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Symbolic methods for studying linear systems of differential and difference equation

Ali El Hajj 1 
Abstract : This thesis is concerned with the development of symbolic algorithms in Computer algebra. We study systems of pseudo-linear equations: a large class of linear functional systems including differential, difference and q-difference systems. The thesis consists of three major parts. In the first part, we are interested in the local analysis of pseudo-linear system near a singularity. We first develop a direct algorithm for computing simple forms of pseudo-linear systems. Whilst direct algorithms for computing simple forms have been already proposed for differential and difference systems, no unifying one for pseudo-linear systems was known prior to our work. Then we show how the reduction to a simple form can be used to compute efficiently local data for pseudo-linear systems. The second part deals with the computation of closed-form solutions. Firstly, we presenta generic and efficient algorithm for computing rational solutions of first order pseudo-linearsystems. Then we develop two new recursive algorithms for computing rational and hypergeometric solutions of partial pseudo-linear systems with arbitrary number of variables. An important contribution of this thesis arises in the implementation of all the algorithms in Maple as part of our freely available package PseudoLinearSystems. In the last part of the thesis, we provide a demonstration of several procedures contained in the package.
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Ali El Hajj. Symbolic methods for studying linear systems of differential and difference equation. Symbolic Computation [cs.SC]. Université de Limoges, 2021. English. ⟨NNT : 2021LIMO0113⟩. ⟨tel-03626516⟩

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