Abstract book

Welcome at the International Conference on Differential and Difference Equations & Applications 2015. The main aim of this conference is to promote, encourage, cooperate, and bring together researchers in the fields of differential and difference equations. All areas of differential & difference equations will be represented with special emphasis on applications. It will be mathematically enriching and socially exciting event. List of registered participants consists of 169 persons from 45 countries. The five-day scientific program runs from May 18 (Monday) till May 22, 2015 (Friday). It consists of invited lectures (plenary lectures and invited lectures in sections) and contributed talks in the following areas: Ordinary differential equations, Partial differential equations, Numerical methods and applications, other topics

Nonlinear differential equations having non-sign-definite weights

In the present PhD thesis we deal with the study of the existence, multiplicity and complex behaviors of solutions for some classes of boundary value problems associated with second order nonlinear ordinary differential equations of the form $u''+f(u)u'+g(t,u)=s,$ or $u''+g(t,u)=0,$ $t\in I$, where $I$ is a bounded interval, $f\colon\mathbb{R}\to\mathbb{R}$ is continuous, $s\in\mathbb{R}$ and $g: I\times \mathbb{R}\to\mathbb{R}$ is a perturbation term characterizing the problems. The results carried out in this dissertation are mainly based on dynamical and topological approaches. The issues we address have arisen in the field of partial differential equations. For this reason, we do not treat only the case of ordinary differential equations, but also we take advantage of some results achieved in the one dimensional setting to give applications to nonlinear boundary value problems associated with partial differential equations. In the first part of the thesis, we are interested on a problem suggested by Antonio Ambrosetti in ``Observations on global inversion theorems'' (2011). In more detail, we deal with a periodic boundary value problem associated with the first differential equation where the perturbation term is given by $g(t,u):=a(t)\phi(u)-p(t)$. We assume that $a,$ $p\in L^{\infty}(I)$ and $\phi\colon\mathbb{R}\to\mathbb{R}$ is a continuous function satisfying $\lim_{|\xi|\to\infty}\phi(\xi)=+\infty$. In this context, if the weight term $a(t)$ is such that $a(t)\geq 0$ for a.e. $t\in I$ and $\int_{I}a(t)\,dt>0$, we generalize the result of multiplicity of solutions given by Fabry, Mawhin and Nakashama in ``A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations'' (1986). We extend this kind of improvement also to more general nonlinear terms under local coercivity conditions. In this framework, we also treat in the same spirit Neumann problems associated with second order ordinary differential equations and periodic problems associated with first order ones. Furthermore, we face the classical case of a periodic Ambrosetti-Prodi problem with a weight term $a(t)$ which is constant and positive. Here, considering in the second differential equation a nonlinearity $g(t,u):=\phi(u)-h(t)$, we provide several conditions on the nonlinearity and the perturbative term that ensure the presence of complex behaviors for the solutions of the associated $T$-periodic problem. We also compare these outcomes with the result of stability carried out by Ortega in ``Stability of a periodic problem of Ambrosetti-Prodi type'' (1990). The case with damping term is discussed as well. In the second part of this work, we solve a conjecture by Yuan Lou and Thomas Nagylaki stated in ``A semilinear parabolic system for migration and selection in population genetics'' (2002). The problem refers to the number of positive solutions for Neumann boundary value problems associated with the second differential equation when the perturbation term is given by $g(t,u):=\lambda w(t)\psi(u)$ with $\lambda>0$, $w\in L^{\infty}(I)$ a sign-changing weight term such that $\int_{I}w(t)\,dt<0$ and $\psi\colon[0,1]\to[0,\infty[$ a non-concave continuous function satisfying $\psi(0)=0=\psi(1)$ and such that the map $\xi\mapsto \psi(\xi)/\xi$ is monotone decreasing. In addition to this outcome, other new results of multiplicity of positive solutions are presented as well, for both Neumann or Dirichlet boundary value problems, by means of a particular choice of indefinite weight terms $w(t)$ and different positive nonlinear terms $\psi(u)$ defined on the interval $[0,1]$ or on the positive real semi-axis $[0,+\infty[$

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio

Existence and Multiplicity of Solutions of Functional Differential Equations

The first part of the memory goes through those discoveries related to Green’s functions. In order to do that, first we recall some general results concerning involutions which will help us understand their remarkable analytic and algebraic properties. Chapter 1 will deal about this subject while Chapter 2 will give a brief overview on differential equations with involutions to set the reader in the appropriate research framework. In Chapter 3 we start working on the theory of Green’s functions for functional differential equations with involutions in the most simple cases: order one problems with constant coefficients and reflection. Here we solve the problem with different boundary conditions, studying the specific characteristics which appear when considering periodic, anti-periodic, initial or arbitrary linear boundary conditions. We also apply some very well known techniques (lower and upper solutions method or Krasnosel’skiĭ’s Fixed Point Theorem, for instance) in order to further derive results. Computing explicitly the Green’s function for a problem with nonconstant coefficients is not simple, not even in the case of ordinary differential equations. We face these obstacles in Chapter 4, where we reduce a new, more general problem containing nonconstant coefficients and arbitrary differentiable involutions, to the one studied in Chapter 3. To end this part of the work, we have Chapter 5, in which we deepen in the algebraic nature of reflections and extrapolate these properties to other algebras. In this way, we do not only generalize the results of Chapter 3 to the case of -th order problems and general twopoint boundary conditions, but also solve functional differential problems in which the Hilbert transform or other adequate operators are involved. The last chapters of this part are about applying the results we have proved so far to some related problems. First, in Chapter 6, setting again the spotlight on some interesting relation between an equation with reflection and an equation with a -Laplacian, we obtain some results concerning the periodicity of solutions of that first problem with reflection. Chapter 7 moves to a more practical setting. It is of the greatest interest to have adequate computer programs in order to derive the Green’s functions obtained in Chapter 5 for, in general, the computations involved are very convoluted. Being so, we present in this chapter such an algorithm, implemented in Mathematica. The reader can find in the appendix the exact code of the program. In the second part of the Thesis we use the fixed point index to solve four different kinds of problems increasing in complexity: a problem with reflection, a problem with deviated arguments (applied to a thermostat model), a problem with nonlinear Neumann boundary conditions and a problem with functional nonlinearities in both the equation and the boundary conditions. As we will see, the particularities of each problem make it impossible to take a common approach to all of the problems studied. Still, there will be important similarities in the different cases which will lead to comparable results

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions