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

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.

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Approximation of Singularly Perturbed Parabolic Reaction-Diffusion Equations with Nonsmooth Data

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2001-0020 ◽

2001 ◽

Vol 1 (3) ◽

pp. 298-315 ◽

Cited By ~ 3

Author(s):

Grigorii Shishkin

Keyword(s):

Reaction Diffusion ◽

Boundary Function ◽

Perturbation Parameter ◽

Continuity Condition ◽

Reaction Diffusion Equations ◽

Singularly Perturbed ◽

Diffusion Type ◽

Local Accuracy ◽

Time Variables ◽

The Right

AbstractIn this paper we consider the Dirichlet problem on a rectangle for singularly perturbed parabolic equations of reaction-diffusion type. The reduced (for ε = 0) equation is an ordinary differential equation with respect to the time variable; the singular perturbation parameter ε may take arbitrary values from the half-interval (0,1]. Assume that sufficiently weak conditions are imposed upon the coefficients and the right-hand side of the equation, and also the boundary function. More precisely, the data satisfy the Hölder continuity condition with a small exponent α and α/2 with respect to the space and time variables. To solve the problem, we use the known ε-uniform numerical method which was developed previously for problems with sufficiently smooth and compatible data. It is shown that the numerical solution converges ε-uniformly. We discuss also the behavior of local accuracy of the scheme in the case where the data of the boundary-value problem are smoother on a part of the domain of definition.

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On a Partial q-Analog of a Singularly Perturbed Problem with Fuchsian and Irregular Time Singularities

Abstract and Applied Analysis ◽

10.1155/2020/7985298 ◽

2020 ◽

Vol 2020 ◽

pp. 1-32 ◽

Cited By ~ 1

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.

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

International Journal of Differential Equations ◽

10.1155/2019/5259130 ◽

2019 ◽

Vol 2019 ◽

pp. 1-10 ◽

Cited By ~ 2

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.

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

Mathematics ◽

10.3390/math8020189 ◽

2020 ◽

Vol 8 (2) ◽

pp. 189 ◽

Cited By ~ 1

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.

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Robust numerical method for a singularly perturbed problem arising in the modelling of enzyme kinetics

BIOMATH ◽

10.11145/j.biomath.2020.08.227 ◽

2020 ◽

Vol 9 (2) ◽

pp. 2008227

Author(s):

John J. H. Miller ◽

Eugene O'Riordan

Keyword(s):

Enzyme Kinetics ◽

Numerical Method ◽

A Priori ◽

Perturbation Parameter ◽

Singularly Perturbed ◽

Singularly Perturbed Problem ◽

Uniform Mesh ◽

Initial Value ◽

Computational Performance ◽

Explicit Bounds

A system of two coupled nonlinear initial value equations, arising in the mathematical modelling of enzyme kinetics, is examined. The system is singularly perturbed and one of the components will contain steep gradients. A priori parameter explicit bounds on the two components are established. A numerical method incorporating a specially constructed piecewise-uniform mesh is used to generate numerical approximations, which are shown to converge pointwise to the continuous solution irrespective of the size of the singular perturbation parameter. Numerical results are presented to illustrate the computational performance of the numerical method. The numerical method is also remarkably simple to implement.

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Analysis of a Streamline-Diffusion Finite Element Method on Bakhvalov-Shishkin Mesh for Singularly Perturbed Problem

Numerical Mathematics Theory Methods and Applications ◽

10.4208/nmtma.2017.y13026 ◽

2017 ◽

Vol 10 (1) ◽

pp. 44-64 ◽

Cited By ~ 1

Author(s):

Yunhui Yin ◽

Peng Zhu ◽

Bin Wang

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Perturbation Parameter ◽

Diffusion Problem ◽

Shishkin Mesh ◽

Singularly Perturbed ◽

Singularly Perturbed Problem ◽

Convection Diffusion ◽

Streamline Diffusion ◽

Element Method

AbstractIn this paper, a bilinear Streamline-Diffusion finite element method on Bakhvalov-Shishkin mesh for singularly perturbed convection – diffusion problem is analyzed. The method is shown to be convergent uniformly in the perturbation parameter ∈ provided only that ∈ ≤ N–1. An convergent rate in a discrete streamline-diffusion norm is established under certain regularity assumptions. Finally, through numerical experiments, we verified the theoretical results.

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Existence and Concentration Behavior of Solutions of the Critical Schrödinger–Poisson Equation in R3

Mathematics ◽

10.3390/math9050464 ◽

2021 ◽

Vol 9 (5) ◽

pp. 464

Author(s):

Jichao Wang ◽

Ting Yu

Keyword(s):

Poisson Equation ◽

Existence Result ◽

Critical Growth ◽

Singularly Perturbed ◽

Singularly Perturbed Problem ◽

Number Of Solutions ◽

Concentration Behavior ◽

The Relationship ◽

Concentration Phenomenon

In this paper, we study the singularly perturbed problem for the Schrödinger–Poisson equation with critical growth. When the perturbed coefficient is small, we establish the relationship between the number of solutions and the profiles of the coefficients. Furthermore, without any restriction on the perturbed coefficient, we obtain a different concentration phenomenon. Besides, we obtain an existence result.

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