Entire solutions of linear systems of moment differential equations and related asymptotic growth at infinity

The general entire solution to a linear system of moment differential equations is obtained in terms of a moment kernel function for generalized summability, and the Jordan decomposition of the matrix defining the problem. The growth at infinity of any solution of the system is also determined, both globally and also following rays to infinity, determining the order and type of such solutions.Agencia Estatal de InvestigaciónUniversidad de Alcal

Architectural form-finding through parametric geometry

This contribution briefly describes the main philosophy of the new book Parametric Geometry of Curves and Surfaces: Architectural Form-Finding (Alberto 2021) and how this philosophy is supported by concepts in the book. The mathematical investigations of such concepts are examined in detail while explaining their novelty and leading application in architectural elements. In addition to this, we focus on the importance of parametric objects from a mathematical point of view in contemporary architecture, and on understanding the geometry underlying the curves and surfaces which conform architectural manifestations. The essence of the book is illustrated by a couple of examples, together with a brief overview of its contents

On the solutions of Okubo-type systems

We show that for certain systems of Okubo type we can find a solution vector, all components of which are expressed in terms of the first one. This first component solves a Volterra integral equation with the kernel expressed in terms of the solutions of reduced Okubo-type systems of smaller dimension. Such a solution is also expressed as a power series about the origin with coefficients satisfying certain recurrence relation. This extends results in [W. Balser, C. Röscheisen, J. Differential Equations, 2009].Agencia Estatal de InvestigaciónUniversidad de Alcal

On parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems

We study a nonlinear initial value Cauchy problem depending upon a complex perturbation parameter ϵ with vanishing initial data at complex time and whose coefficients depend analytically on near the origin in and are bounded holomorphic on some horizontal strip in w.r.t. the space variable. This problem is assumed to be non-Kowalevskian in time t, therefore analytic solutions at cannot be expected in general. Nevertheless, we are able to construct a family of actual holomorphic solutions defined on a common bounded open sector with vertex at 0 in time and on the given strip above in space, when the complex parameter ϵ belongs to a suitably chosen set of open bounded sectors whose union form a covering of some neighborhood Ω of 0 in ⁎ . These solutions are achieved by means of Laplace and Fourier inverse transforms of some common ϵ-depending function on , analytic near the origin and with exponential growth on some unbounded sectors with appropriate bisecting directions in the first variable and exponential decay in the second, when the perturbation parameter belongs to Ω. Moreover, these solutions satisfy the remarkable property that the difference between any two of them is exponentially flat for some integer order w.r.t. ϵ. With the help of the classical Ramis–Sibuya theorem, we obtain the existence of a formal series (generally divergent) in ϵ which is the common Gevrey asymptotic expansion of the built up actual solutions considered above

On parametric multilevel q-Gevrey asymptotics for some linear q-difference-differential equations

We study linear q-difference-differential equations under the action of a perturbation parameter . This work deals with a q-analog of the research made in (Lastra and Malek in Adv. Differ. Equ. 2015:200, 2015) giving rise to a generalization of the work (Malek in Funkc. Ekvacioj, 2015, to appear). This generalization is related to the nature of the forcing term which suggests the use of a q-analog of an acceleration procedure. The proof leans on a q-analog of the so-called Ramis-Sibuya theorem which entails two distinct q-Gevrey orders. The work concludes with an application of the main result when the forcing term solves a related problem

On multiscale Gevrey and q-Gevrey asymptotics for some linear q-difference-differential initial value Cauchy problems

We study the asymptotic behavior of the solutions related to a singularly perturbed q-difference-differential problem in the complex domain. The analytic solution can be splitted according to the nature of the equation and its geometry so that both, Gevrey and q-Gevrey asymptotic phenomena are observed and can be distinguished, relating the analytic and the formal solution. The proof leans on a two level novel version of Ramis-Sibuya theorem under Gevrey and q-Gevrey orders

On parametric Gevrey asymptotics for some initial value problems in two asymmetric complex time variables

We study a family of nonlinear initial value problem for partial differential equations in the complex domain under the action of two asymmetric time variables. Different Gevrey bounds and multisummability results are obtained depending on each element of the family, providing a more complete picture on the asymptotic behavior of the solutions of PDEs in the complex domain in several complex variables. The main results lean on a fixed point argument in certain Banach space in the Borel plane, together with a Borel summability procedure and the action of different Ramis-Sibuya type theorems.Ministerio de Economía, Industria y Competitivida

Parametric Borel summability for linear singularly perturbed Cauchy problems with linear fractional transforms

We consider a family of linear singularly perturbed Cauchy problems which combines partial differential operators and linear fractional transforms. This work is the sequel of a study initiated in [17]. We construct a collection of holomorphic solutions on a full covering by sectors of a neighborhood of the origin in C with respect to the perturbation parameter ϵ. This set is built up through classical and special Laplace transforms along piecewise linear paths of functions which possess exponential or super exponential growth/decay on horizontal strips. A fine structure which entails two levels of Gevrey asymptotics of order 1 and so-called order 1+ is presented. Furthermore, unicity properties regarding the 1+ asymptotic layer are observed and follow from results on summability with respect to a particular strongly regular sequence recently obtained in [13] .Ministerio de Economía, Industria y Competitivida

Gevrey multiscale expansions of singular solutions of PDEs with cubic nonlinearity

We study a singularly perturbed PDE with cubic nonlinearity depending on a complex perturbation parameter ϵ. This is a continuation of the precedent work by the first author. We construct two families of sectorial meromorphic solutions obtained as a small perturbation in ϵ of two branches of an algebraic slow curve of the equation in time scale. We show that the nonsingular part of the solutions of each family shares a common formal power series in ϵ as Gevrey asymptotic expansion which might be different one to each other, in general.Ministerio de Economía, Industria y Competitivida

Boundary layer expansions for initial value problems with two complex time variables

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 Gevrey asymptotic expansions with respect to ϵ in adequate domains. The asymptotic representation leans on the cohomological approach determined by the Ramis-Sibuya theorem.Ministerio de Economía, Industria y CompetitividadComunidad de MadridUniversidad de Alcal