Finding closed form solutions of differential equations
Sprache des Vortragstitels:
Englisch
Original Tagungtitel:
ECCAD 2013
Sprache des Tagungstitel:
Englisch
Original Kurzfassung:
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
Sprache der Kurzfassung:
Englisch
Vortragstyp:
Hauptvortrag / Eingeladener Vortrag auf einer Tagung
Forschungs-