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出现在数学、科学和工程的各种问题中。本课题的主题是cis方程的研究，cis方程是解之间具有收敛整数幂级数的线性微分方程。研究者做了一个意想不到的观察：所有小于4阶的cis方程似乎都可以用代数和超几何函数来解。这一观察结果令人惊讶，因为cis方程在许多研究领域都很常见。目前，寻找这样的解决方案非常耗时。计算机代数系统如Maple和Mathematica经常找不到这些封闭形式的解，导致用户得出封闭形式解很少的错误结论。这个项目的目标是开发一种算法，可以求解每一个阶数小于4的cis方程。为了做到这一点，研究者将建立在之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、贝尔伊图和设计的技术。对社会的好处是多方面的，但却是间接的；计算机算法不能建造桥梁，但它们在设计桥梁、研究海浪、光纤、量子力学、人口动力学等方面都很有用。微分方程的应用清单是漫长而多样的。该项目的目标是开发一类微分方程的完整求解器，这些微分方程在不同的研究领域（如组合学或物理学中的伊辛模型）中非常常见。当所提出的算法被开发出来时，在这些领域工作的研究人员将受益匪浅，并将节省大量时间。

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.