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About this book

This work gathers a selection of outstanding papers presented at the 25th Conference on Differential Equations and Applications / 15th Conference on Applied Mathematics, held in Cartagena, Spain, in June 2017. It supports further research into both ordinary and partial differential equations, numerical analysis, dynamical systems, control and optimization, trending topics in numerical linear algebra, and the applications of mathematics to industry. The book includes 14 peer-reviewed contributions and mainly addresses researchers interested in the applications of mathematics, especially in science and engineering. It will also greatly benefit PhD students in applied mathematics, engineering and physics.

Table of Contents

Frontmatter

Applications of Observability Inequalities

This article presents two observability inequalities for the heat equation over Ω × (0, T). In the first one, the observation is from a subset of positive measure in Ω × (0, T), while in the second, the observation is from a subset of positive surface measure on ∂Ω × (0, T). We will provide some applications for the above-mentioned observability inequalities, the bang-bang property for the minimal time control problems and the bang-bang property for the minimal norm control problems, and also establish new open problems related to observability inequalities and the aforementioned applications.
Jone Apraiz

Optimal Design of Piezoelectric Microactuators: Linear vs Non-linear Modeling

The main point of this work is the comparison between linear and geometrically non-linear elasticity modeling in the field of piezoelectric actuators fabricated at the micro-scale. Manufacturing limitations such as non-symmetrical lamination of the structure or minimum length scale are taken into account during the optimization process. The robust approach implemented in the problem also reduces the sensitivity of the designs to small manufacturing errors.
David Ruiz, José Carlos Bellido, Alberto Donoso

Formulation and Analysis of a Class of Direct Implicit Integration Methods for Special Second-Order I.V.P.s in Predictor-Corrector Modes

A detailed analysis of different formulations of a class of explicit direct integration methods in predictor-corrector modes for solving special second-order initial-value problems has been carried out (Comput. Phys. Comm. 181 (2010) 1833–1841), showing that the adequate combination of the involved formulas led to an increase in the order of the method. In this paper we consider different formulations of the implicit direct integration methods in predictor-corrector modes. An analysis of the accumulated truncation errors is made and the stability analysis is addressed, including the intervals of stability. Some numerical examples are presented to show the performance of the different formulations. These methods may constitute a reliable alternative to other methods in the literature for solving special second order problems.
Higinio Ramos

Application of a Local Discontinuous Galerkin Method to the 1D Compressible Reynolds Equation

In this work we present a numerical method to approximate the solution of the steady-state compressible Reynolds equation with additional first-order slip flow terms. This equation models the hydrodynamic features of read/write processes in magnetic storage devices such as hard disks. The numerical scheme is based on the local discontinuous Galerkin method proposed by Cockburn and Shu (SIAM J Numer Anal 35:2440–2463, 1998), which shows good properties in the presence of internal layers appearing in convection-diffusion problems. Several test examples illustrate the good performance of the method.
Iñigo Arregui, J. Jesús Cendán, María González

Classical Symmetries for Two Special Cases of Unsteady Flow in Nanoporous Rock

In the present paper, we study two equations related to the theory of fluid and gas flow in nanoporous media. Both models are special cases of the basic equation of the unsteady flow in nanoporous rock, the first one is the case of weakly compressible fluid and the second one of the case of isothermic gas flow. Moreover, a generalization of the equations is presented to study thoroughly the physical phenomena. This new generalized equations involve arbitrary functions. Lie method is applied to the model of generalized unsteady flow and the model of generalized isothermic gas flow. Finally a classification in different cases depending on the arbitrary function is shown.
Tamara M. Garrido, Rafael de la Rosa, María Santos Bruzón

Asymptotic Behaviour of Finite Length Solutions in a Thermosyphon Viscoelastic Model

