scholarly journals Onq-Gevrey Asymptotics for Singularly Perturbedq-Difference-Differential Problems with an Irregular Singularity

2012 ◽  
Vol 2012 ◽  
pp. 1-35 ◽  
Author(s):  
Alberto Lastra ◽  
Stéphane Malek
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Expansion ◽  
Power Series ◽  
Formal Power Series ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Complex Domain ◽  
Irregular Singularity ◽  
Formal Power

We study aq-analog of a singularly perturbed Cauchy problem with irregular singularity in the complex domain which generalizes a previous result by Malek in (2011). First, we construct solutions defined in openq-spirals to the origin. By means of aq-Gevrey version of Malgrange-Sibuya theorem we show the existence of a formal power series in the perturbation parameter which turns out to be theq-Gevrey asymptotic expansion (of certain type) of the actual solutions.

scholarly journals On Parametric Gevrey Asymptotics for Singularly Perturbed Partial Differential Equations with Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-18 ◽  
Author(s):  
Alberto Lastra ◽  
Stéphane Malek
Keyword(s):  
Partial Differential Equations ◽  
Power Series ◽  
Differential Equations ◽  
Formal Power Series ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Partial Differential ◽  
Actual Solution ◽  
Gevrey Estimates ◽  
Formal Power

We study a family of singularly perturbed -difference-differential equations in the complex domain. We provide sectorial holomorphic solutions in the perturbation parameter . Moreover, we achieve the existence of a common formal power series in which represents each actual solution and establish -Gevrey estimates involved in this representation. The proof of the main result rests on a new version of the so-called Malgrange-Sibuya theorem regarding -Gevrey asymptotics. A particular Dirichlet like series is studied on the way.

scholarly journals On a Partial q-Analog of a Singularly Perturbed Problem with Fuchsian and Irregular Time Singularities

2020 ◽  
Vol 2020 ◽  
pp. 1-32 ◽  
Author(s):  
Stephane Malek
Keyword(s):  
Asymptotic Expansion ◽  
Differential Equations ◽  
Perturbation Parameter ◽  
Scale Structure ◽  
Singularly Perturbed ◽  
Complex Domain ◽  
Singularly Perturbed Problem ◽  
Time Singularities ◽  
Finite Set ◽  
Double Scale

A family of linear singularly perturbed difference differential equations is examined. These equations stand for an analog of singularly perturbed PDEs with irregular and Fuchsian singularities in the complex domain recently investigated by A. Lastra and the author. A finite set of sectorial holomorphic solutions is constructed by means of an enhanced version of a classical multisummability procedure due to W. Balser. These functions share a common asymptotic expansion in the perturbation parameter, which is shown to carry a double scale structure, which pairs q-Gevrey and Gevrey bounds.

scholarly journals A solution decomposition for a singularly perturbed fourth-order problem

Analysis ◽  
2017 ◽  
Vol 37 (2) ◽  
Author(s):  
Katharina Höhne ◽  
Sebastian Franz ◽  
Marcus Waurick
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Fourth Order ◽  
Singularly Perturbed ◽  
Third Order ◽  
Order Problem ◽  
Unit Square ◽  
Fourth Order Problem ◽  
Formal Power

AbstractWe consider a singularly perturbed fourth-order problem with third-order terms on the unit square. With a formal power series approach, we decompose the solution into solutions of reduced (third-order) problems and various layer parts. The existence of unique solutions for the problem itself and for the reduced third-order problems is also addressed. To our knowledge, this paper is a first attempt for a solution decomposition of such problems.

scholarly journals On Singular Solutions to PDEs with Turning Point Involving a Quadratic Nonlinearity

2017 ◽  
Vol 2017 ◽  
pp. 1-32
Author(s):  
Stéphane Malek
Keyword(s):  
Turning Point ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Quadratic Nonlinearity ◽  
Singular Solutions ◽  
Meromorphic Solutions ◽  
Irregular Singularity ◽  
Irregular Singular Point ◽  
New Feature ◽  
Formal Power

