Title Solving differential equations in terms of special fonctions. Version française


BP 93, 06902 France


Manuel Bronstein


Kovacic's algorithm [1] is able to solve the equation y''(x) = r(x) y(x) in Liouvillian closed form (quadratures) whenever such solutions exist. That algorithm, which has become classical, is implemented in most computer algebra systems. The goal of this internship is to develop and experiment with new methods for computing the solutions of that equation in terms of special functions (for example Bessel or Airy functions) whenever they do not have Liouvillian solutions. Starting from a recent method [2] that can compute some solutions of that type, one will try to determine, in the case of the Airy functions Ai and Bi, their potential arguments f(x). This will be done through solving the differential equation that such an f(x) must satisfy. In a first step, the algorithm used to bound the poles of rational solutions of Riccati equations will be generalized to the equation for f(x). Then, a prototype using those bounds will be implemented in order to find basis of solutions of the form g(x) {Ai,Bi}(f(x)).
It is possible to continue this work towards a doctoral thesis around that topic (other special functions or equations of higher order).

[1] J.Kovacic, An Algorithm for Solving Second Order Linear Homogeneous Differential Equations, Journal of Symbolic Computation 2, 3-43 (1986).
[2] B.Willis, An Extensible Differential Equation Solver, à paraitre dans le SIGSAM Bulletin.


Unix workstation, either one of Axiom, Maple or Mathematica, or Aldor programming language.


3 - 4 months, possible prolongation as a doctoral thesis if desired.


February 1, 2001