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Register a new userFrom formal to actual Puiseux series solutions of algebraic differential equations of first order
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The existence and uniqueness of formal Puiseux series solutions of non-autonomous algebraic differential equations of the first order at a nonsingular point of the equation is proven. The convergence of those Puiseux series is established. Several new examples are provided. Relationships to the celebrated Painleve theorem and lesser-known Petrovics results are discussed in detail.
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Using increasing sequences of real numbers, we generalize the idea of formal moment differentiation first introduced by W. Balser and M. Yoshino. Slight departure from the concept of Gevrey sequences enables us to include a wide variety of operators in our study. Basing our approach on tools such as the Newton polygon and divergent formal norms, we obtain estimates for formal solutions of certain families of generalized linear moment partial differential equations with constant and time variable coefficients.
Generalized summability results are obtained regarding formal solutions of certain families of linear moment integro-differential equations with time variable coefficients. The main result leans on the knowledge of the behavior of the moment derivatives of the elements involved in the problem. A refinement of the main result is also provided giving rise to more accurate results which remain valid in wide families of problems of high interest in practice, such as fractional integro-differential equations.
We explain that in the study of the asymptotic expansion at the origin of a period integral like $gamma$z $omega$/df or of a hermitian period like f =s $rho$.$omega$/df $land$ $omega$ /df the computation of the Bernstein polynomial of the fresco (filtered differential equation) associated to the pair of germs (f, $omega$) gives a better control than the computation of the Bernstein polynomial of the full Brieskorn module of the germ of f at the origin. Moreover, it is easier to compute as it has a better functoriality and smaller degree. We illustrate this in the case where f $in$ C[x 0 ,. .. , x n ] has n + 2 monomials and is not quasi-homogeneous, by giving an explicite simple algorithm to produce a multiple of the Bernstein polynomial when $omega$ is a monomial holomorphic volume form. Several concrete examples are given.
We propose a sufficient condition of the convergence of a Dulac series formally satisfying an algebraic ordinary differential equation (ODE). Such formal solutions of algebraic ODEs appear rather often, in particular, the third, fifth, and sixth Painleve equations possess formal Dulac series solutions, whose convergence follows from the proposed sufficient condition.
We present some distinct asymptotic properties of solutions to Caputo fractional differential equations (FDEs). First, we show that the non-trivial solutions to a FDE can not converge to the fixed points faster than $t^{-alpha}$, where $alpha$ is the order of the FDE. Then, we introduce the notion of Mittag-Leffler stability which is suitable for systems of fractional-order. Next, we use this notion to describe the asymptotical behavior of solutions to FDEs by two approaches: Lyapunovs first method and Lyapunovs second method. Finally, we give a discussion on the relation between Lipschitz condition, stability and speed of decay, separation of trajectories to scalar FDEs.
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