TY - UNPB
T1 - Rational and Algebraic Solutions of First-Order Algebraic ODEs (Thesis Dissertation)
AU - Vo Ngoc, Thieu
PY - 2016
Y1 - 2016
N2 - Main aim: study new algorithms for determining polynomial,rational and algebraic solutions of first-order algebraic ordinary differential equations (AODEs).There is a bunch of solution methods for specific classes of such ODEs.But no decision algorithm for general first-order AODEs, even for seeking specific kinds of solutions such as polynomial,rational or algebraic functions.Our interests are solutions in: algebraic general,rational general,particular rational,polynomial.We approach first-order AODEs from several aspects.By considering the derivative as a new indeterminate,a first-order AODE can be viewed as a hypersurface over the ground field (tools from algebraic geometry are applicable).In particular,we use birational transformations of algebraic hypersurfaces to transform the differential equation to another one for which we hope that it is easier to solve.This geometric approach leads us to a procedure for determining an algebraic general solution of a parametrizable first-order AODE.A general solution contains an arbitrary constant.For the problem of determining a rational general solution in which the constant appears rationally,we propose a decision algorithm for the general class of first-order AODEs.Geometric method is not applicable for studying particular rational solutions. Instead: we study such solutions from combinatorial+algebraic aspects.In combinatorial consideration,poles of coefficients of the differential equation are important in the estimation of candidates for poles of a rational solution and their multiplicities.An algebraic method based on algebraic function field theory is proposed to globally estimate the degree of a rational solution.A combination of these methods leads us to an algorithm for determining all rational solutions for a generic class of first-order AODEs,which covers every first-order AODEs from Kamke's collection.For polynomial solutions the algorithm works for the general class of first-order AODEs.
AB - Main aim: study new algorithms for determining polynomial,rational and algebraic solutions of first-order algebraic ordinary differential equations (AODEs).There is a bunch of solution methods for specific classes of such ODEs.But no decision algorithm for general first-order AODEs, even for seeking specific kinds of solutions such as polynomial,rational or algebraic functions.Our interests are solutions in: algebraic general,rational general,particular rational,polynomial.We approach first-order AODEs from several aspects.By considering the derivative as a new indeterminate,a first-order AODE can be viewed as a hypersurface over the ground field (tools from algebraic geometry are applicable).In particular,we use birational transformations of algebraic hypersurfaces to transform the differential equation to another one for which we hope that it is easier to solve.This geometric approach leads us to a procedure for determining an algebraic general solution of a parametrizable first-order AODE.A general solution contains an arbitrary constant.For the problem of determining a rational general solution in which the constant appears rationally,we propose a decision algorithm for the general class of first-order AODEs.Geometric method is not applicable for studying particular rational solutions. Instead: we study such solutions from combinatorial+algebraic aspects.In combinatorial consideration,poles of coefficients of the differential equation are important in the estimation of candidates for poles of a rational solution and their multiplicities.An algebraic method based on algebraic function field theory is proposed to globally estimate the degree of a rational solution.A combination of these methods leads us to an algorithm for determining all rational solutions for a generic class of first-order AODEs,which covers every first-order AODEs from Kamke's collection.For polynomial solutions the algorithm works for the general class of first-order AODEs.
M3 - Preprint
T3 - RISC Report Series
BT - Rational and Algebraic Solutions of First-Order Algebraic ODEs (Thesis Dissertation)
PB - RISC, JKU
CY - Hagenberg, Linz
ER -