Computation of Mixed Type Functional Differential Boundary Value Problems

This is the published version, also available here: http://dx.doi.org/10.1137/040603425.We study boundary value differential-difference equations where the difference terms may contain both advances and delays. Special attention is paid to connecting orbits, in particular to the modeling of the tails after truncation to a finite interval, and we reformulate these problems as functional differential equations over a bounded domain. Connecting orbits are computed for several such problems including discrete Nagumo equations, an Ising model, and Frenkel--Kontorova type equations. We describe the collocation boundary value problem code used to compute these solutions, and the numerical analysis issues which arise, including linear algebra, boundary functions and conditions, and convergence theory for the collocation approximation on finite intervals

Sistemas de equações diferenciais não lineares de ordem superior em domínios limitados ou não limitados

The Boundary value problems on bounded or unbounded intervals, involving two or more coupled systems of the nonlinear differen- tial equations with full nonlinearities are scarce and have gap in literature. The present work modestly try to fill this gap. The systems covered in the work are essentially of the second- order (except for the first chapter of the first part) with boundary constraints either in bounded or unbounded intervals presented in several forms and conditions (three points, mixed, with functional dependence, homoclinic and heteroclinic). The existence, and in some cases, the localization of the solu- tions is carried out in of Banach space and norms considered, fo- llowing arguments and approaches such as: Schauder’s fixed-point theorem or of Guo–Krasnosel’ski˘ı fixed-point theorem in cones, allied to Green’s function or its estimates, lower and upper solutions, convenient truncatures, the Nagumo condition presented in different forms, concept of equiconvergence, Carathéodory functions and sequences. On the other hand, parallel to the theoretical explanation of this work, there is a range of practical examples and applications involving real phenomena, focusing on the physics, mechanics, bio- logy, forestry, and dynamical systems; A falta ou a raridade de problemas de valor fronteira na literatura, quer em dom´ınios limitados ou ilimitados, envolvendo sistemas de duas ou mais equações n˜ao lineares acopladas com todas as n˜ao linearidades completas, levou à elaboração do presente trabalho. Os sistemas abordados no trabalho s˜ao essencialmente de segunda ordem (exceto o primeiro capítulo da primeira parte) com condições de fronteira em domínios limitados ou ilimitados, de diversos tipos (três pontos, mistas, com condições funcionais, homoclínicas e heteroclínicas). A existência e em alguns casos a localização das soluções dos sistemas è considerada em espaços de Banach, seguindo vários ar- gumentos e abordagens: o teorema de ponto fixo de Schauder ou de Guo–Krasnosel’ski˘ı em cones, aliados a funções de Green ou suas estimativas, sub e sobre-soluções, truncaturas convenientes, a condição de Nagumo apresentada sob várias formas, o conceito de equiconvergência e funções e sucess˜oes de Carath´eodory. Por outro lado, paralelamente àcomponente teórica do trabalho, encontra-se um leque de aplicações e exemplos práticos envolvendo fenómenos reais, com enfoque na física, mecânica, biologia, exploração florestal e sistemas dinâmico

New Trends in Differential and Difference Equations and Applications

This is a reprint of articles from the Special Issue published online in the open-access journal Axioms (ISSN 2075-1680) from 2018 to 2019 (available at https://www.mdpi.com/journal/axioms/special issues/differential difference equations)

Book of Abstracts

USPCAPESFAPESPCNPqINCTMatICMC Summer Meeting on Differentail Equations.\ud São Carlos, Brasil. 3-7 february 2014

Symmetry in Modeling and Analysis of Dynamic Systems

Real-world systems exhibit complex behavior, therefore novel mathematical approaches or modifications of classical ones have to be employed to precisely predict, monitor, and control complicated chaotic and stochastic processes. One of the most basic concepts that has to be taken into account while conducting research in all natural sciences is symmetry, and it is usually used to refer to an object that is invariant under some transformations including translation, reflection, rotation or scaling.The following Special Issue is dedicated to investigations of the concept of dynamical symmetry in the modelling and analysis of dynamic features occurring in various branches of science like physics, chemistry, biology, and engineering, with special emphasis on research based on the mathematical models of nonlinear partial and ordinary differential equations. Addressed topics cover theories developed and employed under the concept of invariance of the global/local behavior of the points of spacetime, including temporal/spatiotemporal symmetries

Traveling waves and pattern formation for spatially discrete bistable reaction-diffusion equations (survey)

Analysis and Stochastic

Neutral Equations of Mixed Type

In this dissertation we consider neutral equations of mixed type. In particular, we con- sider the associated linear Fredholm theory and nerve fiber models that are written as systems of neutral equations of mixed type. In Chapter 2, we extend the existing Fredholm theory for mixed type functional differential equations developed by Mallet-Paret to the case of implicitly defined mixed type functional differential equations. In Chapter 3, we apply the theory to an example arising from modeling signal prop- agation in nerve fibers. In this two-dimensional system, we rely on the Lyapunov- Schmidt method to demonstrate the existence of traveling wave solutions. With the aid of numerical computations, a saddle-node bifurcation was detected. In Chapter 4, we consider an extension of the parallel nerve fiber model examining in Chapter 3 and present the results of a numerical study. In this chapter, an additional form of coupling is examined not considered in the model from Chapter 3. This second type of coupling may be excitatory or inhibitory depending on the sign of the coupling parameter. Within a continuation framework, we employ a pseudo-spectral approach utilizing Chebyshev polynomials as basis functions. The chebfun package, consisting of Chebyshev tools, was utilized to manipulate the polynomials

Coupled systems with Ambrosetti-Prodi-type differential equations

In this paper, we consider some boundary value problems composed by coupled systems of second-order differential equations with full nonlinearities and general functional boundary conditions verifying some monotone assumptions. The arguments apply the lower and upper solutions method, and defining an adequate auxiliary, homotopic, and truncated problem, it is possible to apply topological degree theory as the tool to prove the existence of solution. In short, it is proved that for the parameter values such that there are lower and upper solutions, then there is also, at least, a solution and this solution is localized in a strip bounded by lower and upper solutions. As far as we know, it is the first paper where Ambrosetti-Prodi differential equations are considered in couple systems with different parameters