×

Iterative reproducing kernel method for nonlinear oscillator with discontinuity. (English) Zbl 1202.34074

Summary: The iterative reproducing kernel method is applied to obtain the analytical approximate solution of a nonlinear oscillator with discontinuities. The solution obtained by using the method takes the form of a convergent series with easily computable components. An illustrative example is given to demonstrate the effectiveness of the present method. The results obtained using the scheme presented here show that the numerical scheme is very effective and convenient for the nonlinear oscillator with discontinuities.

MSC:

34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34A36 Discontinuous ordinary differential equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Beléndez, A.; Hernández, A.; Beléndez, T.; Neipp, C.; Márquez, A., Higher accuracy analytical approximations to a nonlinear oscillator with discontinuity by He’s homotopy perturbation method, Physics Letters A, 372, 2010-2016 (2008) · Zbl 1220.70022
[2] Beléndez, A.; Pascual, C.; Ortuño, M.; Beléndez, T.; Gallego, S., Application of a modified He’s homotopy perturbation method to obtain higher-order approximations to a nonlinear oscillator with discontinuities, Nonlinear Analysis. Real World Applications, 10, 1, 416-427 (2009) · Zbl 1167.34327
[3] Beléndez, A.; Pascual, C.; Beléndez, T.; Hernández, A., Solution of an anti-symmetric quadratic nonlinear oscillator by a modified He’s homotopy perturbation method, Nonlinear Analysis. Real World Applications, 10, 416-427 (2009) · Zbl 1154.65349
[4] Wu, B. S.; Sun, W. P.; Lim, C. W., An analytical approximate technique for a class of strongly non-linear oscillators, International Journal of Non-Linear Mechanics, 41, 766-774 (2006) · Zbl 1160.70340
[5] Geng, F. Z., A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems, Applied Mathematics and Computation, 213, 163-169 (2009) · Zbl 1166.65358
[6] Geng, F. Z.; Cui, M. G.; Zhang, B., Method for solving nonlinear initial value problems by combining homotopy perturbation and reproducing kernel Hilbert space methods, Nonlinear Analysis. Real World Applications, 11, 637-644 (2010) · Zbl 1187.34012
[7] 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
[8] 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
[9] Cui, M. G.; Lin, Y. Z., Nonlinear Numerical Analysis in Reproducing Kernel Space (2009), Nova Science Pub. Inc. · Zbl 1165.65300
[10] Berlinet, A.; Thomas-Agnan, Christine, Reproducing Kernel Hilbert Space in Probability and Statistics (2004), Kluwer Academic Publishers · Zbl 1145.62002
[11] Daniel, A., Reproducing Kernel Spaces and Applications (2003), Springer · Zbl 1021.00005
[12] 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
[13] 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
[14] Du, J.; Cui, M. G., Constructive approximation of solution for fourth-order nonlinear boundary value problems, Mathematical Methods in the Applied Sciences, 32, 723-737 (2009) · Zbl 1170.34015
[15] Yao, H. M.; Lin, Y. Z., Solving singular boundary-value problems of higher even-order, Journal of Computational and Applied Mathematics, 223, 703-713 (2009) · Zbl 1181.65108
[16] Xie, S. S.; Heo, S.; Kim, S.; Woo, G.; Yi, S., Numerical solution of one-dimensional Burgers’ equation using reproducing kernel function, Journal of Computational and Applied Mathematics, 214, 417-434 (2008) · Zbl 1140.65069
[17] 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.