@Article{AAMM-17-1789,
author = {Cao , Yanhua and Luo , Song},
title = {Two Novel Space-Time Polynomial Particular Solutions Methods for the Time-Dependent Schrödinger Equation},
journal = {Advances in Applied Mathematics and Mechanics},
year = {2025},
volume = {17},
number = {6},
pages = {1789--1812},
abstract = {<p style="text-align: justify;">The conventional method of polynomial particular solutions is only applicable to partial differential equations on the real number field. Building on this
 method, this paper proposes two novel approaches for numerical simulation of the
 time-dependent Schrödinger equation. Under the assumption of treating the time variable as an ordinary spatial variable, the first approach approximates the real and imaginary parts of the equation using two different sets of linear combinations, which are
 then substituted into the corresponding original governing equations to form a coupled differential equation. The second approach is derived based on the basic form
 of complex coefficient partial differential equations and involves polynomial particular solutions with imaginary terms. Using the same numerical examples, compared to
 conventional finite difference method, Fourier spectral method and radial basis function collocation method, this algorithm&#39;s stability is not limited by the grid ratio of
 the time step and spatial step. It is not only simple and feasible but also suitable for
 solving high-dimensional problems.</p>},
issn = {2075-1354},
doi = {https://doi.org/10.4208/aamm.OA-2023-0217},
url = {http://global-sci.org/intro/article_detail/aamm/24495.html}
}