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

具有有理函数系数的线性微分方程式在数学、科学和工程中广泛存在。这个项目的主题是研究独联体方程，它是解中含有收敛的整数幂级数的线性微分方程组。这位研究人员做了一个意想不到的观察：所有低于4阶的CIS方程似乎都是代数函数和超几何函数可解的。这一观察结果令人惊讶，因为逆方程在许多研究领域都很常见。目前，寻找这样的解是耗时的。像Maple和数学这样的计算机代数系统经常找不到这些闭合形式的解，导致用户得出闭合形式解很少见的错误结论。这个项目的目标是开发算法来解决所有低于4阶的独联体方程。为了做到这一点，研究人员将建立在以前与他的研究生所做的工作的基础上，并将使用模曲线、数论、Belyi映射和dessins d‘enfantings中的技术。对社会的好处是多方面的，但间接的；计算机算法不建立桥梁，但它们对设计桥梁、研究海浪、光纤、量子力学、人口动力学等有用。微分方程的应用列表很长，而且种类繁多。这个项目的目标是为一类在不同研究领域中非常常见的微分方程开发一个完整的解算器，例如组合学或物理学中的伊辛模型。在这些领域工作的研究人员将大大受益，并将节省大量时间，当拟议的算法已经开发出来的时候。

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.