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

**Location**

INRIA, Project
CAFÉ

BP 93, 06902 France

**Information**

**Description**

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.

**Tools**

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

**Duration**

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