AF:Small: Linear Differential Equations with a Convergent Integer Series Solution

AF:Small：具有收敛整数级数解的线性微分方程

• 批准号: 1319547
• 负责人: Mark van Hoeij
• 金额: $ 47.94万
• 依托单位: Florida State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1319547&HistoricalAwards=false
• 关键词: AF; Small; Linear; Differential; Equations

Linear differential equations with rational function coefficients appear in a wide variety of problems in mathematics, science, and engineering. The topic in this project is the study of CIS-equations, which are linear differential equations with a Convergent Integer power Series among their solutions. The investigator has made an unexpected observation: All CIS-equations of order less than 4 appear to be solvable in terms of algebraic and hypergeometric functions. This observation is surprising because CIS-equations are common in many areas of research.At the moment, finding such solutions is time consuming. Computer algebra systems such as Maple and Mathematica often fail to find these closed form solutions, leading users to the incorrect conclusion that closed form solutions are rare. The goal in this project is to develop algorithms that will solve every CIS-equation of order less than 4. To do this, the investigator will build on prior work done with his graduate students, and will use techniques from modular curves, number theory, Belyi maps and dessins d'enfants.The benefit to society is manifold but indirect; computer algorithms do not build bridges, but they are useful for designing bridges, studying ocean waves, fiber optics, quantum mechanics, population dynamics, etc., the list of applications of differential equations is long and diverse. The goal in this project is to develop a complete solver for a class of differential equations that is very common in diverse fields of study such as combinatorics, or the Ising model in physics. Researchers working in such areas will benefit significantly and will save much time when the proposed algorithms have been developed.

具有合理功能系数的线性微分方程出现在数学，科学和工程方面的各种问题中。该项目的主题是顺式方程的研究，它们是线性微分方程，其解决方案中具有收敛整数功率序列。 研究人员已经意外的观察结果：在代数和超几何函数方面，所有小于4的顺序方程似乎都是可解决的。这一观察结果令人惊讶，因为在许多研究领域中，顺式方程很常见。在此刻，找到这种解决方案很耗时。诸如Maple和Mathematica之类的计算机代数系统常常找不到这些封闭式解决方案，这使用户得出了不正确的结论，即封闭式解决方案很少见。 该项目的目标是开发算法，以解决少于4的订单的每个顺式方程。为此，研究人员将以与他的研究生进行的先前工作为基础，并将使用模块化曲线，数字理论，Belyi Map和Dessins D'Enfants的技术。计算机算法不会建造桥梁，但它们对于设计桥梁，研究海浪，光纤，量子力学，人口动态等有用，差分方程的应用清单较长而多样。 该项目的目的是为一类微分方程的完整求解器开发一个在各种研究领域（例如组合学领域或物理学中的Ising模型）中非常普遍的求解器。 在此类领域工作的研究人员将大大受益，并在开发拟议算法的情况下节省很多时间。