@Article{AAMM-7-6,
author = {Sha, Jie and Zhang, Lixiang and Wu, Chuijie},
title = {A Novel Low-Dimensional Method for Analytically Solving Partial Differential Equations},
journal = {Advances in Applied Mathematics and Mechanics},
year = {2015},
volume = {7},
number = {6},
pages = {754--779},
abstract = {<p style="text-align: justify;">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.</p>},
issn = {2075-1354},
doi = {https://doi.org/10.4208/aamm.2014.m671},
url = {https://global-sci.com/article/73421/a-novel-low-dimensional-method-for-analytically-solving-partial-differential-equations}
}