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论文类型： 期刊论文

发表时间： 2015-12-01

发表刊物： ADVANCES IN APPLIED MATHEMATICS AND MECHANICS

收录刊物： SCIE、Scopus

卷号： 7

期号： 6

页面范围： 754-779

ISSN号： 2070-0733

关键字： Low-dimensional system model; partial differential equation; analytical solution; posterior optimal truncated method

摘要： This paper is concerned with a low-dimensional dynamical system model for analytically solving partial differential equations (PDEs). The model proposed is based on a posterior optimal truncated weighted residue (POT-WR) method, by which an infinite dimensional PDE is optimally truncated and analytically solved in required condition of accuracy. To end that, a POT-WR condition for PDE under consideration is used as a dynamically optimal control criterion with the solving process. A set of bases needs to be constructed without any reference database in order to establish a space to describe low-dimensional dynamical system that is required. The Lagrangian multiplier is introduced to release the constraints due to the Galerkin projection, and a penalty function is also employed to remove the orthogonal constraints. According to the extreme principle, a set of ordinary differential equations is thus obtained by taking the variational operation of the generalized optimal function. A conjugate gradient algorithm by FORTRAN code is developed to solve the ordinary differential equations. The two examples of one-dimensional heat transfer equation and nonlinear Burgers' equation show that the analytical results on the method proposed are good agreement with the numerical simulations and analytical solutions in references, and the dominant characteristics of the dynamics are well captured in case of few bases used only.