Non-liouvillian solutions for second order Linear ODEs

ABSTRACT

There exist sound literature and algorithms for computing Liouvillian solutions for the important problem of linear ODEs with rational coefficients. Taking as sample the 363 second order equations of that type found in Kamke's book, for instance, 51% of them admit Liouvillian solutions and so are solvable using Kovacic's algorithm. On the other hand, special function solutions not admitting Liouvillian form appear frequently in mathematical physics, but there are not so general algorithms for computing them. In this paper we present an algorithm for computing special function solutions which can be expressed using the ;_2;F;_1;, ;_1;F;_1; or ;_0;F;_1; hypergeometric functions. The algorithm is easy to implement in the framework of a computer algebra system and systematically solves 91% of the 363 Kamke's linear ODE examples mentioned.

REFERENCES

• 1: James B. Seaborn, Hypergeometric functions and their applications, Springer-Verlag, London, 1991
• 2: Milton Abramowitz, Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,, Dover Publications, Incorporated, 1974
• 3: L. Chan supervised by E. S. Cheb-Terrab. "On Solving second order linear ODEs admitting non-Liouvillian solutions." Report for NSERC University research award, Department of Mathematics, Simon Fraser University (2001).
• 4: Niky Kamran , Peter J. Olver, Equivalence of differential operators, SIAM Journal on Mathematical Analysis, v.20 n.5, p.1172-1185, Sep. 1989 [doi>10.1137/0520077]
• 5: Marko Petkovšek , Bruno Salvy, Finding all hypergeometric solutions of linear differential equations, Proceedings of the 1993 international symposium on Symbolic and algebraic computation, p.27-33, July 06-08, 1993, Kiev, Ukraine [doi>10.1145/164081.164087]
• 6: K. von Bülow, "Equivalence methods for second order linear differential equations", M. Sc. Thesis, Faculty of Mathematics, University of Waterloo (2000).
• 7: Barton L. Willis, An extensible differential equation solver, ACM SIGSAM Bulletin, v.35 n.1, p.3-7, March 2001 [doi>10.1145/504331.504332]
• 8: Manuel Bronstein , Sébastien Lafaille, Solutions of linear ordinary differential equations in terms of special functions, Proceedings of the 2002 international symposium on Symbolic and algebraic computation, p.23-28, July 07-10, 2002, Lille, France [doi>10.1145/780506.780510]
• 9: E. L. Ince, "Ordinary Differential Equations", Dover Publications (1956).
• 10: P. Olver, "Equivalence, Invariants and Symmetry", Cambridge Unversity Press (1995).
• 11: E. W. Weisstein, "Concise Encyclopedia of Mathematics", second edition, CRC Press (1999).
• 12: E. Kamke, "Differentialgleichungen", N.Y. Chelsea Publ. Co. (1947).
• 13: G. Labahn. "Methods for Solving Linear ODEs in Maple." University of Waterloo Tech Report (2000).
• 14: M. van Hoeij, http://web.math.fsu.edu/~hoeij/daisy/lib/DE-tools/src/RiemannPsols (1997).
• 15: E. S. Cheb-Terrab, "ODE trends in computer algebra: four linear and nonlinear challenges", proceedings of the Maple Summer Workshop, Waterloo, Canada (2002).
• 16: S. Y. Slavyanov and W.Lay. "Special Functions, A Unified Theory Based on Singularities", Oxford Mathematical Monographs
• 17: E. S. Cheb-Terrab, "Computing Mathieu function solutions for linear ODEs", http://lie.uwaterloo.ca/odetools/mathieu_func-tion_solutions.html (2003).
• 18: E. S. Cheb-Terrab, D. Frenkel, R. Portugal, "Mathieu equations: series and exact solutions", in preparation (2004).

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION

General Terms:
Algorithms, Design, Theory

Keywords:
hypergeometric solutions, linear ordinary differential equations, non-Liouvillian solutions