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Solving the forced Duffing equation with integral boundary conditions in the reproducing kernel space. (English) Zbl 1213.65106

The authors propose a new method to solve the forced Duffing equation with integral boundary conditions. The idea is to construct the numerical solution by an iterative scheme in the corresponding reproducing kernel space. Then, uniform convergence towards the exact solution is shown. Some examples based on linear and nonlinear second-order equations are provided to demonstrate the efficiency and robustness of the method.

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
42B05 Fourier series and coefficients in several variables
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)

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References:

[1] DOI: 10.1007/s00365-006-0657-0 · Zbl 1159.42309 · doi:10.1007/s00365-006-0657-0
[2] Ahmad B., Comm. Appl. Nonlinear Anal. 4 pp 67– (2007)
[3] DOI: 10.1016/j.nonrwa.2007.05.005 · Zbl 1154.34311 · doi:10.1016/j.nonrwa.2007.05.005
[4] DOI: 10.1006/jdeq.2002.4166 · Zbl 1025.34015 · doi:10.1006/jdeq.2002.4166
[5] Alpay, D. 2003. ”Reproducing Kernel Spaces and Applications”. Springer. · Zbl 1021.00005
[6] DOI: 10.1016/j.jmaa.2007.10.033 · Zbl 1223.47012 · doi:10.1016/j.jmaa.2007.10.033
[7] Benchohra M., Rocky Mountain J. Math.
[8] DOI: 10.1016/S0168-9274(01)00139-8 · Zbl 1003.65102 · doi:10.1016/S0168-9274(01)00139-8
[9] DOI: 10.1016/j.na.2007.12.007 · Zbl 1169.34310 · doi:10.1016/j.na.2007.12.007
[10] Bouziani A., Kobe J. Math. 15 pp 47– (1998)
[11] DOI: 10.1016/j.amc.2003.11.035 · Zbl 1068.65100 · doi:10.1016/j.amc.2003.11.035
[12] DOI: 10.1016/j.physa.2008.03.003 · doi:10.1016/j.physa.2008.03.003
[13] DOI: 10.1016/j.jmaa.2008.03.023 · Zbl 1144.45002 · doi:10.1016/j.jmaa.2008.03.023
[14] DOI: 10.1016/j.nonrwa.2006.06.004 · Zbl 1124.35030 · doi:10.1016/j.nonrwa.2006.06.004
[15] DOI: 10.1016/j.cam.2006.04.037 · Zbl 1149.65057 · doi:10.1016/j.cam.2006.04.037
[16] Ding T., Proc. Amer. Math. Soc. 86 pp 47– (1982) · doi:10.1090/S0002-9939-1982-0663864-1
[17] DOI: 10.1016/0309-1708(91)90055-S · doi:10.1016/0309-1708(91)90055-S
[18] Fasshauer, G. E. 2007. ”Meshfree Approximation Methods with MATLAB. With 1 CD-ROM (Windows, Macintosh and UNIX)”. Hackensack, NJ: World Scientific Publishing Co. Pvt. Ltd. · Zbl 1123.65001
[19] DOI: 10.1007/s007910050030 · Zbl 1067.76624 · doi:10.1007/s007910050030
[20] DOI: 10.1016/j.amc.2007.03.016 · Zbl 1193.34017 · doi:10.1016/j.amc.2007.03.016
[21] Ionkin N. I., Diff. Uravn. 13 pp 294– (1977)
[22] Jiang W., Appl. Math. Comput. 202 pp 667– (2008)
[23] DOI: 10.1016/j.jmaa.2003.09.020 · Zbl 1134.34322 · doi:10.1016/j.jmaa.2003.09.020
[24] Ladde, G. S., Lakshmikantham, V. and Vatsala, A. S. 1985. ”Monotone Iterative Techniques for Nonlinear Differential Equations”. Boston: Pitman. · Zbl 0658.35003
[25] DOI: 10.1016/S0096-3003(02)00370-3 · Zbl 1034.47030 · doi:10.1016/S0096-3003(02)00370-3
[26] DOI: 10.1016/S0362-546X(97)00524-5 · Zbl 0937.34053 · doi:10.1016/S0362-546X(97)00524-5
[27] DOI: 10.1016/j.camwa.2006.01.012 · Zbl 1140.34406 · doi:10.1016/j.camwa.2006.01.012
[28] DOI: 10.1155/S1048953303000054 · Zbl 1035.35085 · doi:10.1155/S1048953303000054
[29] DOI: 10.1137/0524004 · Zbl 0810.35033 · doi:10.1137/0524004
[30] DOI: 10.1006/jmaa.1997.5793 · Zbl 0915.34032 · doi:10.1006/jmaa.1997.5793
[31] DOI: 10.1155/BVP.2005.93 · Zbl 1143.65388 · doi:10.1155/BVP.2005.93
[32] DOI: 10.1006/jmaa.2001.7474 · Zbl 0988.34032 · doi:10.1006/jmaa.2001.7474
[33] DOI: 10.1016/j.amc.2007.04.029 · Zbl 1193.65134 · doi:10.1016/j.amc.2007.04.029
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