利用谱投影与泰勒展开全局求解隐式半线性微分方程的两类组合方法
Two combined methods for the global solution of implicit semilinear differential equations with the use of spectral projectors and Taylor expansions
AI总结：
本文提出两类基于谱投影、泰勒展开和有限差分的组合数值方法，用于全局求解隐式半线性微分代数方程，并证明其收敛性及在较弱非线性限制下的有效性。
摘要：
本文获得了两类用于求解半线性微分代数方程（DAEs）的组合数值方法，并证明了它们的收敛性。对这两类方法进行了比较分析，并就其在各种情形下的应用有效性得出了结论。与其他已知方法相比，所获得的方法对DAE的非线性部分要求更弱的限制条件。此外，所获得的方法能够在任意给定的时间区间上计算DAEs的近似解，因此能够对由DAEs描述的数学模型进行全局动力学数值分析。文中提供了展示所开发方法能力的示例。为了构造这些方法，我们使用了谱投影、泰勒展开和有限差分。由于所使用的谱投影易于计算，因此应用这些方法时无需进行额外的解析变换。
英文摘要：
Two combined numerical methods for solving semilinear differential-algebraic equations (DAEs) are obtained and their convergence is proved. The comparative analysis of these methods is carried out and conclusions about the effectiveness of their application in various situations are made. In comparison with other known methods, the obtained methods require weaker restrictions for the nonlinear part of the DAE. Also, the obtained methods enable to compute approximate solutions of the DAEs on any given time interval and, therefore, enable to carry out the numerical analysis of global dynamics of mathematical models described by the DAEs. The examples demonstrating the capabilities of the developed methods are provided. To construct the methods we use the spectral projectors, Taylor expansions and finite differences. Since the used spectral projectors can be easily computed, to apply the methods it is not necessary to carry out additional analytical transformations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
