An Algorithm Computing the Regular Formal Solutions of a System of Linear Differential Equations

Abstract. We consider the following problem: given a linear differential system with formal Laurent series coefficients, we want to decide whether the system has non-zero Laurent series solutions, and find all such solutions if they exist. Let us also assume we need only a given positive integer number l of initial terms of these series solutions. How many initial terms of the coefficients of the original system should we use to construct what we need?Supposing that the series coefficients of the original systems are represented algorithmically, we show that these questions are undecidable in general. However, they are decidable in the scalar case and in the case when we know in advance that a given system has an invertible leading matrix. We use our results in order to improve some functionality of the Maple [17] package ISOLDE [11].

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“…(i) The statements of this proposition follow from the results in [7]. The algorithm presented therein computes regular formal solutions (i.e.…”

Section: Denotes the Ith Row Of The Matrix A(x)mentioning

confidence: 94%

“…If this polynomial I S (λ) is non-zero then we shall say that the system S is simple and refer to the polynomial I S (λ) as the indicial polynomial of S as in [7,Definition 2.1]. By extension, a system of the form (9) will be called simple if the corresponding system (10) is simple.…”

Section: Denotes the Ith Row Of The Matrix A(x)mentioning

confidence: 99%

Higher-Order Linear Differential Systems with Truncated Coefficients

Абрамов¹,

Barkatou²,

Pflügel³

2011

Computer Algebra in Scientific Computing

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“…Moreover, unlike the case of m > 1, algorithms to related problems leading to the construction of the formal solutions and computation of the exponential parts have been developed by various authors (see, e.g., [Barkatou, 1997;Barkatou et al, 2013Barkatou et al, , 1999Barkatou et al, , 2009 and references therein). The package ISOLDE [Barkatou et al, 2013] written in the computer algebra system Maple is dedicated to the symbolic resolution of ODS and more generally linear functional matrix equations.…”

Section: Proposition 1 the Eigenvalues Ofmentioning

confidence: 99%

“…Our first contribution is an explicit method to compute the x i -exponential parts. Alongside their importance in the asymptotic theory, their computation reduces the ultimate task of formal reduction to constructing a basis of the C-space of regular solutions (see, for m = 1, [Barkatou et al, 1999]). Rank reduction as well can complement our work into full formal reduction.…”

Section: Proposition 1 the Eigenvalues Ofmentioning

confidence: 99%

Formal solutions of a class of Pfaffian systems in two variables

Maddah

Barkatou

Abbas

2014

Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation

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In this article, we present an algorithm which computes a fundamental matrix of formal solutions of completely integrable Pfaffian systems with normal crossings in two variables, based on [Barkatou, 1997]. A first step was set in [Barkatou et al., 2006] where the problem of rank reduction was tackled via the approach in [Levelt, 1991]. We give instead a Moser-based approach following [Barkatou et al., 2014;Barkatou, 1995]. And, as a complementary step, we associate to our problem a system of ordinary linear singular differential equations from which the formal invariants can be efficiently derived via the package ISOLDE [Barkatou et al., 2013], implemented in the computer algebra system Maple.

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“…In [28], Singer established the general framework for finding Liouvillian solutions for general linear homogeneous ODEs. Many other interesting results on finding Liouvillian solutions of linear ODEs were reported in [1,4,5,6,7,8,12,22,30,31,33,34]. In [21], Li and Schwarz gave the first method to find rational solutions for a class of partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

Rational general solutions of algebraic ordinary differential equations

Feng

Gao

2004

Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation

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We give a necessary and sufficient condition for an algebraic ODE to have a rational type general solution. For an autonomous first order ODE, we give an algorithm to compute a rational general solution if it exists. The algorithm is based on the relation between rational solutions of the first order ODE and rational parametrizations of the plane algebraic curve defined by the first order ODE and Padé approximants.

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An Algorithm Computing the Regular Formal Solutions of a System of Linear Differential Equations

Abstract: We propose a method for computing the regular singular formal solutions of a linear differential system in the neighbourhood of a singular point. This algorithm avoids the use of cyclic vectors and has been implemented † in the computer algebra system Maple.

Cited by 44 publications

References 15 publications

Higher-Order Linear Differential Systems with Truncated Coefficients

Higher-Order Linear Differential Systems with Truncated Coefficients

Formal solutions of a class of Pfaffian systems in two variables

Rational general solutions of algebraic ordinary differential equations

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