A thermosyphon, in the engineering literature, is a device composed of a closed loop containing a fluid whose motion is driven by several actions, such as gravity and natural convection. In this work we prove some results about the asymptotic behaviour for solutions of a closed loop thermosyphon model with a viscoelastic fluid in the interior (Jiménez-Casas et al., Discrete Conti Dynam Syst (9th AIMS Conference Sool) 2013:375–384, 2013; Chaotic Model Simul 2:281–288, 2013). In this model a viscoelastic fluid described by the Maxwell constitutive equation is considered, this kind of fluids present elastic-like behavior and memory effects. Their dynamics are governed by a coupled differential nonlinear systems. In several previous works we have shown chaos in the fluid, even with this kind of viscoelastic fluid (Jiménez-Casas and Castro, Electron J Differ Equ (Conference 22), 53–61, 2015; Yasapan et al., Abstr Appl Anal 2013, Article ID: 748683, 2013; Discrete Conti Dynam Syst Ser B 20:3267–3299, 2015 among others). In this model, we consider a prescribed heat flux like Rodríguez-Bernal and Van Vleck (SIAM J Appl Math 58:1072–1093, 1998), Jiménez-Casas and Ovejero (Appl Math Comput 124:289–318, 2001) among others (all of them with Newtonian fluids). This work is, in some sense, a generalization of some previous results on standard (Newtonian) fluids obtained by Rodríguez-Bernal and Van Vleck (SIAM J Appl Math 58:1072–1093, 1998), when we consider a viscoelastic fluid.
Ángela Jiménez-Casas

Conservation Laws and Potential Symmetries for a Generalized Gardner Equation

In this paper, a generalized Gardner equation with nonlinear terms of any order has been analyzed from the point of view of group transformations and conservation laws. The generalized Gardner equation appears in many areas of physics and it is widely used to model a great variety of wave phenomena in plasma and solid state. By using the direct method of the multipliers, we have obtained an exhaustive classification of all low-order conservation laws which the generalized Gardner equation admits. Then, taking into account these conserved vectors we have determined the associated potential systems and we have searched for potential symmetries of the equation. Furthermore, we have determined and examined its first-level and second-level potential systems. From the first-level potential system we have found two new nonlocal conserved vectors.
Rafael de la Rosa, Tamara M. Garrido, María Santos Bruzón

On a Nonlocal Boussinesq System for Internal Wave Propagation

In this paper we are concerned with a nonlocal system to model the propagation of internal waves in a two-layer interface problem with rigid lid assumption and under a Boussinesq regime for both fluids. The main goal is to investigate aspects of well-posedness of the Cauchy problem for the deviation of the interface and the velocity, as well as the existence of solitary wave solutions and some of their properties.
Angel Durán

Subdivision Schemes and Multiresolution Analyses: Focus on the Shifted Lagrange and Shifted PPH Schemes

Subdivision schemes have been extensively developed since the eighties with very powerful applications for surface generation. To be implemented for compression, subdivision schemes have to be coupled with decimation operators sharing some consistency relation and with detail operators. The flexibility of subdivision schemes (they can be non-stationary, position or zone dependent, non-linear,…) makes that the construction of consistent decimation operators is a difficult task. In this paper, following the first results introduced in Kui et al. (On the coupling of decimation operator with subdivision schemes for multi-scale analysis. In: Lecture notes in computer science, vol. 10521. Springer, Berlin, pp. 162–185, 2016), we present the construction of multiresolution analyses connected to general subdivision schemes with detailed application to a non-interpolatory linear scheme called shifted Lagrange (Dyn et al., A C2 four-point subdivision scheme with fourth order accuracy and its extensions. In: Mathematical methods for curves and surfaces: Tromsø 2004. Citeseer, 2005) and its non-linear version called shifted PPH (Amat et al., Math. Comput. 80:959–959, 2011).
Zhiqing Kui, Jean Baccou, Jacques Liandrat

Modelling Sparse Saliency Maps on Manifolds: Numerical Results and Applications

Saliency detection is an image processing task which aims at automatically estimating visually salient object regions in a digital image mimicking human visual attention and eyes fixation. A number of different computational approaches for visual saliency estimation has recently appeared in Computer and Artificial Vision. Relevant and new applications can be found everywhere varying from automatic image segmentation and understanding, localization and quantification for biomedical and aerial images to fast video tracking and surveillance. In this contribution, we present a new variational model on finite dimensional manifolds generated by some characteristic features of the data. A Primal-Dual method is implemented for the numerical resolution showing promising preliminary results.
Euardo Alcaín, Ana Isabel Muñoz, Iván Ramírez, Emanuele Schiavi

