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、科学和工程中的各种问题中。本计画的主题是CIS-方程的研究，CIS-方程是一个解间有收敛幂级数的线性微分方程。 研究者有了一个意想不到的发现：所有阶数小于4的CIS方程似乎都可以用代数和超几何函数来求解。这个观察结果令人惊讶，因为CIS方程在许多研究领域都很常见，目前，找到这样的解是很耗时的。计算机代数系统，如Maple和Mathematica往往无法找到这些封闭形式的解决方案，导致用户错误的结论，封闭形式的解决方案是罕见的。 这个项目的目标是开发算法，将解决每一个CIS方程的阶数小于4。 为了做到这一点，研究人员将建立在以前的工作与他的研究生，并将使用技术，从模曲线，数论，贝利地图和dessins d 'enfants。社会的好处是多方面的，但间接的;计算机算法不建桥，但他们是有用的桥梁设计，研究海浪，光纤，量子力学，人口动力学等，微分方程的应用范围很广。 这个项目的目标是为一类微分方程开发一个完整的求解器，这类微分方程在不同的研究领域中非常常见，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受益匪浅，并将节省大量的时间时，所提出的算法已经开发。

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.