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.
