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Wavelet solution of a class of two-dimensional nonlinear boundary value problems

Xiaojing Liu1, Jizeng Wang1,2, Youhe Zhou1,2
Key Laboratory of Mechanics on Disaster and Environment in Western China, the Ministry of Education, and School of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, Gansu 730000, China.
Corresponding author. Email: jzwang@lzu.edu.cn (J. Wang); zhouyh@lzu.edu.cn (Y. Zhou).

Computer Modeling in Engineering & Sciences 2013, 92(5), 493-505. https://doi.org/10.3970/cmes.2013.092.493

Abstract

By combining techniques of boundary extension and Coiflet-type wavelet expansion, an approximation scheme for a function defined on a two-dimensional bounded space is proposed. In this wavelet approximation, each expansion coefficient can be directly obtained by a single-point sampling of the function. And the boundary values and derivatives of the bounded function can be embedded in the modified wavelet basis. Based on this approximation scheme, a modified wavelet Galerkin method is developed for solving two-dimensional nonlinear boundary value problems, in which the interpolating property makes the solution of such strong nonlinear problems very effective and accurate. As an example, we have applied the proposed method to the solution of two-dimensional Bratu-like equations. Results demonstrate an excellent numerical accuracy, a 2.5-order convergence rate for the present wavelet method, and a very good capability in dealing with the nonlinear boundary value problems with multiple solution branches. Interestingly, unlike most existing methods, numerical errors of the present solutions are not sensitive to the nonlinear intensity of the two-dimensional equations.

Keywords

modified wavelet Galerkin method, two-dimensional, strong nonlinearity, boundary value problems, Bratu-like equation

Cite This Article

Liu, X., Wang, J., Zhou, Y. (2013). Wavelet solution of a class of two-dimensional nonlinear boundary value problems. CMES-Computer Modeling in Engineering & Sciences, 92(5), 493–505.



This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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