Weyl代数的分数阶微积分与Fuchsian微分方程
Fractional calculus of Weyl algebra and Fuchsian differential equations
摘要：
给出了多项式系数线性常微分方程的合流、邻接关系和Katz中卷积的统一解释，并将其推广到偏微分方程.其解的整表示和级数展开式也在我们的解释范围内。作为对Riemann球面上Fuchsian微分方程的应用，我们构造了具有给定谱型的Fuchsian微分方程的万有模型，特别地，我们构造了对应于刚性局部系统的单个无显奇点的常微分方程，其存在性是Katz提出的一个公开问题.进而得到了刚性Fuchsian微分方程联络问题的显式解及其不可约的充要条件。我们给出了许多用分数阶微积分计算的例子。
英文摘要：
We give a unified interpretation of confluences, contiguity relations and Katz's middle convolutions for linear ordinary differential equations with polynomial coefficients and their generalization to partial differential equations. The integral representations and series expansions of their solutions are also within our interpretation. As an application to Fuchsian differential equations on the Riemann sphere, we construct a universal model of Fuchsian differential equations with a given spectral type, in particular, we construct single ordinary differential equations without apparent singularities corresponding to the rigid local systems, whose existence was an open problem presented by Katz. Furthermore we obtain an explicit solution to the connection problem for the rigid Fuchsian differential equations and the necessary and sufficient condition for their irreducibility. We give many examples calculated by our fractional calculus.
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