Abstract
Galois theory has now produced algorithms for solving linear ordinary differential and difference equations in closed form. In addition, recent algorithmic advances have made those algorithms effective and implementable in computer algebra systems. After introducing the relevant parts of the theory, we describe the latest algorithms for solving such equations.
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References
S. A. Abramov, M. Bronstein, and M. Petkovsek. On polynomial solutions of linear operator equations. In Proceedings of ISSAC’95, pages 290–296. ACM Press, 1995.
R. Bomboy. Solutions Liouvilliennes des équations aux différences finies linéaires. Thèse de mathématiques, Université de Nice, 2001.
D. Boucher. Sur les équations différentielles paramétrées, une application aux systèmes hamiltoniens. Thèse de mathématiques, Université de Limoges, 2000.
M. Bronstein. On solutions of linear ordinary differential equations in their coefficient field. J. Symbolic Computation, 13(4):413–440, April 1992.
M. Bronstein. On solutions of linear ordinary difference equations in their coefficient field. Journal of Symbolic Computation, 29(6):841–877, June 2000.
M. Bronstein and A. Fredet. Solving linear ordinary differential equations over C(x,of t(x)dx). In S. Dooley, editor, Proceedings of ISSAC’99, pages 173–179. ACM Press, 1999.
M. Bronstein, T. Mulders, and J.-A. Weil. On symmetric powers of differential operators. In Proceedings of ISSAC’97, pages 156–163. ACM Press, 1997.
M. Bronstein and M. Petkovsek. An introduction to pseudo-linear algebra. Theoretical Computer Science, 157:3–33, 1996.
F. T. Cope. Formal solutions of irregular linear differential equations II, American Journal of Mathematics, 58:130–140, 1936.
P. A. Hendriks and M. F. Singer. Solving difference equations in finite terms. J. Symbolic Computation, 27(3):239–260, March 1999.
M. van Hoeij. Factorization of differential operators with rational functions coefficients. J. Symbolic Computation, 24(5):537–562, November 1997.
M. Van Hoeij, J.-F. Ragot, F. Ulmer, and J.-A. Weil. Liouvillian solutions of linear differential equations of order three and higher. J. Symbolic Computation, 28(4 and 5):589–610, October/November 1999.
M. van Hoeij and J.-A. Weil. An algorithm for computing invariants of differential galois groups. Journal of Pure and Applied Algebra, 117,118:353–379, 1997.
E. L. Ince. Ordinary Differential Equations. Dover Publications Inc., 1956.
C. Jordan. Mémoire sur les équations différentielles linéaires à intégrale algébrique. Journal für Mathematik, 84:89–215, 1878.
Irving Kaplansky. An introduction to differential algebra. Hermann, Paris, 1957.
E. R. Kolchin. Algebraic matrix groups and the Picard—Vessiot theory of homogeneous linear ordinary differential equations. Annals of Mathematics, 49:1–42, 1948.
E. R. Kolchin. Differential algebra and algebraic groups. Academic Press, New York and London, 1973.
O. Ore. Theory of non-commutative polynomials. Annals of Mathematics, 34:480–508, 1933.
M. Petkovsek. Hypergeometric solutions of linear recurrences with polynomial coefficients. J. Symbolic Computation, 14(2 and 3):243–264, 1992.
M. Petkovsek, H. S. Wilf, and D. Zeilberger. A = B. A. K. Peters, Wellesley, 1996.
M. Put. Galois theory of differential equations, algebraic groups and Lie algebras. J. Symbolic Computation, 28(4 and 5):441–472, 1999.
A. Seidenberg. Abstract differential algebra and the analytic case. Proceedings of the American Mathematical Society, 9:159–164, 1958.
A. Seidenberg. Abstract differential algebra and the analytic case II. Proceedings of the American Mathematical Society, 23:689–691, 1969.
M. F. Singer. Liouvillian solutions of nth order homogeneous linear differential equations. American Journal of Mathematics, 103:661–682, 1981.
M. F. Singer. Liouvillian solution of linear differential equations with Liouvillian coefficients. J. Symbolic Computation, 11(3):251–274, March 1991.
M. F. Singer. Testing reducibility of linear differential operators: a group theoretic perspective. Applicable Algebra in Engineering, Communication and Computing, 7:77–104, 1996.
M. F. Singer and F. Ulmer. Liouvillian and algebraic solutions of second and third order linear differential equations. J. Symbolic Computation, 16(1):37–74, July 1993.
M. F. Singer and F. Ulmer. Linear differential equations and products of linear forms. Journal of Pure and Applied Algebra, 117 Si 118:549–563, 1997.
M. F. Singer and M. Put. Galois Theory of Difference Equations. LNM 1666. Springer, 1997.
F. Ulmer. On Liouvillian solutions of linear differential equations. Applicable Algebra in Engineering, Communication and Computing, 2:171–193, 1992.
F. Ulmer. Irreducible linear differential equations of prime order. J. Symbolic Computation, 18(4):385–401, October 1994.
F. Ulmer and J.-A. Weil. Note on Kovacic’s algorithm. J. Symbolic Computation, 22(2):179–200, August 1996.
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Bronstein, M. (2001). Computer Algebra Algorithms for Linear Ordinary Differential and Difference equations. In: Casacuberta, C., Miró-Roig, R.M., Verdera, J., Xambó-Descamps, S. (eds) European Congress of Mathematics. Progress in Mathematics, vol 202. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8266-8_9
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DOI: https://doi.org/10.1007/978-3-0348-8266-8_9
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