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 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 ASYMPTOTIC EXPANSION OF SOLUTION OF GENERAL BVP WITH INITIAL JUMPS FOR HIGHER-ORDER SINGULARLY PERTURBED INTEGRO-DIFFERENTIAL EQUATION

2018 ◽  
pp. 28-36
Author(s):  
Dauylbayev M. ◽  
Atakhan N. ◽  
Mirzakulova A.E.
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Higher Order ◽  
Singularly Perturbed ◽  
Jump Phenomenon ◽  
Integro Differential Equation ◽  
Asymptotic Expansion Of Solution ◽  
Initial Jump

In this article we constructed an asymptotic expansion of the solution undivided boundary value problem for singularly perturbed integro-differential equations with an initial jump phenomenon m – th order. We obtain the theorem about estimation of the remainder term’s asymptotic with any degree of accuracy in the smallparameter.

scholarly journals Solution Of Singularly Perturbed Delay Differential Equations Using Liouville Green Transformation

2021 ◽  
Vol 12 (4) ◽  
pp. 325-335
Author(s):  
M. Adilaxmi , Et. al.
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Perturbation Problem ◽  
Delay Differential ◽  
Singularly Perturbation ◽  
The Given

This paper envisages the use of Liouville Green Transformation to find the solution of singularly perturbed delay differential equations. First, using Taylor series, the given singularly perturbed delay differential equation is approximated by an asymptotically equivalent singularly perturbation problem. Then the Liouville Green Transformation is applied to get the solution. The method is demonstrated by implementing several model examples by taking various values for the delay parameter and perturbation parameter.

scholarly journals On a q-Analog of a Singularly Perturbed Problem of Irregular Type with Two Complex Time Variables

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 924 ◽  
Author(s):  
Alberto Lastra ◽  
Stéphane Malek
Keyword(s):  
Growth Rate ◽  
Asymptotic Relation ◽  
Perturbation Parameter ◽  
Analytic Solutions ◽  
Point Of View ◽  
Singularly Perturbed ◽  
Lambert W Function ◽  
Singularly Perturbed Problem ◽  
Time Variables ◽  
Asymptotic Point

The analytic solutions of a family of singularly perturbed q-difference-differential equations in the complex domain are constructed and studied from an asymptotic point of view with respect to the perturbation parameter. Two types of holomorphic solutions, the so-called inner and outer solutions, are considered. Each of them holds a particular asymptotic relation with the formal ones in terms of asymptotic expansions in the perturbation parameter. The growth rate in the asymptotics leans on the - 1 -branch of Lambert W function, which turns out to be crucial.

Asymptotics beyond all orders and Stokes lines in nonlinear differential-difference equations

2001 ◽  
Vol 12 (4) ◽  
pp. 433-463 ◽  
Author(s):  
J. R. KING ◽  
S. J. CHAPMAN
Keyword(s):  
Asymptotic Expansion ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Difference Equations ◽  
Singularly Perturbed ◽  
Time Dependent ◽  
Stokes Line ◽  
Linear Ordinary Differential Equations ◽  
Stokes Lines ◽  
Asymptotics Beyond All Orders

A technique for calculating exponentially small terms beyond all orders in singularly perturbed difference equations is presented. The approach is based on the application of a WKBJ-type ansatz to the late terms in the naive asymptotic expansion and the identification of Stokes lines, and is closely related to the well-known Stokes line smoothing phenomenon in linear ordinary differential equations. The method is illustrated by application to examples and the results extended to time-dependent differential-difference problems.

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.

Numerical Method for a Class of Nonlinear Singularly Perturbed Delay Differential Equations Using Parametric Cubic Spline

2018 ◽  
Vol 19 (3-4) ◽  
pp. 357-365 ◽  
Author(s):  
A. S. V. Ravi Kanth ◽  
P. Murali Mohan Kumar
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Delay Differential Equations ◽  
Perturbation Parameter ◽  
Cubic Spline ◽  
Singularly Perturbed ◽  
Delay Term ◽  
Second Order Convergence ◽  
Delay Differential ◽  
Parametric Cubic Spline

AbstractIn this paper, we study the numerical method for a class of nonlinear singularly perturbed delay differential equations using parametric cubic spline. Quasilinearization process is applied to convert the nonlinear singularly perturbed delay differential equations into a sequence of linear singularly perturbed delay differential equations. When the delay is not sufficiently smaller order of the singular perturbation parameter, the approach of expanding the delay term in Taylor’s series may lead to bad approximation. To handle the delay term, we construct a special type of mesh in such a way that the term containing delay lies on nodal points after discretization. The parametric cubic spline is presented for solving sequence of linear singularly perturbed delay differential equations. The error analysis of the method is presented and shows second-order convergence. The effect of delay parameter on the boundary layer behavior of the solution is discussed with two test examples.

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