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Journal of Applied Mathematics
Volume 2013, Article ID 248246, 10 pages
Research Article

HAM-Based Adaptive Multiscale Meshless Method for Burgers Equation

College of Information and Electrical Engineering, China Agricultural University, 17 Qinghua Donglu Road, East Campus, Haidian District, Post box 53, Beijing 100083, China

Received 25 January 2013; Revised 2 August 2013; Accepted 2 August 2013

Academic Editor: Hui-Shen Shen

Copyright © 2013 Shu-Li Mei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Based on the multilevel interpolation theory, we constructed a meshless adaptive multiscale interpolation operator (MAMIO) with the radial basis function. Using this operator, any nonlinear partial differential equations such as Burgers equation can be discretized adaptively in physical spaces as a nonlinear matrix ordinary differential equation. In order to obtain the analytical solution of the system of ODEs, the homotopy analysis method (HAM) proposed by Shijun Liao was developed to solve the system of ODEs by combining the precise integration method (PIM) which can be employed to get the analytical solution of linear system of ODEs. The numerical experiences show that HAM is not sensitive to the time step, and so the arithmetic error is mainly derived from the discrete in physical space.