Formal Algorithmic Elimination for PDEs
Formal Algorithmic Elimination for PDEs
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DOI：
10.1145/2930889.2930941
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发表时间：
2014-10
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影响因子：
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通讯作者：
D. Robertz
中科院分区：
文献类型：
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作者：
D. Robertz
类似于代数几何中多项式环的根理想与簇之间的对应关系，微分代数中的根微分理想与其解析解集之间的对应关系已经建立。本教程讨论了涉及符号计算的这种对应关系的各个方面。特别是，介绍了托马斯分解方法。它将一个多项式非线性偏微分方程组分解成许多所谓的简单微分系统，这些系统的解集形成原始解集的一个分区。每个简单系统的幂级数解都可以直接求出。相反地，某些解析函数的集合可以用偏微分方程和不等式来进行隐式描述。解决相关的微分消元问题的策略和应用程序的微分方程的符号求解。托马斯分解方法的Maple实现是免费提供的。
Similarly to the correspondence between radical ideals of a polynomial ring and varieties in algebraic geometry, a correspondence between radical differential ideals and their analytic solution sets has been established in differential algebra. This tutorial discusses aspects of this correspondence involving symbolic computation. In particular, an introduction to the Thomas decomposition method is given. It splits a system of polynomially nonlinear partial differential equations into finitely many so-called simple differential systems whose solution sets form a partition of the original solution set. The power series solutions of each simple system can be determined in a straightforward way. Conversely, certain sets of analytic functions admit an implicit description in terms of partial differential equations and inequations. Strategies for solving related differential elimination problems and applications to symbolic solving of differential equations are presented. A Maple implementation of the Thomas decomposition method is freely available.