Solutions of linear ordinary differential equations in terms of special functions

ABSTRACT

We describe a new algorithm for computing special function solutions of the form y(x) = m(x)F(ξ(x)) of second order linear ordinary differential equations, where m(x) is an arbitrary Liouvillian function, ξ(x) is an arbitrary rational function, and F satisfies a given second order linear ordinary differential equation. Our algorithm, which is based on finding an appropriate point transformation between the equation defining F and the one to solve, is able to find all rational transformations for a large class of functions F, in particular (but not only) the 0F1 and 1F1 special functions of mathematical physics, such as Airy, Bessel, Kummer and Whittaker functions. It is also able to identify the values of the parameters entering those special functions, and can be generalized to equations of higher order.

REFERENCES

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INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms
          Subjects: Analysis of algorithms

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.1 Numerical Algorithms and Problems
          Subjects: Computation of transforms (e.g., fast Fourier transform)
  F.3 LOGICS AND MEANINGS OF PROGRAMS
      F.3.3 Studies of Program Constructs
          Subjects: Functional constructs

General Terms:
Algorithms, Measurement