Stéphane Malek
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View article: On some $q$-analog of an initial value problem with infinite order irregular singularity and Mahler transforms
On some $q$-analog of an initial value problem with infinite order irregular singularity and Mahler transforms Open
A family of $q$-difference-differential equations in two complex variables is studied, under the action of a so-called Mahler transform on time variable. The appearance of a leading formal $q$-difference operator of irregular type in the e…
View article: Ramified asymptotic expansions for some linear initial value problem with Mahler transforms
Ramified asymptotic expansions for some linear initial value problem with Mahler transforms Open
We focus on a linear partial differential initial value problem in complex time t and complex space combined with time acting Mahler transforms. Bounded holomorphic solutions on bounded sectors edged at the origin in time and on horizontal…
View article: Parametric formal Gevrey asymptotic expansions in two complex time variable problems
Parametric formal Gevrey asymptotic expansions in two complex time variable problems Open
The analytic and formal solutions to a family of singularly perturbed partial differential equations in the complex domain involving two complex time variables are considered. The analytic continuation properties of the solution of an auxi…
View article: On parametric $$0-$$Gevrey asymptotic expansions in two levels for some linear partial $$q-$$difference-differential equations
On parametric $$0-$$Gevrey asymptotic expansions in two levels for some linear partial $$q-$$difference-differential equations Open
A novel asymptotic representation of the analytic solutions to a family of singularly perturbed $$q-$$ difference-differential equations in the complex domain is obtained. Such asymptotic relation shows two different levels associated …
View article: On Summable Formal Power Series Solutions to Some Initial Value Problem with Infinite Order Irregular Singularity and Mahler Transforms
On Summable Formal Power Series Solutions to Some Initial Value Problem with Infinite Order Irregular Singularity and Mahler Transforms Open
In this paper, we examine a nonlinear partial differential equation in complex time t and complex space z combined with so-called Mahler transforms acting on time. This equation is endowed with a leading term represented by some infinite o…
View article: On parametric $0$-Gevrey asymptotic expansions in two levels for some linear partial $q$-difference-differential equations
On parametric $0$-Gevrey asymptotic expansions in two levels for some linear partial $q$-difference-differential equations Open
A novel asymptotic representation of the analytic solutions to a family of singularly perturbed $q-$difference-differential equations in the complex domain is obtained. Such asymptotic relation shows two different levels associated to the …
View article: Parametric asymptotic expansions and confluence for Banach valued solutions to some singularly perturbed nonlinear q-difference-differential Cauchy problem
Parametric asymptotic expansions and confluence for Banach valued solutions to some singularly perturbed nonlinear q-difference-differential Cauchy problem Open
We investigate a singularly perturbed q-difference differential Cauchy problem with polynomial coefficients in complex time and space with quadratic nonlinearity. We construct local holomorphic solutions on sectors in the complex plane wit…
View article: Double‐Scale Expansions for a Logarithmic Type Solution to a <i>q</i>‐Analog of a Singular Initial Value Problem
Double‐Scale Expansions for a Logarithmic Type Solution to a <i>q</i>‐Analog of a Singular Initial Value Problem Open
We examine a linear q −difference differential equation, which is singular in complex time t at the origin. Its coefficients are polynomial in time and bounded holomorphic on horizontal strips in one complex space variable. The equation un…
View article: Gevrey versus q−Gevrey Asymptotic Expansions for Some Linear q−Difference-Differential Cauchy Problem
Gevrey versus q−Gevrey Asymptotic Expansions for Some Linear q−Difference-Differential Cauchy Problem Open
The asymptotic behavior of the analytic solutions of a family of singularly perturbed q-difference-differential equations in the complex domain is studied. Different asymptotic expansions with respect to the perturbation parameter and to t…
View article: Gevrey versus q-Gevrey asymptotic expansions for some linear q-difference-differential Cauchy problem
Gevrey versus q-Gevrey asymptotic expansions for some linear q-difference-differential Cauchy problem Open
The asymptotic behavior of the analytic solutions of a family of singularly perturbed q-difference-differential equations in the complex domain is studied. Different asymptotic expansions with respect to the perturbation parameter and to t…
View article: Double-Scale Expansions for a Logarithmic Type Solution to a Q-analog of a Singular Initial Value Problem
Double-Scale Expansions for a Logarithmic Type Solution to a Q-analog of a Singular Initial Value Problem Open
We examine a linear q-difference differential equation which is singular in complex time t at the origin. Its coefficients are polynomial in time and bounded holomorphic on horizontal strips in one complex space variable. The equation unde…
View article: On q−Gevrey Asymptotics for Logarithmic Type Solutions in Singularly Perturbed q−Difference-Differential Equations
On q−Gevrey Asymptotics for Logarithmic Type Solutions in Singularly Perturbed q−Difference-Differential Equations Open
A family of singularly perturbed q-difference-differential equations under the action of a small complex perturbation parameter is studied. The action of the formal monodromy around the origin is present in the equation, which suggests the…
View article: On q-Gevrey asymptotics for logarithmic type solutions in singularly perturbed q-difference-differential equations
On q-Gevrey asymptotics for logarithmic type solutions in singularly perturbed q-difference-differential equations Open
A family of singularly perturbed q-difference-differential equations under the action of a small complex perturbation parameter is studied. The action of the formal monodromy around the origin is present in the equation, which suggests the…
View article: Gevrey Asymptotics for Logarithmic-Type Solutions to Singularly Perturbed Problems with Nonlocal Nonlinearities
