Results 1  10
of
44
An algorithm for solving second order linear homogeneous differential equations
 J. Symbolic Comput
, 1986
"... The Galois group tells us a lot about a linear homogeneous differential equation specifically whether or not it has “closedform” solutions. Using it, we have been able to develop an algorithm for finding “closedform ” solutions. First we will compute the Galois group of some very simple equatio ..."
Abstract

Cited by 138 (1 self)
 Add to MetaCart
(Show Context)
The Galois group tells us a lot about a linear homogeneous differential equation specifically whether or not it has “closedform” solutions. Using it, we have been able to develop an algorithm for finding “closedform ” solutions. First we will compute the Galois group of some very simple equations. We then will solve a more complicated one, using the techniques of the algorithm. This example illustrates how the algorithm was discovered and the kinds of calculations used by it. 1
Galoisian Obstructions to integrability of Hamiltonian Systems
, 2001
"... An inconvenience of all the known galoisian formulations of Ziglin’s nonintegrability theory is the Fuchsian condition at the singular points of the variational equations. We avoid this restriction. Moreover we prove that a necessary condition for meromorphic complete integrability (in Liouville se ..."
Abstract

Cited by 67 (11 self)
 Add to MetaCart
An inconvenience of all the known galoisian formulations of Ziglin’s nonintegrability theory is the Fuchsian condition at the singular points of the variational equations. We avoid this restriction. Moreover we prove that a necessary condition for meromorphic complete integrability (in Liouville sense) is that the identity component of the Galois group of the variational equation (in the complex domain) must be abelian. We test the efficacy of these new approaches on some examples. We will give some non academic applications in two following papers.
On Solutions Of Linear Ordinary Differential Equations In Their Coefficient Field
 Journal of Symbolic Computation
, 1991
"... this paper, we consider the following specific subproblem in this area: given a differential field k, g 2 k, and a linear ordinary differential operator L with coefficients in k, can we decide in a finite number of steps whether L(y) = g has a solution in k, and in the affirmative, can we find one ( ..."
Abstract

Cited by 44 (6 self)
 Add to MetaCart
this paper, we consider the following specific subproblem in this area: given a differential field k, g 2 k, and a linear ordinary differential operator L with coefficients in k, can we decide in a finite number of steps whether L(y) = g has a solution in k, and in the affirmative, can we find one (or all) such solution(s)?
Liouvillian solutions of linear differential equations of order three and higher
 JOURNAL OF SYMBOLIC COMPUTATION
, 1999
"... Singer and Ulmer (1997) gave an algorithm to compute Liouvillian (“closedform”) solutions of homogeneous linear differential equations. However, there were several efficiency problems that made computations often not practical. In this paper we address these problems. We extend the algorithm in van ..."
Abstract

Cited by 32 (10 self)
 Add to MetaCart
Singer and Ulmer (1997) gave an algorithm to compute Liouvillian (“closedform”) solutions of homogeneous linear differential equations. However, there were several efficiency problems that made computations often not practical. In this paper we address these problems. We extend the algorithm in van Hoeij and Weil (1997) to compute semiinvariants and a theorem in Singer and Ulmer (1997) in such a way that, by computing one semiinvariant that factors into linear forms, one gets all coefficients of the minimal polynomial of an algebraic solution of the Riccati equation, instead of only one coefficient. These coefficients come “for free ” as a byproduct of our algorithm for computing semiinvariants. We specifically detail the algorithm in the cases of equations of order three (order two equations are handled by the algorithm of Kovacic (1986), see also Ulmer and Weil (1996) or Fakler (1997)). In the appendix, we present several methods to decide when a multivariate polynomial depending on parameters can admit linear factors, which is a necessary ingredient in the algorithm.
On Symmetric Powers of Differential Operators
 Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (ISSAC’97
, 1997
"... We present alternative algorithms for computing symmetric powers of linear ordinary differential operators. Our algorithms are applicable to operators with coefficients in arbitrary integral domains and become faster than the traditional methods for symmetric powers of sufficiently large order, or o ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
(Show Context)
We present alternative algorithms for computing symmetric powers of linear ordinary differential operators. Our algorithms are applicable to operators with coefficients in arbitrary integral domains and become faster than the traditional methods for symmetric powers of sufficiently large order, or over sufficiently complicated coefficient domains. The basic ideas are also applicable to other computations involving cyclic vector techniques, such as exterior powers of differential or difference operators. Introduction Let R be an integral domain of characteristic 0, D be a derivation on R, K be the quotient field of R and R[@; D] be the corresponding ring of linear differential operators with coefficients in R. Let L = P n i=0 a i @ i 2 R[@; D] with n ? 0 and an 6= 0, Y0 ; : : : ; Yn\Gamma1 be indeterminates, and consider the extension of D to K[Y0 ; : : : ; Yn\Gamma1 ] given by DY i = Y i+1 for 0 i ! n \Gamma 1 and DYn\Gamma1 = \Gammaa \Gamma1 n P n\Gamma1 i=0 a i Y i . For an...
Solving second order linear differential equations with Klein’s theorem
 In ISSAC’05
, 2005
"... Given a second order linear differential equations with coefficients in a field k = C(x), the Kovacic algorithm finds all Liouvillian solutions, that is, solutions that one can write in terms of exponentials, logarithms, integration symbols, algebraic extensions, and combinations thereof. A theorem ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
(Show Context)
Given a second order linear differential equations with coefficients in a field k = C(x), the Kovacic algorithm finds all Liouvillian solutions, that is, solutions that one can write in terms of exponentials, logarithms, integration symbols, algebraic extensions, and combinations thereof. A theorem of Klein states that, in the most interesting cases of the Kovacic algorithm (i.e when the projective differential Galois group is finite), the differential equation must be a pullback (a change of variable) of a standard hypergeometric equation. This provides a way to represent solutions of the differential equation in a more compact way than the format provided by the Kovacic algorithm. Formulas to make Klein’s theorem effective were given in [4, 2, 3]. In this paper we will give a simple algorithm based on such formulas. To make the algorithm more easy to implement for various differential fields k, we will give a variation on the earlier formulas, namely we will base the formulas on invariants of the differential Galois group instead of semiinvariants. 1.
Irreducible Linear Differential Equations of Prime Order
 J. Symb. Comp
, 1995
"... this paper we consider linear differential equations of the form ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
this paper we consider linear differential equations of the form