numerical solution of integro-differential equation, Appl math comput, 188, 1124-1130, (2007) · Zbl 1118.65131 [17] Mohsen, A.; El-Gamel, M., On the numerical solution of linear and nonlinear Volterra integral and integro-differential equations, Appl math comput, 217, 3330-3337, (2010) · Zbl 1204.65158 [18] El-Gamel, M.; Zayed, A., Sinc – galerkin method for solving nonlinear boundary-value problems, Comput math appl, 48, 1285-1298, (2004) · Zbl 1072.65111 [19] Revelli, R.; Ridolfi, L., Sinc collocation-interpolation method for the simulation of nonlinear waves, Comput math appl, 46, 1443-1453, (2003) · Zbl 1049.65107 [20] Mohsen, A.; El-Gamel, M., A sinc-collocation method for the linear Fredholm integro-differential equations, Z angew mat phys, 58, 380-390, (2007) · Zbl 1116.65131 [21] Mohsen, A.; El-Gamel, M., On the Galerkin and collocation methods for two-point boundary value problems using sinc bases, Comput math appl, 56, 930-941, (2008) · Zbl 1155.65365 [22] Muhammad, M.; Nurmuhammad, A.; Mori, M.; Sugihara, M., Numerical solution of integral equations by means of the sinc-collocation method based on the double exponential transformation, J comput appl math, 177, 269-286, (2005) · Zbl 1072.65168 [23] Sababheh, M.S.; Al-khaled, A.M.N., Some convergence results on sinc interpolation, J inequal pure appl math, 4, 32-48, (2003) · Zbl 1069.41005 [24] Hajji MA. Multi-point special boundary-value problems and applications to fluid flow through porous media, In: Proceedings of the International Multi-Conference of Engineers and Computer Scientists 2009, vol II, Hong Kong. [25] Il’in, V.A.; Moiseev, E.I., Nonlocal boundary value problem of the first kind for a sturm – liouville operator in its differential and finite difference aspects, Differ equ, 23, 7, 803-810, (1987) · Zbl 0668.34025 [26] Il’in, V.A.; Moiseev, E.I., Nonlocal boundary value problem of the second kind for a sturm – liouville operator, Differ equ, 23, 8, 979-987, (1987) · Zbl 0668.34024 [27] Gupta, C.P., Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation, J math anal appl, 168, 540-551, (1992) · Zbl 0763.34009 [28] Ma, R., A survey on nonlocal boundary value problems, Appl math E-notes, 7, 257-279, (2007) · Zbl 1163.34300 [29] Moshinsky, M., Sobre los problems de condiciones a la frontiera en una dimension de carac-teristicas discontinuas, Bol soc mat mexicana, 7, 1-25, (1950) [30] Timoshenko, S., Theory of elastic stability, (1961), McGraw-Hill New York [31] Geng, F.; Cui, M., Multi-point boundary value problem for optimal bridge design, Int J comput math, 87, 1051-1056, (2010) · Zbl 1192.65109 [32] Bai, C.; Fang, J., Existence of multiple positive solutions for nonlinear m-point boundary value problems, J math anal appl, 281, 76-85, (2003) · Zbl 1030.34026 [33] Lentini, M.; Pereyra, V., A variable order finite difference method for nonlinear multi-point boundary value problems, Math comput, 28, 981-1003, (1974) · Zbl 0308.65054 [34] Dehghan, M., Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math comput simulat, 71, 16-30, (2006) · Zbl 1089.65085 [35] Zou, Y.; Hu, Q.; Zhang, R., On numerical studies of multi-point boundary value problems and its fold bifurcation, Appl math comput, 185, 527-537, (2007) · Zbl 1112.65069 [36] Tatari, M.; Dehghan, M., The use of the Adomian decomposition method for solving multi-point boundary value problems, Phys scripta, 73, 672-676, (2006) [37] Dehghan M, Shakeri F. A semi-numerical technique for solving the multi-point boundary value problems and engineering applications. Int J Numer Methods Heat Fluid Flow, in press [38] Ali, J.; Islam, S.; Zaman, G., The solution of multi-point boundary value problems by the optimal homotopy asymptotic method, Comput math appl, 59, 2000-2006, (2010) · Zbl 1189.65154 [39] Tatari M, Dehghan M, An efficient method for solving multi-point boundary value problems and applications in physics, J Vibr Contr, in press, [40] Haque, M.; Baluch, M.H.; Mohsen, M.F.N., Solution of multiple point, nonlinear boundary value problems by method of weighted residuals, Int J comput math, 19, 69-84, (1986) · Zbl 0653.65059
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