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 the Parametric Stokes Phenomenon for Solutions of Singularly Perturbed Linear Partial Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-86 ◽  
Author(s):  
Stéphane Malek
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Formal Solution ◽  
Sufficient Conditions ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Stokes Phenomenon ◽  
Partial Differential ◽  
Borel Transform ◽  
Linear Partial Differential Equations

We study a family of singularly perturbed linear partial differential equations with irregular type in the complex domain. In a previous work, Malek (2012), we have given sufficient conditions under which the Borel transform of a formal solution to the above mentioned equation with respect to the perturbation parameter converges near the origin in and can be extended on a finite number of unbounded sectors with small opening and bisecting directions, say , for some integer . The proof rests on the construction of neighboring sectorial holomorphic solutions to the first mentioned equation whose differences have exponentially small bounds in the perturbation parameter (Stokes phenomenon) for which the classical Ramis-Sibuya theorem can be applied. In this paper, we introduce new conditions for the Borel transform to be analytically continued in the larger sectors , where it develops isolated singularities of logarithmic type lying on some half lattice. In the proof, we use a criterion of analytic continuation of the Borel transform described by Fruchard and Schäfke (2011) and is based on a more accurate description of the Stokes phenomenon for the sectorial solutions mentioned above.

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 Fractional-Order Logistic Differential Equation with Mittag–Leffler-Type Kernel

2021 ◽  
Vol 5 (4) ◽  
pp. 273
Author(s):  
Iván Area ◽  
Juan J. Nieto
Keyword(s):  
Differential Equation ◽  
Power Series ◽  
Differential Equations ◽  
Formal Power Series ◽  
Fractional Order ◽  
Numerical Approximations ◽  
Formal Power

In this paper, we consider the Prabhakar fractional logistic differential equation. By using appropriate limit relations, we recover some other logistic differential equations, giving representations of each solution in terms of a formal power series. Some numerical approximations are implemented by using truncated series.

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