Linear Elimination in Chemical Reaction Networks

We consider dynamical systems arising in biochemistry and systems biology that model the evolution of the concentrations of biochemical species described by chemical reactions. These systems are typically confined to an invariant linear subspace of \({\mathbb R}^n\). The steady states of the system are solutions to a system of polynomial equations for which only non-negative solutions are of interest. Here we study the set of non-negative solutions and provide a method for simplification of this polynomial system by means of linear elimination of variables. We take a graphical approach. The interactions among the species are represented by an edge labelled graph. Subgraphs with only certain labels correspond to sets of species concentrations that can be eliminated from the steady state equations using linear algebra. To assess positivity of the eliminated variables in terms of the non-eliminated variables, a multigraph is introduced that encodes the connections between the eliminated species in the reactions. We give graphical conditions on the multigraph that ensure the eliminated variables are expressed as positive functions of the non-eliminated variables. We interpret these conditions in terms of the reaction network. The results are illustrated by examples.
Meritxell Sáez, Elisenda Feliu, Carsten Wiuf

Minimal Set of Generators of Controllability Space for Singular Linear Systems

In recent years, there has been increasing the interest in the descriptive analysis of singular (also called generalized) systems in the form \(E\dot x(t)=Ax(t)\) because they play important roles in mathematical modelling problems permeating many aspects of daily life arising in a wide range of applications. Considerable advances have been obtained in the description of their structural and dynamical properties. However, much less effort has been devoted to studying the exact controllability measuring the minimum set of controls that are needed to steer the whole system \(E\dot x(t)=Ax(t)\) toward any desired state. In this paper, we focus the study on the obtention of the set of all B making the system \(E\dot x(t)=Ax(t)+Bu(t)\) controllable.
María Isabel García-Planas

On Stability of Discontinuous Galerkin Approximations to Anisotropic Stokes Equations

This work delves into the numerical approximation of Anisotropic Stokes equations (with small vertical diffusion coefficient), which is a generalization of the Hydrostatic Stokes equations (with zero vertical diffusion). It is known that the Ladyzhenskaya-Babuška-Brezzi condition is not sufficient to stabilize usual finite elements approximations, because a new stability condition appears. Here we extend to the Anisotropic Stokes equations the new approach given in Guillén González et al. (On stability of discontinuous galerkin approximations to the hydrostatic Stokes equations, 2018, submitted) for the Hydrostatic case. This approach is a symmetric interior penalty discontinuous Galerkin method (SIP DG) with adequate stability terms, approximating both velocity and pressure in the same Finite Element (FE) space (\(\ensuremath {\mathcal {P}_{k}}\)-discontinuous). Stability and well-posedness of this method is proven. Finally, we show some numerical tests in agreement with our numerical analysis.
Francisco Guillén-González, María Victoria Redondo-Neble, José Rafael Rodríguez-Galván

Numerical Simulation of Wear-Related Problems in a Blast Furnace Runner

Two hydrodynamic problems related to the wear suffered by refractory linings at blast furnace runners during a stage of the steelmaking process are proposed. A thermo-hydrodynamic model is posed with the scope of finding the position of the critical isotherms inside the solid refractory layers. The computational domain is based on a runner at the ArcelorMittal Company, where the three-phase flow of slag, hot metal and air is solved using the SST K − ω turbulence model and the VOF method. Radiation heat transfer is accounted for using the S2S model. The impact of a jet of hot metal falling from the blast furnace on the runner is also analyzed using a similar hydrodynamic model. Shear stress, which is the main driving factor of the erosion rate, is computed at the impinging zone. Both models are solved using ANSYS Fluent.
Patricia Barral, Begoña Nicolás, Luis Javier Pérez-Pérez, Peregrina Quintela
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