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.

A second-order finite difference scheme for singularly perturbed initial value problem on layer-adapted meshes

2019 ◽  
Vol 10 (03) ◽  
pp. 1950016
Author(s):  
S. R. Sahu ◽  
J. Mohapatra
Keyword(s):  
Numerical Solution ◽  
Initial Value Problem ◽  
Perturbation Parameter ◽  
Second Order ◽  
Singularly Perturbed ◽  
Hybrid Scheme ◽  
Initial Value ◽  
Second Order Convergence ◽  
Theoretical Results ◽  
Numerical Derivative

In this paper, a second-order singularly perturbed initial value problem is considered. A hybrid scheme which is a combination of a cubic spline and a modified midpoint upwind scheme is proposed on various types of layer-adapted meshes. The error bounds are established for the numerical solution and for the scaled numerical derivative in the discrete maximum norm. It is observed that the numerical solution and the scaled numerical derivative are of second-order convergence on the layer-adapted meshes irrespective of the perturbation parameter. To show the performance of the proposed method, it is applied on few test examples which are in agreement with the theoretical results. Furthermore, existing results are also compared to show the robustness of the proposed scheme.

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.

Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations

1990 ◽  
Vol 427 (1872) ◽  
pp. 221-239 ◽  
Keyword(s):  
Cauchy Problem ◽  
Perfect Fluid ◽  
Initial Value Problem ◽  
Einstein Equations ◽  
Initial Value ◽  
Fluid Source ◽  
Null Hypersurfaces ◽  
The Cauchy Problem ◽  
Well Posed ◽  
Characteristic Initial Value Problem

A method is described by means of which the characteristic initial value problem can be reduced to the Cauchy problem and examples are given of how it can be used in practice. As an application it is shown that the characteristic initial value problem for the Einstein equations in vacuum or with perfect fluid source is well posed when data are given on two transversely intersecting null hypersurfaces. A new discussion is given of the freely specifiable data for this problem.

An almost second order uniformly convergent scheme for a singularly perturbed initial value problem

2013 ◽  
Vol 67 (2) ◽  
pp. 457-476 ◽  
Author(s):  
Zhongdi Cen ◽  
Fevzi Erdogan ◽  
Aimin Xu
Keyword(s):  
Initial Value Problem ◽  
Second Order ◽  
Singularly Perturbed ◽  
Initial Value ◽  
Uniformly Convergent

Solving Initial Value Problem by Matching Asymptotic Expansions

2012 ◽  
Vol 72 (1) ◽  
pp. 405-416 ◽  
Author(s):  
Yuri Skrynnikov
Keyword(s):  
Asymptotic Expansions ◽  
Initial Value Problem ◽  
Initial Value

scholarly journals Convergence of Global Solutions to the Cauchy Problem for the Replicator Equation in Spatial Economics

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Minoru Tabata ◽  
Nobuoki Eshima
Keyword(s):  
Cauchy Problem ◽  
Initial Value Problem ◽  
Equilibrium Solution ◽  
Global Solutions ◽  
Initial Time ◽  
Spatial Economics ◽  
Replicator Equation ◽  
Unique Global Solution ◽  
Initial Value ◽  
The Cauchy Problem

We study the initial-value problem for the replicator equation of theN-region Core-Periphery model in spatial economics. The main result shows that if workers are sufficiently agglomerated in a region at the initial time, then the initial-value problem has a unique global solution that converges to the equilibrium solution expressed by full agglomeration in that region.

On a singularly perturbed initial value problem in the case of a double root of the degenerate equation

2013 ◽  
Vol 83 ◽  
pp. 1-11 ◽  
Author(s):  
V.F. Butuzov ◽  
N.N. Nefedov ◽  
L. Recke ◽  
K.R. Schneider
Keyword(s):  
Initial Value Problem ◽  
Singularly Perturbed ◽  
Degenerate Equation ◽  
Double Root ◽  
Initial Value

scholarly journals Some Mathematical Aspects of f(R)-Gravity with Torsion: Cauchy Problem and Junction Conditions

Universe ◽  
2019 ◽  
Vol 5 (12) ◽  
pp. 224 ◽  
Author(s):  
Stefano Vignolo
Keyword(s):  
Cauchy Problem ◽  
Initial Value Problem ◽  
Sufficient Conditions ◽  
Well Posedness ◽  
Junction Conditions ◽  
Field Equations ◽  
Initial Value ◽  
The Cauchy Problem ◽  
General Conditions

We discuss the Cauchy problem and the junction conditions within the framework of f ( R ) -gravity with torsion. We derive sufficient conditions to ensure the well-posedness of the initial value problem, as well as general conditions to join together on a given hypersurface two different solutions of the field equations. The stated results can be useful to distinguish viable from nonviable f ( R ) -models with torsion.

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