scholarly journals On Boundary Layer Expansions for a Singularly Perturbed Problem with Confluent Fuchsian Singularities

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 189 ◽  
Author(s):  
Stephane Malek
Keyword(s):  
Boundary Layer ◽  
Laplace Transform ◽  
Asymptotic Expansions ◽  
Perturbation Parameter ◽  
Fourier Integral ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Irregular Singularity ◽  
Gevrey Order ◽  
Logarithmic Dependence

We consider a family of nonlinear singularly perturbed PDEs whose coefficients involve a logarithmic dependence in time with confluent Fuchsian singularities that unfold an irregular singularity at the origin and rely on a single perturbation parameter. We exhibit two distinguished finite sets of holomorphic solutions, so-called outer and inner solutions, by means of a Laplace transform with special kernel and Fourier integral. We analyze the asymptotic expansions of these solutions relatively to the perturbation parameter and show that they are (at most) of Gevrey order 1 for the first set of solutions and of some Gevrey order that hinges on the unfolding of the irregular singularity for the second.

scholarly journals On Inner Expansions for a Singularly Perturbed Cauchy Problem with Confluent Fuchsian Singularities

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 976
Author(s):  
Stephane Malek
Keyword(s):  
Cauchy Problem ◽  
Laplace Transform ◽  
Asymptotic Expansions ◽  
Initial Value Problem ◽  
Perturbation Parameter ◽  
Laplace Transforms ◽  
Singularly Perturbed ◽  
Initial Value ◽  
Logarithmic Kernel

A nonlinear singularly perturbed Cauchy problem with confluent Fuchsian singularities is examined. This problem involves coefficients with polynomial dependence in time. A similar initial value problem with logarithmic reliance in time has recently been investigated by the author, for which sets of holomorphic inner and outer solutions were built up and expressed as a Laplace transform with logarithmic kernel. Here, a family of holomorphic inner solutions are constructed by means of exponential transseries expansions containing infinitely many Laplace transforms with special kernel. Furthermore, asymptotic expansions of Gevrey type for these solutions relatively to the perturbation parameter are established.

THREE-STEP TAYLOR GALERKIN METHOD FOR SINGULARLY PERTURBED GENERALIZED HODGKIN–HUXLEY EQUATION

2010 ◽  
Vol 01 (02) ◽  
pp. 257-276
Author(s):  
VIVEK SANGWAN ◽  
B. V. RATHISH KUMAR ◽  
S. V. S. S. N. V. G. K. MURTHY ◽  
MOHIT NIGAM
Keyword(s):  
Boundary Layer ◽  
Finite Element Method ◽  
Finite Element ◽  
Singular Perturbation ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Galerkin Finite Element Method ◽  
Third Order ◽  
Galerkin Finite Element ◽  
Huxley Equation

A numerical study is carried out for the singularly perturbed generalized Hodgkin–Huxley equation. The equation is nonlinear which mimics the ionic processes at a real nerve membrane. A small parameter called singular perturbation parameter is introduced in the highest order derivative term. Keeping other parameters fixed, as this singular perturbation parameter approaches to zero, a boundary layer occurs in the solution. Three-step Taylor Galerkin finite element method is employed on a piecewise uniform Shishkin mesh to solve the equation. To procure more accurate temporal differencing, the method employs forward-time Taylor series expansion including time derivatives of third order which are evaluated from the governing singularly perturbed generalized Hodgkin–Huxley equation. This yields a generalized time-discretized equation which is successively discretized in space by means of the standard Bubnov–Galerkin finite element method. The method is third-order accurate in time. The code based on the purposed scheme has been validated against the cases for which the exact solution is available. It is also observed that for the Singularly Perturbed Generalized Hodgkin–Huxley equation, the boundary layer in the solution manifests not only by varying the singular perturbation parameter but also by varying the other parameters appearing in the model.

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 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.

SINGULARLY PERTURBED PROBLEM WITH DOUBLE BOUNDARY LAYER

2021 ◽  
Vol 1 (1) ◽  
pp. 102-109
Author(s):  
Dilmurat Abdillazhanovich Tursunov ◽  
Gulbayra Abdimalikovna Omaralieva ◽  
Makhfuzakhon Ibrakhimzhanovna Mamatbuvaeva ◽  
Shahzadakhan Adylzhanovna Ramankulova
Keyword(s):  
Boundary Layer ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem

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 Solving Linear Second-Order Singularly Perturbed Differential Difference Equations via Initial Value Method

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Wondwosen Gebeyaw Melesse ◽  
Awoke Andargie Tiruneh ◽  
Getachew Adamu Derese
Keyword(s):  
Taylor Series Expansion ◽  
Perturbation Parameter ◽  
Second Order ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Test Problems ◽  
Initial Value ◽  
Correction Problem ◽  
The Given ◽  
Theoretical Results

In this paper, an initial value method for solving a class of linear second-order singularly perturbed differential difference equation containing mixed shifts is proposed. In doing so, first, the given problem is modified in to an equivalent singularly perturbed problem by approximating the term containing the delay and advance parameters using Taylor series expansion. From the modified problem, two explicit initial value problems which are independent of the perturbation parameter are produced; namely, the reduced problem and the boundary layer correction problem. These problems are then solved analytically and/or numerically, and those solutions are combined to give an approximate solution to the original problem. An error estimate for this method is derived using maximum norm. Several test problems are considered to illustrate the theoretical results. It is observed that the present method approximates the exact solution very well.

scholarly journals A comparison between MMAE and SCEM for solving singularly perturbed linear boundary layer problems

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 2135-2148
Author(s):  
Süleyman Cengizci
Keyword(s):  
Boundary Layer ◽  
Asymptotic Expansions ◽  
Present Method ◽  
Numerical Experiments ◽  
Expansion Method ◽  
Singularly Perturbed ◽  
Linear Boundary ◽  
Point Boundary ◽  
Matching Process ◽  
Boundary Layer Problems

In this study, we propose an efficient method so-called Successive Complementary Expansion Method (SCEM), that is based on generalized asymptotic expansions, for approximating to the solutions of singularly perturbed two-point boundary value problems. In this easy-applicable method, in contrast to the well-known method the Method of Matched Asymptotic Expansions (MMAE), any matching process is not required to obtain uniformly valid approximations. The key point: A uniformly valid approximation is adopted first, and complementary functions are obtained imposing the corresponding boundary conditions. An illustrative and two numerical experiments are provided to show the implementation and numerical properties of the present method. Furthermore, MMAE results are also obtained in order to compare the numerical robustnesses of the methods.

Sign in / Sign up

Export Citation Format

Share Document