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 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.

An 𝜀-uniform method for a class of singularly perturbed parabolic problems with Robin boundary conditions having boundary turning point

2018 ◽  
Vol 13 (01) ◽  
pp. 2050025
Author(s):  
G. Janani Jayalakshmi ◽  
A. Tamilselvan
Keyword(s):  
Turning Point ◽  
Perturbation Parameter ◽  
Rectangular Domain ◽  
Singularly Perturbed ◽  
Parabolic Problems ◽  
Supremum Norm ◽  
Parabolic Differential Equations ◽  
Type Boundary ◽  
Robin Boundary ◽  
Type Boundary Condition

A class of singularly perturbed parabolic differential equations with Robin type boundary condition having boundary turning point is propounded on a rectangular domain in the [Formula: see text]-[Formula: see text] plane. A numerical method comprising a standard upwind finite difference scheme is formulated on a rectangular piecewise uniform fitted mesh [Formula: see text] and it is proved to be [Formula: see text]-uniform. Furthermore, it is shown that the errors are bounded in the supremum norm by [Formula: see text], where [Formula: see text] is a constant independent of [Formula: see text], [Formula: see text] and the perturbation parameter [Formula: see text]. Numerical results are given to illustrate the analytical results.

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 A Fitted Mesh Method for a Class of Singularly Perturbed Parabolic Problems with a Boundary Turning Point

2003 ◽  
Vol 3 (3) ◽  
pp. 361-372 ◽  
Author(s):  
Raymond K. Dunne ◽  
Eugene O'Riordan ◽  
Gregorii I. Shishkin
Keyword(s):  
Turning Point ◽  
Perturbation Parameter ◽  
Rectangular Domain ◽  
Singularly Perturbed ◽  
Parabolic Problems ◽  
Parabolic Boundary ◽  
Convection Diffusion ◽  
Upwind Finite Difference ◽  
Singularly Perturbed Parabolic Problems ◽  
Diffusion Problems

AbstractA class of singularly perturbed time-dependent convection-diffusion problems with a boundary turning point is examined on a rectangular domain. The solution of problems from this class possesses a parabolic boundary layer in the neighborhood of one of the sides of the domain. Classical numerical methods on uniform meshes are known to be inadequate for problems with boundary layers. A numerical method consisting of a standard upwind finite difference operator on a fitted mesh is constructed. It is proved that the numerical approximations generated by this method converge uniformly with respect to the singular perturbation parameter. Numerical results are presented that verify computationally the theoretical result.

scholarly journals Regularized Asymptotics of the Solution of the Singularly Perturbed First Boundary Value Problem on the Semiaxis for a Parabolic Equation with a Rational “Simple” Turning Point

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 405
Author(s):  
Alexander Yeliseev ◽  
Tatiana Ratnikova ◽  
Daria Shaposhnikova
Keyword(s):  
Parabolic Equation ◽  
Maximum Principle ◽  
Boundary Value Problem ◽  
Regularization Method ◽  
Turning Point ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Asymptotic Convergence ◽  
The Maximum Principle ◽  
Regularized Asymptotics

The aim of this study is to develop a regularization method for boundary value problems for a parabolic equation. A singularly perturbed boundary value problem on the semiaxis is considered in the case of a “simple” rational turning point. To prove the asymptotic convergence of the series, the maximum principle is used.

A Parameter Robust Finite Element Method for Fourth Order Singularly Perturbed Problems

2017 ◽  
Vol 17 (2) ◽  
pp. 337-349 ◽  
Author(s):  
Christos Xenophontos
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Fourth Order ◽  
Hermite Polynomials ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
The Finite Element Method ◽  
Singularly Perturbed Problems ◽  
Exponentially Graded ◽  
Element Method

AbstractWe consider fourth order singularly perturbed problems in one-dimension and the approximation of their solution by the h version of the finite element method. In particular, we use piecewise Hermite polynomials of degree ${p\geq 3}$ defined on an exponentially graded mesh. We show that the method converges uniformly, with respect to the singular perturbation parameter, at the optimal rate when the error is measured in both the energy norm and a stronger, ‘balanced’ norm. Finally, we illustrate our theoretical findings through numerical computations, including a comparison with another scheme from the literature.

THE PARAMETER-ROBUST NUMERICAL METHOD BASED ON DEFECT-CORRECTION TECHNIQUE FOR SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS WITH LAYER BEHAVIOR

2010 ◽  
Vol 07 (04) ◽  
pp. 573-594 ◽  
Author(s):  
JUGAL MOHAPATRA ◽  
SRINIVASAN NATESAN
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Perturbation Parameter ◽  
Correction Method ◽  
Singularly Perturbed ◽  
Defect Correction ◽  
Central Difference ◽  
Correction Technique ◽  
Delay Differential

In this article, we consider a defect-correction method based on finite difference scheme for solving a singularly perturbed delay differential equation. We solve the equation using upwind finite difference scheme on piecewise-uniform Shishkin mesh, then apply the defect-correction technique that combines the stability of the upwind scheme and the higher-order central difference scheme. The method is shown to be convergent uniformly in the perturbation parameter and almost second-order convergence measured in the discrete maximum norm is obtained. Numerical results are presented, which are in agreement with the theoretical findings.

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