The Algebro-Geometric Method for Solving Algebraic Differential Equations — A Survey
This paper presents the algebro-geometric method for computing explicit formula solutions for algebraic differential equations (ADEs). An algebraic differential equation is a polynomial relation between a function, some of its partial derivatives, and the variables in which the function is defined. Regarding all these quantities as unrelated variables, the polynomial relation leads to an algebraic relation defining a hypersurface on which the solution is to be found. A solution in a certain class of functions, such as rational or algebraic functions, determines a parametrization of the hypersurface in this class. So in the algebro-geometric method the author first decides whether a given ADE can be parametrized with functions from a given class; and in the second step the author tries to transform a parametrization into one respecting also the differential conditions. This approach is relatively well understood for rational and algebraic solutions of single algebraic ordinary differential equations (AODEs). First steps are taken in a generalization to systems and to partial differential equations.
KeywordsAlgebraic differential equation exact solution parametrization of curves
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- Ritt J F, Differential Algebra, American Mathematical Society, 1950.Google Scholar
- Magid A R, Lectures on Differential Galois Theory, American Mathematical Society, 1997.Google Scholar
- Behloul D and Cheng S S, Computation of rational solutions for a first-order nonlinear differential equation, Electronic Journal of Differential Equations (EJDE), 2011, 1–16.Google Scholar
- Fuchs L, Über Differentialgleichungen, deren Integrale feste Verzweigungspunkte besitzen, Sitzungsberichte der Königlich Preuβischen Akademie der Wissenschaften zu Berlin, 1884, 11(3): 251–273.Google Scholar
- Hubert E, The general solution of an ordinary differential equation, Proc. Internat. Symposium on Symbolic and Algebraic Computation (ISSAC 1996), Ed. by Lakshman Y N, ACM Press, New York, 1996, 189–195.Google Scholar
- Aroca J M, Cano J, Feng R, et al., Algebraic General Solutions of Algebraic Ordinary Differential Equations, Proceedings of the 30th International Symposium on Symbolic and Algebraic Computation (ISSAC 05), Ed. by Kauers M, 29–36, ACM Press, New York, 2005.Google Scholar
- Grasegger G and Winkler F, Symbolic solutions of first order algebraic ODEs, Computer Algebra and Polynomials, Eds. by Gutierrez J, Schicho J, and Weimann M, LNCS 8942: 94–104, Springer Switzerland, 2015.Google Scholar