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An iterative HAM approach for nonlinear boundary value problems in a semi-infinite domain
Authors:Yinlong Zhao  Zhiliang Lin  Shijun Liao
Affiliation:1. Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China;2. State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China;3. School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China;4. Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, Saudi Arabia
Abstract:In this paper, we propose an iterative approach to increase the computation efficiency of the homotopy analysis method (HAM), a analytic technique for highly nonlinear problems. By means of the Schmidt–Gram process (Arfken et al., 1985)  [15], we approximate the right-hand side terms of high-order linear sub-equations by a finite set of orthonormal bases. Based on this truncation technique, we introduce the MMth-order iterative HAM by using each MMth-order approximation as a new initial guess. It is found that the iterative HAM is much more efficient than the standard HAM without truncation, as illustrated by three nonlinear differential equations defined in an infinite domain as examples. This work might greatly improve the computational efficiency of the HAM and also the Mathematica package BVPh for nonlinear BVPs.
Keywords:Homotopy analysis method   Truncation technique   Iteration technique   Orthonormal functions   Approximate analytical solutions
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