Wavelet-Galerkin quasilinearization method for nonlinear boundary value problems. (English) Zbl 1474.65257

Summary: A numerical method is proposed by wavelet-Galerkin and quasilinearization approach for nonlinear boundary value problems. Quasilinearization technique is applied to linearize the nonlinear differential equation and then wavelet-Galerkin method is implemented to linearized differential equations. In each iteration of quasilinearization technique, solution is updated by wavelet-Galerkin method. In order to demonstrate the applicability of proposed method, we consider the various nonlinear boundary value problems.


65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65T60 Numerical methods for wavelets
Full Text: DOI


[1] Mathews, J. H.; Fink, K. D., Numerical Methods Using MATLAB (1999), New York, NY, USA: Prentice Hall, New York, NY, USA
[2] Beylkin, G., On the representation of operators in bases of compactly supported wavelets, SIAM Journal on Numerical Analysis, 29, 6, 1716-1740 (1992) · Zbl 0766.65007
[3] Latto, A.; Resnikoff, H. L.; Tenenbaum, E., The evaluation of connection coefficients of compactly supported wavelets, Proceedings of the French-USA Workshop on Wavelets and Turbulence, Princeton University, June 1991, New York, NY, USA: Springer, New York, NY, USA
[4] Chen, M.; Hwang, C.; Shih, Y., The computation of wavelet-Galerkin approximation on a bounded interval, International Journal for Numerical Methods in Engineering, 39, 17, 2921-2944 (1996) · Zbl 0884.76058
[5] Restrepo, J. M.; Leaf, G. K., Inner product computations using periodized Daubechies wavelets, International Journal for Numerical Methods in Engineering, 40, 19, 3557-3578 (1997) · Zbl 0906.65138
[6] Amaratunga, K.; Williams, J. R.; Qian, S.; Weiss, J., Wavelet-Galerkin solutions for one dimensional partial differential equations, IESL Technical Report, 9205 (1992), Intelligent Engineering Systems Laboratory, M. I. T .
[7] Mishra, V.; Sabina, Wavelet Galerkin solutions of ordinary differential equations, International Journal of Mathematical Analysis, 5, 9, 407-424 (2011) · Zbl 1236.65084
[8] Jianhua, S.; Xuming, Y.; Biquan, Y.; Yuantong, S., Wavelet-Galerkin solutions for differential equations, Wuhan University. Journal of Natural Sciences, 3, 4, 403-406 (1998) · Zbl 0931.65082
[9] Xu, J.; Shann, W. C., Galerkin-Wavelet methods for two-point boundary value problems, Numerische Mathematik, 63, 1, 123-144 (1992) · Zbl 0771.65050
[10] Qian, S.; Weiss, J., Wavelets and the numerical solution of boundary value problems, Applied Mathematics Letters, 6, 1, 47-52 (1993) · Zbl 0769.65083
[11] Qian, S.; Weiss, J., Wavelets and the numerical solution of partial differential equations, Journal of Computational Physics, 106, 1, 155-175 (1993) · Zbl 0771.65072
[12] El-Gamel, M., Comparison of the solutions obtained by Adomian decomposition and wavelet-Galerkin methods of boundary-value problems, Applied Mathematics and Computation, 186, 1, 652-664 (2007) · Zbl 1117.65136
[13] Scheider, A. K., Implementation of Wavelet Galerkin method for boundary value problems [M.S. thesis] (1998), New York, NY, USA: Rochester Institute of Technology, New York, NY, USA
[14] Kalaba, R., On nonlinear differential equations, the maximum operation, and monotone convergence, Journal of Mathematics and Mechanics, 8, 519-574 (1959) · Zbl 0092.07703
[15] Bellman, R. E.; Kalaba, R. E., Quasilinearization and Nonlinear Boundary-Value Problems (1965), New York, NY, USA: American Elsevier Publishing, New York, NY, USA · Zbl 0139.10702
[16] Conte, S. D.; de Boor, C., Elementary Numerical Analysis (1981), McGraw-Hill International Editions
[17] Deeba, E.; Khuri, S. A.; Xie, S., An algorithm for solving boundary value problems, Journal of Computational Physics, 159, 2, 125-138 (2000) · Zbl 0959.65091
[18] Mohyud-din, S. T., Solution of Troesch’s problem using He’s polynomials, Revista de la Unión Matemática Argentina, 52, 1, 143-148 (2011) · Zbl 1368.65127
[19] Jiwari, R., A Haar wavelet quasilinearization approach for numerical simulation of Burgers’ equation, Computer Physics Communications, 183, 11, 2413-2423 (2012) · Zbl 1302.35337
[20] Kaur, H.; Mittal, R. C.; Mishra, V., Haar wavelet quasilinearization approach for solving nonlinear boundary value problems, The American Journal of Computational Mathematics, 1, 176-182 (2011)
[21] Saeed, U.; Rehman, M., Haar wavelet-quasilinearization technique for fractional nonlinear differential equations, Applied Mathematics and Computation, 220, 630-648 (2013) · Zbl 1329.65173
[22] Daubechies, I., Orthonormal bases of compactly supported wavelets, Communications on Pure and Applied Mathematics, 41, 7, 909-996 (1988) · Zbl 0644.42026
[23] Daubechies, I., Ten Lectures on Wavelets. Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics (1992), Philadelphia, Pa, USA: SIAM, Philadelphia, Pa, USA · Zbl 0776.42018
[24] Soman, K. P.; Ramachandran, K. I., Insight into Wavelets from Theory to Practice (2005), New Delhi, India: PHI Learning Pvt., New Delhi, India
[25] Dianfeng, L. U.; Ohyoshi, T.; ZHU, L., Treatment of Boundary Condition in the Application of wavelet-Galerkin Method to a SH Wave Problem (1996), Akita, Japan: Akita University, Akita, Japan
[26] Lee, E. S., Quasilinearization and Invariant Imbedding (1968), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0212.17602
[27] Mandelzweig, V. B.; Tabakin, F., Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs, Computer Physics Communications, 141, 2, 268-281 (2001) · Zbl 0991.65065
[28] Erturk, V. S.; Momani, S., Differential transform method for obtaining positive solutions for two-point nonlinear boundary value problems, International Journal Mathematical Manuscripts, 1, 65-72 (2007)
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