Finding closed form solutions of differential equations

Description

We consider ordinary linear differential equations with polynomial coefficients. Each such equation has a finite dimensional vector space of solutions, but usually none of these solutions can be expressed in closed form. We discuss the problem of finding out for a given specific differential equation whether one (or some, or all) of its solutions admit a closed form representation. After recalling the classical algorithms for finding polynomial and rational solutions, we turn to hyperexponential solutions. These are solutions that can be written in the form $\exp(u(x))v_1(x)^{e_1}\cdots v_k(x)^{e_k}$ for certain rational functions $u,v_1,\dots,v_k$ and constants $e_1,\dots,e_k$. The first algorithm for finding such solutions was proposed by Beke at the end of the 19th century. His algorithm is very costly. A more efficient algorithm was given at the end of the 20th century by Mark van Hoeij. We will sketch the basic ideas of these two algorithms and then present a new algorithm based on effective analytic continuation, which was recently found by the speaker in joint work with F. Johansson and M. Mezzarobba

Period	27 Apr 2013
Event title	ECCAD 2013
Event type	Conference
Location	United StatesShow on map