Gevrey Asymptotics for Logarithmic-Type Solutions to Singularly Perturbed Problems with Nonlocal Nonlinearities Open
We investigate a family of nonlinear partial differential equations which are singularly perturbed in a complex parameter ϵ and singular in a complex time variable t at the origin. These equations combine differential operators of Fuch…
View article: Gevrey Asymptotics for Logarithmic Type Solutions to Singularly Perturbed Problems with Nonlocal Nonlinearities
Gevrey Asymptotics for Logarithmic Type Solutions to Singularly Perturbed Problems with Nonlocal Nonlinearities Open
We investigate a family of nonlinear partial differential equations which are singularly perturbed in a complex parameter and singular in a complex time variable at the origin. These equations combine differential operators of Fuchsian typ…
View article: Small Divisors Effects in Some Singularly Perturbed Initial Value Problem with Irregular Singularity
Small Divisors Effects in Some Singularly Perturbed Initial Value Problem with Irregular Singularity Open
We examine a nonlinear initial value problem both singularly perturbed in a complex parameter and singular in complex time at the origin. The study undertaken in this paper is the continuation of a joined work with A. Lastra published in 2…
View article: Asymptotics and Confluence for a Singular Nonlinear<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:mi>q</a:mi></a:math>-Difference-Differential Cauchy Problem
Asymptotics and Confluence for a Singular Nonlinearq-Difference-Differential Cauchy Problem Open
We examine a family of nonlinear q− difference-differential Cauchy problems obtained as a coupling of linear Cauchy problems containing dilation q− difference operators, recently investigated by the author, and quasilinear Kowalevski type …
View article: Small Divisors Effects in Some Singularly Perturbed Initial Value Problem With Irregular Singularity
Small Divisors Effects in Some Singularly Perturbed Initial Value Problem With Irregular Singularity Open
We examine a nonlinear initial value problem both singularly perturbed in a complex parameter and singular in complex time at the origin. The study undertaken in this paper is the continuation of a joined work with A. Lastra published in 2…
View article: Asymptotics and Confluence for a Singular Nonlinear Q-Difference-Differential Cauchy Problem
Asymptotics and Confluence for a Singular Nonlinear Q-Difference-Differential Cauchy Problem Open
We examine a family of nonlinear q-difference-differential Cauchy problems obtained as a coupling of linear Cauchy problems containing dilation q-difference operators, recently investigated by the author, and quasi-linear Kowalevski type p…
View article: Double-Scale Gevrey Asymptotics for Logarithmic Type Solutions to Singularly Perturbed Linear Initial Value Problems
Double-Scale Gevrey Asymptotics for Logarithmic Type Solutions to Singularly Perturbed Linear Initial Value Problems Open
We examine a family of linear partial differential equations both singularly perturbed in a complex parameter and singular in complex time at the origin. These equations entail forcing terms which combine polynomial and logarithmic type fu…
View article: Parametric Gevrey asymptotics in two complex time variables through truncated Laplace transforms
Parametric Gevrey asymptotics in two complex time variables through truncated Laplace transforms Open
This work is devoted to the study of a family of linear initial value problems of partial differential equations in the complex domain, dealing with two complex time variables. The use of a truncated Laplace-like transformation in the cons…
View article: On Inner Expansions for a Singularly Perturbed Cauchy Problem with Confluent Fuchsian Singularities
On Inner Expansions for a Singularly Perturbed Cauchy Problem with Confluent Fuchsian Singularities Open
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…
View article: On Boundary Layer Expansions for a Singularly Perturbed Problem with Confluent Fuchsian Singularities
On Boundary Layer Expansions for a Singularly Perturbed Problem with Confluent Fuchsian Singularities Open
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 perturbat…
View article: On a Partial q-Analog of a Singularly Perturbed Problem with Fuchsian and Irregular Time Singularities
On a Partial q-Analog of a Singularly Perturbed Problem with Fuchsian and Irregular Time Singularities Open
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 b…
View article: Boundary layer expansions for initial value problems with two complex time variables
Boundary layer expansions for initial value problems with two complex time variables Open
We study a family of partial differential equations in the complex domain, under the action of a complex perturbation parameter ϵ . We construct inner and outer solutions of the problem and relate them to asymptotic representations via Gev…
View article: Boundary layer expansions for initial value problems with two complex time variables
Boundary layer expansions for initial value problems with two complex time variables Open
We study a family of partial differential equations in the complex domain, under the action of a complex perturbation parameter ϵ. We construct inner and outer solutions of the problem and relate them to asymptotic representations via Gevr…
View article: On a q—Analog of a Singularly Perturbed Problem of Irregular Type with Two Complex Time Variables
On a q—Analog of a Singularly Perturbed Problem of Irregular Type with Two Complex Time Variables Open
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 holo…
View article: On the multiple-scale analysis for some linear partial q-difference and differential equations with holomorphic coefficients
On the multiple-scale analysis for some linear partial q-difference and differential equations with holomorphic coefficients Open
The analytic and formal solutions of certain family of\n$q$-difference-differential equations under the action of a complex\nperturbation parameter is considered. The previous study of the last two\nauthors provides information in the case…
View article: On a $q-$analog of a singularly perturbed problem of irregular type with\n two complex time variables
On a $q-$analog of a singularly perturbed problem of irregular type with\n two complex time variables Open
Analytic solutions and their formal asymptotic expansions for a family of the\nsingularly perturbed $q-$difference-differential equations in the complex\ndomain are constructed. They stand for a $q-$analog of the singularly perturbed\npart…