Numerical solutions of partial differential equations by discrete homotopy analysis method |
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Authors: | Hongqing Zhu Huazhong Shu |
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Affiliation: | a Department of Electronics and Communications Engineering, East China University of Science and Technology, Shanghai 200237, China b School of Computer Science and Engineering, Southeast University, Nanjing 210096, China |
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Abstract: | 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 ?, 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. |
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Keywords: | Discrete homotopy analysis method Crank-Nicolson Burgers&rsquo equations Finite difference scheme Diffusion equation Convergence region |
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