Numerical solutions of partial differential equations by discrete homotopy analysis method

作者：Zhu, Hongqing^*; Shu, Huazhong; Ding, Meiyu

来源：Applied Mathematics and Computation, 2010, 216(12): 3592-3605.

DOI：10.1016/j.amc.2010.05.005

• Discrete homotopy analysis method
• Crank-Nicolson
• Burgers' equations
• Finite difference scheme
• Diffusion equation
• Convergence region

This paper introduces a discrete homotopy analysis method (DHAM) to obtain approximate solutions of linear or nonlinear partial differential equations (PDEs). The DHAM can take the many advantages of the continuous homotopy analysis method. The proposed DHAM also contains the auxiliary parameter (h) over bar, which provides a simple way to adjust and control the convergence region of solution series. The convergence of the DHAM is proved under some reasonable hypotheses, which provide the theoretical basis of the DHAM for solving nonlinear problems. Several examples, including a simple diffusion equation and two-dimensional Burgers' equations, are given to investigate the features of the DHAM. The numerical results obtained by this method have been compared with the exact solutions. It is shown that they are in good agreement with each other.

• 出版日期2010-8-15
• 单位东南大学; 华东理工大学