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Title: Solving second-order singularly perturbed ODE by the collocation method based on energetic Robin boundary functions (English)
Author: Liu, Chein-Shan
Author: Li, Botong
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 64
Issue: 6
Year: 2019
Pages: 679-693
Summary lang: English
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Category: math
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Summary: For a second-order singularly perturbed ordinary differential equation (ODE) under the Robin type boundary conditions, we develop an energetic Robin boundary functions method (ERBFM) to find the solution, which automatically satisfies the Robin boundary conditions. For the non-singular ODE the Robin boundary functions consist of polynomials, while the normalized exponential trial functions are used for the singularly perturbed ODE. The ERBFM is also designed to preserve the energy, which can quickly find accurate numerical solutions for the highly singularly perturbed problems by a simple collocation technique. (English)
Keyword: singularly perturbed ODE
Keyword: Robin boundary function
Keyword: energetic Robin boundary function
Keyword: collocation method
MSC: 34B60
idZBL: 07144733
idMR: MR4042433
DOI: 10.21136/AM.2019.0066-19
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Date available: 2019-12-09T11:48:45Z
Last updated: 2020-07-02
Stable URL: http://hdl.handle.net/10338.dmlcz/147928
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Reference: [1] Ascher, U. M., Mattheij, R. M. M., Russell, R. D.: Numerical Solution of Boundary Value Problems for Ordinary Differential Equations.Classics in Applied Mathematics 13, SIAM, Society for Industrial and Applied Mathematics, Philadelphia (1995). Zbl 0843.65054, MR 1351005, 10.1137/1.9781611971231
Reference: [2] Awoke, A., Reddy, Y. N.: An exponentially fitted special second-order finite difference method for solving singular perturbation problems.Appl. Math. Comput. 190 (2007), 1767-1782. Zbl 1122.65377, MR 2339770, 10.1016/j.amc.2007.02.051
Reference: [3] Bender, C. M., Orszag, S. A.: Advanced Mathematical Methods for Scientists and Engineers.International Series in Pure and Applied Mathematics, McGraw-Hill Book, New York (1978). Zbl 0417.34001, MR 0538168
Reference: [4] Cash, J. R.: Numerical integration of nonlinear two-point boundary value problems using iterated deferred corrections. I. A survey and comparison of some one-step formulae.Comput. Math. Appl., Part A 12 (1986), 1029-1048. Zbl 0618.65071, MR 0862027, 10.1016/0898-1221(86)90009-X
Reference: [5] Cash, J. R.: On the numerical integration of nonlinear two-point boundary value problems using iterated deferred corrections. II. The development and analysis of highly stable deferred correction formulae.SIAM J. Numer. Anal. 25 (1988), 862-882. Zbl 0658.65070, MR 0954789, 10.1137/0725049
Reference: [6] Cash, J. R., Wright, R. W.: Continuous extensions of deferred correction schemes for the numerical solution of nonlinear two-point boundary value problems.Appl. Numer. Math. 28 (1998), 227-244. Zbl 0926.65078, MR 1655162, 10.1016/S0168-9274(98)00045-2
Reference: [7] Doğan, N., Ertürk, V. S., Akı{n}, Ö.: Numerical treatment of singularly perturbed two-point boundary value problems by using differential transformation method.Discrete Dyn. Nat. Soc. 2012 (2012), Article ID 579431, 10 pages. Zbl 1244.65119, MR 2914044, 10.1155/2012/579431
Reference: [8] Ilicasu, F. O., Schultz, D. H.: High-order finite-difference techniques for linear singular perturbation boundary value problems.Comput. Math. Appl. 47 (2004), 391-417. Zbl 1168.76343, MR 2048192, 10.1016/S0898-1221(04)90033-8
Reference: [9] Kadalbajoo, M. K., Aggarwal, V. K.: Fitted mesh B-spline collocation method for solving self-adjoint singularly perturbed boundary value problems.Appl. Math. Comput. 161 (2005), 973-987. Zbl 1073.65062, MR 2113530, 10.1016/j.amc.2003.12.078
Reference: [10] Keller, H. B.: Numerical Methods for Two-Point Boundary Value Problems.Blaisdell Publishing Company, Waltham (1968). Zbl 0172.19503, MR 0230476
Reference: [11] Khuri, S. A., Sayfy, A.: Self-adjoint singularly perturbed boundary value problems: an adaptive variational approach.Math. Methods Appl. Sci. 36 (2013), 1070-1079. Zbl 1290.65064, MR 3066728, 10.1002/mma.2664
Reference: [12] Lin, T.-C., Schultz, D. H., Zhang, W.: Numerical solutions of linear and nonlinear singular perturbation problems.Comput. Math. Appl. 55 (2008), 2574-2592. Zbl 1142.65306, MR 2416027, 10.1016/j.camwa.2007.09.011
Reference: [13] Liu, C.-S.: Efficient shooting methods for the second-order ordinary differential equations.CMES, Comput. Model. Eng. Sci. 15 (2006), 69-86. Zbl 1152.65453, MR 2265974
Reference: [14] Liu, C.-S.: The Lie-group shooting method for singularly perturbed two-point boundary value problems.CMES, Comput. Model. Eng. Sci. 15 (2006), 179-196. Zbl 1152.65452, MR 2269392
Reference: [15] Liu, C.-S.: An equilibrated method of fundamental solutions to choose the best source points for the Laplace equation.Eng. Anal. Bound. Elem. 36 (2012), 1235-1245. Zbl 1352.65637, MR 2913110, 10.1016/j.enganabound.2012.03.001
Reference: [16] Liu, C.-S.: The Lie-group shooting method for solving nonlinear singularly perturbed boundary value problems.Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 1506-1521. Zbl 1244.65113, MR 2855443, 10.1016/j.cnsns.2011.09.029
Reference: [17] Liu, C.-S., Li, B.: Reconstructing a second-order Sturm-Liouville operator by an energetic boundary function iterative method.Appl. Math. Lett. 73 (2017), 49-55. Zbl 1375.65100, MR 3659907, 10.1016/j.aml.2017.04.023
Reference: [18] Liu, C.-S., Liu, D., Jhao, W.-S.: Solving a singular beam equation by using a weak-form integral equation method.Appl. Math. Lett. 64 (2017), 51-58. Zbl 1388.74062, MR 3564739, 10.1016/j.aml.2016.08.010
Reference: [19] Patidar, K. C.: High order parameter uniform numerical method for singular perturbation problems.Appl. Math. Comput. 188 (2007), 720-733. Zbl 1119.65070, MR 2327398, 10.1016/j.amc.2006.10.040
Reference: [20] Reddy, Y. N., Chakravarthy, P. Pramod: An initial-value approach for solving singularly perturbed two-point boundary value problems.Appl. Math. Comput. 155 (2004), 95-110. Zbl 1058.65079, MR 2078097, 10.1016/S0096-3003(03)00763-X
Reference: [21] Varner, T. N., Choudhury, S. R.: Non-standard difference schemes for singular perturbation problems revisited.Appl. Math. Comput. 92 (1998), 101-123. Zbl 0942.65081, MR 1625458, 10.1016/S0096-3003(97)10025-X
Reference: [22] Vigo-Aguiar, J., Natesan, S.: An efficient numerical method for singular perturbation problems.J. Comput. Appl. Math. 192 (2006), 132-141. Zbl 1095.65068, MR 2226996, 10.1016/j.cam.2005.04.042
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