SIAM Journal on Applied Mathematics, Volume 80, Issue 2, Page 753-771, January 2020.
In various fields of applied mathematics, e.g., electrostatics, heat conduction, fluid mechanics, elastostatics, etc., boundary-value problems involving regions described in spheroidal, toroidal, and bispherical coordinate systems reduce to a system of second-order difference equations, whose solution, $\{{x}_n\}_{n=0}^\infty$ with ${x}_n\in\mathbb{R}^m$, should vanish asymptotically, i.e., $\lim_{n\to\infty}{x}_n=0$. There are several methods for constructing such $\{{x}_n\}_{n=0}^\infty$. However, in general, those methods do not guarantee $\lim_{n\to\infty}{x}_n=0$. Moreover, in actual computations, they yield an approximate solution $\{\widehat{x}_n\}_{n=0}^N$ different from the truncated true solution $\{{x}_n\}_{n=0}^N$ and coinciding with the solution of the system being truncated at $N$ with ${x}_{N+1}$ set to 0. This work establishes sufficient conditions for the existence of an asymptotically vanishing solution to the system and provides the rate of convergence of the solution to the truncated system. Those results are used to analyze systems of second-order difference equations arising in the boundary-value problems in electrostatics, heat conduction, fluid mechanics, and elastostatics when a medium contains an inhomogeneity having the shape of either a torus or two unequal spheres. In particular, when $m=1$ in those problems, $\|\widehat{x}_n - {x}_n\|$ decays exponentially as $N\to\infty$.

中文翻译：

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应用数学中的矩阵差分方程

SIAM应用数学杂志，第80卷，第2期，第753-771页，2020年1月。
在应用数学的各个领域，例如静电学，热传导，流体力学，弹性静力学等，涉及球面，环面和双球面坐标系中描述的区域的边值问题都简化为二阶差分方程组。解决方案，$ \ {{x} _n \} _ {n = 0} ^ \ infty $与$ {x} _n \ in \ mathbb {R} ^ m $，应渐近消失，即$ \ lim_ {n \到\ infty} {x} _n = 0 $。有几种方法可以构造这样的$ \ {{x} _n \} _ {n = 0} ^ \ infty $。但是，通常，这些方法不能保证$ \ lim_ {n \ to \ infty} {x} _n = 0 $。此外，在实际计算中，它们产生的近似解$ \ {\ widehat {x} _n \} _ {n = 0} ^ N $与截断后的真实解$ \ {{x} _n \} _ {n = 0} ^ N $，并且与将$ {x} _ {N + 1} $设置为0时系统在$ N $处被截断的解一致。这项工作为系统的渐近消失解的存在建立了充分的条件，并为截断系统提供了解的收敛速度。当介质包含圆环或两个不相等球体形状的不均匀性时，这些结果用于分析由静电，导热，流体力学和弹性静力学的边值问题引起的二阶差分方程组。特别是，当$ m = 1 $出现这些问题时，$ \ | \ widehat {x} _n-{x} _n \ | $呈指数衰减，即$ N \ to \ infty $。当介质包含圆环或两个不相等球体形状的不均匀性时，这些结果用于分析由静电，导热，流体力学和弹性静力学的边值问题引起的二阶差分方程组。特别是，当$ m = 1 $出现这些问题时，$ \ | \ widehat {x} _n-{x} _n \ | $呈指数衰减，即$ N \ to \ infty $。当介质包含圆环或两个不相等球体形状的不均匀性时，这些结果用于分析由静电，导热，流体力学和弹性静力学的边值问题引起的二阶差分方程组。特别是，当$ m = 1 $出现这些问题时，$ \ | \ widehat {x} _n-{x} _n \ | $呈指数衰减，即$ N \ to \ infty $。