We study a singularly perturbed PDE with quadratic nonlinearity depending on a complex perturbation parameter ϵ. The problem involves an irregular singularity in time, as in a recent work of the author and A. Lastra, but possesses also, as a new feature, a turning point at the origin in C. We construct a family of sectorial meromorphic solutions obtained as a small perturbation in ϵ of a slow curve of the equation in some time scale. We show that the nonsingular parts of these solutions share common formal power series (that generally diverge) in ϵ as Gevrey asymptotic expansion of some order depending on data arising both from the turning point and from the irregular singular point of the main problem.

scholarly journals Generating Asymptotics for Factorially Divergent Sequences

10.37236/5999 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Michael Borinsky
Keyword(s):  
Asymptotic Expansion ◽  
Power Series ◽  
Formal Power Series ◽  
Asymptotic Expansions ◽  
Series Form ◽  
Chord Diagrams ◽  
Algebraic Properties ◽  
Chain Rules ◽  
Formal Power ◽  
And Inversion

The algebraic properties of formal power series, whose coefficients show factorial growth and admit a certain well-behaved asymptotic expansion, are discussed. It is shown that these series form a subring of $\mathbb{R}[[x]]$. This subring is also closed under composition and inversion of power series. An `asymptotic derivation' is defined which maps a power series to the asymptotic expansion of its coefficients. Product and chain rules for this derivation are deduced. With these rules asymptotic expansions of the coefficients of implicitly defined power series can be obtained. The full asymptotic expansions of the number of connected chord diagrams and the number of simple permutations are given as examples.

scholarly journals Rational solutions of linear differential equations

1989 ◽  
Vol 46 (2) ◽  
pp. 184-196 ◽  
Author(s):  
J.-P. Bezivin ◽  
P. Robba
Keyword(s):  
Differential Operator ◽  
Power Series ◽  
Formal Power Series ◽  
Linear Differential Operator ◽  
Zero Solution ◽  
Irregular Singularity ◽  
Rational Solutions ◽  
Linear Differential ◽  
Almost All ◽  
Formal Power

AbstractLet L be a linear differential operator with rational coefficients such that 0 is not an irregular singularity of L and that for sufficiently many p's the equation Lv = 0 has no zero solution mod p. We show that if u is a formal power series whose coefficients are p-adic integers for almost all p and if Lu is rational, then u too is rational.

scholarly journals On transformations of formal power series

2003 ◽  
Vol 184 (2) ◽  
pp. 369-383 ◽  
Author(s):  
Manfred Droste ◽  
Guo-Qiang Zhang
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Formal Power

scholarly journals On the Square Root of a Bell Matrix

2021 ◽  
Vol 76 (1) ◽  
Author(s):  
Donatella Merlini
Keyword(s):  
Power Series ◽  
Closed Form ◽  
Formal Power Series ◽  
Computational Method ◽  
Square Root ◽  
Riordan Arrays ◽  
Formal Power

AbstractIn the context of Riordan arrays, the problem of determining the square root of a Bell matrix $$R={\mathcal {R}}(f(t)/t,\ f(t))$$ R = R ( f ( t ) / t , f ( t ) ) defined by a formal power series $$f(t)=\sum _{k \ge 0}f_kt^k$$ f ( t ) = ∑ k ≥ 0 f k t k with $$f(0)=f_0=0$$ f ( 0 ) = f 0 = 0 is presented. It is proved that if $$f^\prime (0)=1$$ f ′ ( 0 ) = 1 and $$f^{\prime \prime }(0)\ne 0$$ f ″ ( 0 ) ≠ 0 then there exists another Bell matrix $$H={\mathcal {R}}(h(t)/t,\ h(t))$$ H = R ( h ( t ) / t , h ( t ) ) such that $$H*H=R;$$ H ∗ H = R ; in particular, function h(t) is univocally determined by a symbolic computational method which in many situations allows to find the function in closed form. Moreover, it is shown that function h(t) is related to the solution of Schröder’s equation. We also compute a Riordan involution related to this kind of matrices.

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